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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.12069">arXiv:2411.12069</a> <span> [<a href="https://arxiv.org/pdf/2411.12069">pdf</a>, <a href="https://arxiv.org/format/2411.12069">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Matroid Secretary via Labeling Schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=B%C3%A9rczi%2C+K">Krist贸f B茅rczi</a>, <a href="/search/cs?searchtype=author&query=Livanos%2C+V">Vasilis Livanos</a>, <a href="/search/cs?searchtype=author&query=Soto%2C+J">Jos茅 Soto</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.12069v1-abstract-short" style="display: inline;"> The Matroid Secretary Problem (MSP) is one of the most prominent settings for online resource allocation and optimal stopping. A decision-maker is presented with a ground set of elements $E$ revealed sequentially and in random order. Upon arrival, an irrevocable decision is made in a take-it-or-leave-it fashion, subject to a feasibility constraint on the set of selected elements captured by a matr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12069v1-abstract-full').style.display = 'inline'; document.getElementById('2411.12069v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.12069v1-abstract-full" style="display: none;"> The Matroid Secretary Problem (MSP) is one of the most prominent settings for online resource allocation and optimal stopping. A decision-maker is presented with a ground set of elements $E$ revealed sequentially and in random order. Upon arrival, an irrevocable decision is made in a take-it-or-leave-it fashion, subject to a feasibility constraint on the set of selected elements captured by a matroid defined over $E$. The decision-maker only has ordinal access to compare the elements, and the goal is to design an algorithm that selects every element of the optimal basis with probability at least $伪$ (i.e., $伪$-probability-competitive). While the existence of a constant probability-competitive algorithm for MSP remains a major open question, simple greedy policies are at the core of state-of-the-art algorithms for several matroid classes. We introduce a flexible and general algorithmic framework to analyze greedy-like algorithms for MSP based on constructing a language associated with the matroid. Using this language, we establish a lower bound on the probability-competitiveness of the algorithm by studying a corresponding Poisson point process that governs the words' distribution in the language. Using our framework, we break the state-of-the-art guarantee for laminar matroids by settling the probability-competitiveness of the greedy-improving algorithm to be exactly $1-\ln(2) \approx 0.3068$. For graphic matroids, we show a probability-competitiveness of $0.2693$ when the underlying graph has no parallel edges and a guarantee of $0.2504$ for general graphs, also breaking the state-of-the-art factor of $0.25$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12069v1-abstract-full').style.display = 'none'; document.getElementById('2411.12069v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.23869">arXiv:2410.23869</a> <span> [<a href="https://arxiv.org/pdf/2410.23869">pdf</a>, <a href="https://arxiv.org/format/2410.23869">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Theoretical Economics">econ.TH</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> New Combinatorial Insights for Monotone Apportionment </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Cembrano%2C+J">Javier Cembrano</a>, <a href="/search/cs?searchtype=author&query=Correa%2C+J">Jos茅 Correa</a>, <a href="/search/cs?searchtype=author&query=Schmidt-Kraepelin%2C+U">Ulrike Schmidt-Kraepelin</a>, <a href="/search/cs?searchtype=author&query=Tsigonias-Dimitriadis%2C+A">Alexandros Tsigonias-Dimitriadis</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.23869v1-abstract-short" style="display: inline;"> The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states' populations? This--seemingly simple--task has led to a rich literature and has become well known in the context of the US House of Representatives. In this paper, we connect the design of monotone apportionment methods to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23869v1-abstract-full').style.display = 'inline'; document.getElementById('2410.23869v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.23869v1-abstract-full" style="display: none;"> The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states' populations? This--seemingly simple--task has led to a rich literature and has become well known in the context of the US House of Representatives. In this paper, we connect the design of monotone apportionment methods to classic problems from discrete geometry and combinatorial optimization and explore the extent to which randomization can enhance proportionality. We first focus on the well-studied family of stationary divisor methods, which satisfy the strong population monotonicity property, and show that this family produces only a slightly superlinear number of different outputs as a function of the number of states. While our upper and lower bounds leave a small gap, we show that--surprisingly--closing this gap would solve a long-standing open problem from discrete geometry, known as the complexity of $k$-levels in line arrangements. The main downside of divisor methods is their violation of the quota axiom, i.e., every state should receive $\lfloor q_i\rfloor$ or $\lceil q_i\rceil$ seats, where $q_i$ is the proportional share of the state. As we show that randomizing over divisor methods can only partially overcome this issue, we propose a relaxed version of divisor methods in which the total number of seats may slightly deviate from the house size. By randomizing over them, we can simultaneously satisfy population monotonicity, quota, and ex-ante proportionality. Finally, we turn our attention to quota-compliant methods that are house-monotone, i.e., no state may lose a seat when the house size is increased. We provide a polyhedral characterization based on network flows, which implies a simple description of all ex-ante proportional randomized methods that are house-monotone and quota-compliant. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23869v1-abstract-full').style.display = 'none'; document.getElementById('2410.23869v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.12756">arXiv:2410.12756</a> <span> [<a href="https://arxiv.org/pdf/2410.12756">pdf</a>, <a href="https://arxiv.org/ps/2410.12756">ps</a>, <a href="https://arxiv.org/format/2410.12756">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Prophet Upper Bounds for Online Matching and Auctions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Soto%2C+J">Jos茅 Soto</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.12756v1-abstract-short" style="display: inline;"> In the online 2-bounded auction problem, we have a collection of items represented as nodes in a graph and bundles of size two represented by edges. Agents are presented sequentially, each with a random weight function over the bundles. The goal of the decision-maker is to find an allocation of bundles to agents of maximum weight so that every item is assigned at most once, i.e., the solution is a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.12756v1-abstract-full').style.display = 'inline'; document.getElementById('2410.12756v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.12756v1-abstract-full" style="display: none;"> In the online 2-bounded auction problem, we have a collection of items represented as nodes in a graph and bundles of size two represented by edges. Agents are presented sequentially, each with a random weight function over the bundles. The goal of the decision-maker is to find an allocation of bundles to agents of maximum weight so that every item is assigned at most once, i.e., the solution is a matching in the graph. When the agents are single-minded (i.e., put all the weight in a single bundle), we recover the maximum weight prophet matching problem under edge arrivals (a.k.a. prophet matching). In this work, we provide new and improved upper bounds on the competitiveness achievable by an algorithm for the general online 2-bounded auction and the (single-minded) prophet matching problems. For adversarial arrival order of the agents, we show that no algorithm for the online 2-bounded auction problem achieves a competitiveness larger than $4/11$, while no algorithm for prophet matching achieves a competitiveness larger than $\approx 0.4189$. Using a continuous-time analysis, we also improve the known bounds for online 2-bounded auctions for random order arrivals to $\approx 0.5968$ in the general case, a bound of $\approx 0.6867$ in the IID model, and $\approx 0.6714$ in prophet-secretary model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.12756v1-abstract-full').style.display = 'none'; document.getElementById('2410.12756v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.03304">arXiv:2410.03304</a> <span> [<a href="https://arxiv.org/pdf/2410.03304">pdf</a>, <a href="https://arxiv.org/format/2410.03304">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Theoretical Economics">econ.TH</span> </div> </div> <p class="title is-5 mathjax"> Proportionality in Multiple Dimensions to Design Electoral Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Cembrano%2C+J">Javier Cembrano</a>, <a href="/search/cs?searchtype=author&query=Correa%2C+J">Jos茅 Correa</a>, <a href="/search/cs?searchtype=author&query=D%C3%ADaz%2C+G">Gonzalo D铆az</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.03304v1-abstract-short" style="display: inline;"> How to elect the representatives in legislative bodies is a question that every modern democracy has to answer. This design task has to consider various elements so as to fulfill the citizens' expectations and contribute to the maintenance of a healthy democracy. The notion of proportionality, in that the support of a given idea in the house should be nearly proportional to its support in the gene… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03304v1-abstract-full').style.display = 'inline'; document.getElementById('2410.03304v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.03304v1-abstract-full" style="display: none;"> How to elect the representatives in legislative bodies is a question that every modern democracy has to answer. This design task has to consider various elements so as to fulfill the citizens' expectations and contribute to the maintenance of a healthy democracy. The notion of proportionality, in that the support of a given idea in the house should be nearly proportional to its support in the general public, lies at the core of this design task. In the last decades, demographic aspects beyond political support have been incorporated by requiring that they are also fairly represented in the body, giving rise to a multidimensional version of the apportionment problem. In this work, we provide an axiomatic justification for a recently proposed notion of multidimensional proportionality and extend it to encompass two relevant constraints often used in electoral systems: a threshold on the number of votes that a list needs in order to be eligible and the election of the most-voted candidate in each district. We then build upon these results to design methods based on multidimensional proportionality. We use the Chilean Constitutional Convention election (May 15-16, 2021) results as a testing ground -- where the dimensions are given by political lists, districts, and genders -- and compare the apportionment obtained under each method according to three criteria: proportionality, representativeness, and voting power. While local and global methods exhibit a natural trade-off between local and global proportionality, including the election of most-voted candidates on top of methods based on 3-dimensional proportionality allows us to incorporate both notions while ensuring higher levels of representativeness and a balanced voting power. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03304v1-abstract-full').style.display = 'none'; document.getElementById('2410.03304v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.11224">arXiv:2408.11224</a> <span> [<a href="https://arxiv.org/pdf/2408.11224">pdf</a>, <a href="https://arxiv.org/format/2408.11224">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Theoretical Economics">econ.TH</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Optimal Guarantees for Online Selection Over Time </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Perez-Salazar%2C+S">Sebastian Perez-Salazar</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.11224v1-abstract-short" style="display: inline;"> Prophet inequalities are a cornerstone in optimal stopping and online decision-making. Traditionally, they involve the sequential observation of $n$ non-negative independent random variables and face irrevocable accept-or-reject choices. The goal is to provide policies that provide a good approximation ratio against the optimal offline solution that can access all the values upfront -- the so-call… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.11224v1-abstract-full').style.display = 'inline'; document.getElementById('2408.11224v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.11224v1-abstract-full" style="display: none;"> Prophet inequalities are a cornerstone in optimal stopping and online decision-making. Traditionally, they involve the sequential observation of $n$ non-negative independent random variables and face irrevocable accept-or-reject choices. The goal is to provide policies that provide a good approximation ratio against the optimal offline solution that can access all the values upfront -- the so-called prophet value. In the prophet inequality over time problem (POT), the decision-maker can commit to an accepted value for $蟿$ units of time, during which no new values can be accepted. This creates a trade-off between the duration of commitment and the opportunity to capture potentially higher future values. In this work, we provide best possible worst-case approximation ratios in the IID setting of POT for single-threshold algorithms and the optimal dynamic programming policy. We show a single-threshold algorithm that achieves an approximation ratio of $(1+e^{-2})/2\approx 0.567$, and we prove that no single-threshold algorithm can surpass this guarantee. With our techniques, we can analyze simple algorithms using $k$ thresholds and show that with $k=3$ it is possible to get an approximation ratio larger than $\approx 0.602$. Then, for each $n$, we prove it is possible to compute the tight worst-case approximation ratio of the optimal dynamic programming policy for instances with $n$ values by solving a convex optimization program. A limit analysis of the first-order optimality conditions yields a nonlinear differential equation showing that the optimal dynamic programming policy's asymptotic worst-case approximation ratio is $\approx 0.618$. Finally, we extend the discussion to adversarial settings and show an optimal worst-case approximation ratio of $\approx 0.162$ when the values are streamed in random order. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.11224v1-abstract-full').style.display = 'none'; document.getElementById('2408.11224v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.17767">arXiv:2406.17767</a> <span> [<a href="https://arxiv.org/pdf/2406.17767">pdf</a>, <a href="https://arxiv.org/ps/2406.17767">ps</a>, <a href="https://arxiv.org/format/2406.17767">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Splitting Guarantees for Prophet Inequalities via Nonlinear Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Brustle%2C+J">Johannes Brustle</a>, <a href="/search/cs?searchtype=author&query=Perez-Salazar%2C+S">Sebastian Perez-Salazar</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.17767v1-abstract-short" style="display: inline;"> The prophet inequality is one of the cornerstone problems in optimal stopping theory and has become a crucial tool for designing sequential algorithms in Bayesian settings. In the i.i.d. $k$-selection prophet inequality problem, we sequentially observe $n$ non-negative random values sampled from a known distribution. Each time, a decision is made to accept or reject the value, and under the constr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.17767v1-abstract-full').style.display = 'inline'; document.getElementById('2406.17767v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.17767v1-abstract-full" style="display: none;"> The prophet inequality is one of the cornerstone problems in optimal stopping theory and has become a crucial tool for designing sequential algorithms in Bayesian settings. In the i.i.d. $k$-selection prophet inequality problem, we sequentially observe $n$ non-negative random values sampled from a known distribution. Each time, a decision is made to accept or reject the value, and under the constraint of accepting at most $k$. For $k=1$, Hill and Kertz [Ann. Probab. 1982] provided an upper bound on the worst-case approximation ratio that was later matched by an algorithm of Correa et al. [Math. Oper. Res. 2021]. The worst-case tight approximation ratio for $k=1$ is computed by studying a differential equation that naturally appears when analyzing the optimal dynamic programming policy. A similar result for $k>1$ has remained elusive. In this work, we introduce a nonlinear system of differential equations for the i.i.d. $k$-selection prophet inequality that generalizes Hill and Kertz's equation when $k=1$. Our nonlinear system is defined by $k$ constants that determine its functional structure, and their summation provides a lower bound on the optimal policy's asymptotic approximation ratio for the i.i.d. $k$-selection prophet inequality. To obtain this result, we introduce for every $k$ an infinite-dimensional linear programming formulation that fully characterizes the worst-case tight approximation ratio of the $k$-selection prophet inequality problem for every $n$, and then we follow a dual-fitting approach to link with our nonlinear system for sufficiently large values of $n$. As a corollary, we use our provable lower bounds to establish a tight approximation ratio for the stochastic sequential assignment problem in the i.i.d. non-negative regime. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.17767v1-abstract-full').style.display = 'none'; document.getElementById('2406.17767v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.03687">arXiv:2405.03687</a> <span> [<a href="https://arxiv.org/pdf/2405.03687">pdf</a>, <a href="https://arxiv.org/format/2405.03687">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Monotone Randomized Apportionment </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Correa%2C+J">Jos茅 Correa</a>, <a href="/search/cs?searchtype=author&query=G%C3%B6lz%2C+P">Paul G枚lz</a>, <a href="/search/cs?searchtype=author&query=Schmidt-Kraepelin%2C+U">Ulrike Schmidt-Kraepelin</a>, <a href="/search/cs?searchtype=author&query=Tucker-Foltz%2C+J">Jamie Tucker-Foltz</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.03687v1-abstract-short" style="display: inline;"> Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.03687v1-abstract-full').style.display = 'inline'; document.getElementById('2405.03687v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.03687v1-abstract-full" style="display: none;"> Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed to overcome this impossibility by randomizing the apportionment, which can achieve quota as well as perfect proportionality and monotonicity -- at least in terms of the expected number of seats awarded to each party. Still, the correlations between the seats awarded to different parties may exhibit bizarre non-monotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these non-monotonicities can cause paradoxes, including incentives for strategic voting. In this paper, we propose monotonicity axioms ruling out these paradoxes, and study which of them can be satisfied jointly with Grimmett's axioms. Essentially, we require that, if a set of parties all receive more votes, the probability of those parties jointly receiving more seats should increase. Our work draws on a rich literature on unequal probability sampling in statistics (studied as dependent randomized rounding in computer science). Our main result shows that a sampling scheme due to Sampford (1967) satisfies Grimmett's axioms and a notion of higher-order correlation monotonicity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.03687v1-abstract-full').style.display = 'none'; document.getElementById('2405.03687v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.11084">arXiv:2402.11084</a> <span> [<a href="https://arxiv.org/pdf/2402.11084">pdf</a>, <a href="https://arxiv.org/ps/2402.11084">ps</a>, <a href="https://arxiv.org/format/2402.11084">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> The Competition Complexity of Prophet Inequalities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Brustle%2C+J">Johannes Brustle</a>, <a href="/search/cs?searchtype=author&query=Correa%2C+J">Jos茅 Correa</a>, <a href="/search/cs?searchtype=author&query=D%C3%BCtting%2C+P">Paul D眉tting</a>, <a href="/search/cs?searchtype=author&query=Ezra%2C+T">Tomer Ezra</a>, <a href="/search/cs?searchtype=author&query=Feldman%2C+M">Michal Feldman</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.11084v2-abstract-short" style="display: inline;"> We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the $(1-\varepsilon)$-competition complexity of different types of online algorithms. This metric asks for the smallest $k$ such that the expected value of the online algorithm on $k$ copies of the original instance, is at least a $(1-\varepsilon)$-approximation to the expected… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.11084v2-abstract-full').style.display = 'inline'; document.getElementById('2402.11084v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.11084v2-abstract-full" style="display: none;"> We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the $(1-\varepsilon)$-competition complexity of different types of online algorithms. This metric asks for the smallest $k$ such that the expected value of the online algorithm on $k$ copies of the original instance, is at least a $(1-\varepsilon)$-approximation to the expected offline optimum on a single copy. We show that block threshold algorithms, which set one threshold per copy, are optimal and give a tight bound of $k = 螛(\log \log 1/\varepsilon)$. This shows that block threshold algorithms approach the offline optimum doubly-exponentially fast. For single threshold algorithms, we give a tight bound of $k = 螛(\log 1/\varepsilon)$, establishing an exponential gap between block threshold algorithms and single threshold algorithms. Our model and results pave the way for exploring resource-augmented prophet inequalities in combinatorial settings. In line with this, we present preliminary findings for bipartite matching with one-sided vertex arrivals, as well as in XOS combinatorial auctions. Our results have a natural competition complexity interpretation in mechanism design and pricing applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.11084v2-abstract-full').style.display = 'none'; document.getElementById('2402.11084v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C59; 68W27; 62L15; 60G40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.00890">arXiv:2311.00890</a> <span> [<a href="https://arxiv.org/pdf/2311.00890">pdf</a>, <a href="https://arxiv.org/format/2311.00890">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Online Combinatorial Assignment in Independence Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Marinkovic%2C+J">Javier Marinkovic</a>, <a href="/search/cs?searchtype=author&query=Soto%2C+J+A">Jos茅 A. Soto</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.00890v1-abstract-short" style="display: inline;"> We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matching… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00890v1-abstract-full').style.display = 'inline'; document.getElementById('2311.00890v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.00890v1-abstract-full" style="display: none;"> We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matchings of a hypergraph we recover the combinatorial auction problem, where every node represents an item to be sold, and every edge represents a bundle of items. For combinatorial auctions, Kesselheim et al. showed upper bounds of O(loglog(k)/log(k)) and $O(\log \log(n)/\log(n))$ on the competitiveness of any online algorithm, even in the random order model, where $k$ is the maximum bundle size and $n$ is the number of items. We provide an exponential improvement on these upper bounds to show that the competitiveness of any online algorithm in the prophet IID setting is upper bounded by $O(\log(k)/k)$, and $O(\log(n)/\sqrt{n})$. Furthermore, using linear programming, we provide new and improved guarantees for the $k$-bounded online combinatorial auction problem (i.e., bundles of size at most $k$). We show a $(1-e^{-k})/k$-competitive algorithm in the prophet IID model, a $1/(k+1)$-competitive algorithm in the prophet-secretary model using a single sample per agent, and a $k^{-k/(k-1)}$-competitive algorithm in the secretary model. Our algorithms run in polynomial time and work in more general independence systems where the offline combinatorial assignment problem admits the existence of a polynomial-time randomized algorithm that we call certificate sampler. We show that certificate samplers have a nice interplay with random order models, and we also provide new polynomial-time competitive algorithms for some classes of matroids, matroid intersections, and matchoids. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00890v1-abstract-full').style.display = 'none'; document.getElementById('2311.00890v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2111.06163">arXiv:2111.06163</a> <span> [<a href="https://arxiv.org/pdf/2111.06163">pdf</a>, <a href="https://arxiv.org/format/2111.06163">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> A 2-Approximation for the Bounded Treewidth Sparsest Cut Problem in FPT Time </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Cohen-Addad%2C+V">Vincent Cohen-Addad</a>, <a href="/search/cs?searchtype=author&query=M%C3%B6mke%2C+T">Tobias M枚mke</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2111.06163v1-abstract-short" style="display: inline;"> In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.06163v1-abstract-full').style.display = 'inline'; document.getElementById('2111.06163v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2111.06163v1-abstract-full" style="display: none;"> In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta, Talwar and Witmer [STOC 2013] obtained a 2-approximation with polynomial running time for fixed k, and the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in FPT time, remained open. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in FPT time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in FPT time. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.06163v1-abstract-full').style.display = 'none'; document.getElementById('2111.06163v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, 2 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.04137">arXiv:2108.04137</a> <span> [<a href="https://arxiv.org/pdf/2108.04137">pdf</a>, <a href="https://arxiv.org/ps/2108.04137">ps</a>, <a href="https://arxiv.org/format/2108.04137">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computers and Society">cs.CY</span> </div> </div> <p class="title is-5 mathjax"> Apportionment with Parity Constraints </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Mathieu%2C+C">Claire Mathieu</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2108.04137v1-abstract-short" style="display: inline;"> In the classic apportionment problem the goal is to decide how many seats of a parliament should be allocated to each party as a result of an election. The divisor methods provide a way of solving this problem by defining a notion of proportionality guided by some rounding rule. Motivated by recent challenges in the context of electoral apportionment, we consider the question of how to allocate th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04137v1-abstract-full').style.display = 'inline'; document.getElementById('2108.04137v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.04137v1-abstract-full" style="display: none;"> In the classic apportionment problem the goal is to decide how many seats of a parliament should be allocated to each party as a result of an election. The divisor methods provide a way of solving this problem by defining a notion of proportionality guided by some rounding rule. Motivated by recent challenges in the context of electoral apportionment, we consider the question of how to allocate the seats of a parliament under parity constraints between candidate types (e.g. equal number of men and women elected) while at the same time satisfying party proportionality. We consider two different approaches for this problem. The first mechanism, that follows a greedy approach, corresponds to a recent mechanism used in the Chilean Constitutional Convention 2021 election. We analyze this mechanism from a theoretical point of view. The second mechanism follows the idea of biproportionality introduced by Balinski and Demange [Math. Program. 1989, Math. Oper. Res. 1989]. In contrast with the classic biproportional method by Balinski and Demange, this mechanism is ruled by two levels of proportionality: Proportionality is satisfied at the level of parties by means of a divisor method, and then biproportionality is used to decide the number of candidates allocated to each type and party. We provide a theoretical analysis of this mechanism, making progress on the theoretical understanding of methods with two levels of proportionality. A typical benchmark used in the context of two-dimensional apportionment is the fair share (a.k.a matrix scaling), which corresponds to an ideal fractional biproportional solution. We provide lower bounds on the distance between these two types of solutions, and we explore their consequences in the context of two-dimensional apportionment. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04137v1-abstract-full').style.display = 'none'; document.getElementById('2108.04137v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.08077">arXiv:2107.08077</a> <span> [<a href="https://arxiv.org/pdf/2107.08077">pdf</a>, <a href="https://arxiv.org/format/2107.08077">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Science and Game Theory">cs.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> </div> </div> <p class="title is-5 mathjax"> The Convergence Rates of Blockchain Mining Games: A Markovian Approach </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Jofr%C3%A9%2C+A">Alejandro Jofr茅</a>, <a href="/search/cs?searchtype=author&query=Pardo%2C+A">Angel Pardo</a>, <a href="/search/cs?searchtype=author&query=Salas%2C+D">David Salas</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a>, <a href="/search/cs?searchtype=author&query=Verschae%2C+J">Jos茅 Verschae</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.08077v1-abstract-short" style="display: inline;"> Understanding the strategic behavior of miners in a blockchain is of great importance for its proper operation. A common model for mining games considers an infinite time horizon, with players optimizing asymptotic average objectives. Implicitly, this assumes that the asymptotic behaviors are realized at human-scale times, otherwise invalidating current models. We study the mining game utilizing M… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.08077v1-abstract-full').style.display = 'inline'; document.getElementById('2107.08077v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.08077v1-abstract-full" style="display: none;"> Understanding the strategic behavior of miners in a blockchain is of great importance for its proper operation. A common model for mining games considers an infinite time horizon, with players optimizing asymptotic average objectives. Implicitly, this assumes that the asymptotic behaviors are realized at human-scale times, otherwise invalidating current models. We study the mining game utilizing Markov Decision Processes. Our approach allows us to describe the asymptotic behavior of the game in terms of the stationary distribution of the induced Markov chain. We focus on a model with two players under immediate release, assuming two different objectives: the (asymptotic) average reward per turn and the (asymptotic) percentage of obtained blocks. Using tools from Markov chain analysis, we show the existence of a strategy achieving slow mixing times, exponential in the policy parameters. This result emphasizes the imperative need to understand convergence rates in mining games, validating the standard models. Towards this end, we provide upper bounds for the mixing time of certain meaningful classes of strategies. This result yields criteria for establishing that long-term averaged functions are coherent as payoff functions. Moreover, by studying hitting times, we provide a criterion to validate the common simplification of considering finite states models. For both considered objectives functions, we provide explicit formulae depending on the stationary distribution of the underlying Markov chain. In particular, this shows that both mentioned objectives are not equivalent. Finally, we perform a market share case study in a particular regime of the game. More precisely, we show that an strategic player with a sufficiently large processing power can impose negative revenue on honest players. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.08077v1-abstract-full').style.display = 'none'; document.getElementById('2107.08077v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.04674">arXiv:2107.04674</a> <span> [<a href="https://arxiv.org/pdf/2107.04674">pdf</a>, <a href="https://arxiv.org/ps/2107.04674">ps</a>, <a href="https://arxiv.org/format/2107.04674">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Preserving Diversity when Partitioning: A Geometric Approach </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Perez-Salazar%2C+S">Sebastian Perez-Salazar</a>, <a href="/search/cs?searchtype=author&query=Torrico%2C+A">Alfredo Torrico</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.04674v1-abstract-short" style="display: inline;"> Diversity plays a crucial role in multiple contexts such as team formation, representation of minority groups and generally when allocating resources fairly. Given a community composed by individuals of different types, we study the problem of partitioning this community such that the global diversity is preserved as much as possible in each subgroup. We consider the diversity metric introduced by… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.04674v1-abstract-full').style.display = 'inline'; document.getElementById('2107.04674v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.04674v1-abstract-full" style="display: none;"> Diversity plays a crucial role in multiple contexts such as team formation, representation of minority groups and generally when allocating resources fairly. Given a community composed by individuals of different types, we study the problem of partitioning this community such that the global diversity is preserved as much as possible in each subgroup. We consider the diversity metric introduced by Simpson in his influential work that, roughly speaking, corresponds to the inverse probability that two individuals are from the same type when taken uniformly at random, with replacement, from the community of interest. We provide a novel perspective by reinterpreting this quantity in geometric terms. We characterize the instances in which the optimal partition exactly preserves the global diversity in each subgroup. When this is not possible, we provide an efficient polynomial-time algorithm that outputs an optimal partition for the problem with two types. Finally, we discuss further challenges and open questions for the problem that considers more than two types. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.04674v1-abstract-full').style.display = 'none'; document.getElementById('2107.04674v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1902.10271">arXiv:1902.10271</a> <span> [<a href="https://arxiv.org/pdf/1902.10271">pdf</a>, <a href="https://arxiv.org/ps/1902.10271">ps</a>, <a href="https://arxiv.org/format/1902.10271">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> On the extension complexity of scheduling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Tiwary%2C+H+R">Hans Raj Tiwary</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a>, <a href="/search/cs?searchtype=author&query=Wiese%2C+A">Andreas Wiese</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1902.10271v1-abstract-short" style="display: inline;"> Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful approach for this is the construction of an extended formulation, which is a linear program in a higher dimensional space whose projection yields the original lin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.10271v1-abstract-full').style.display = 'inline'; document.getElementById('1902.10271v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1902.10271v1-abstract-full" style="display: none;"> Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful approach for this is the construction of an extended formulation, which is a linear program in a higher dimensional space whose projection yields the original linear program. For many problems it is known that a small extended formulation cannot exist. However, most of these problems are either $\mathsf{NP}$-hard (like TSP), or only quite complicated polynomial time algorithms are known for them (like for the matching problem). In this work we study the minimum makespan problem on identical machines in which we want to assign a set of $n$ given jobs to $m$ machines in order to minimize the maximum load over the machines. We prove that the canonical formulation for this problem has extension complexity $2^{惟(n/\log n)}$, even if each job has size 1 or 2 and the optimal makespan is 2. This is a case that a trivial greedy algorithm can solve optimally! For the more powerful configuration integer program, we even prove a lower bound of $2^{惟(n)}$. On the other hand, we show that there is an abstraction of the configuration integer program admitting an extended formulation of size $f(\text{opt})\cdot \text{poly}(n,m)$. In addition, we give an $O(\log n)$-approximate integral formulation of polynomial size, even for arbitrary processing times and for the far more general setting of unrelated machines. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.10271v1-abstract-full').style.display = 'none'; document.getElementById('1902.10271v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.08539">arXiv:1811.08539</a> <span> [<a href="https://arxiv.org/pdf/1811.08539">pdf</a>, <a href="https://arxiv.org/ps/1811.08539">ps</a>, <a href="https://arxiv.org/format/1811.08539">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a>, <a href="/search/cs?searchtype=author&query=Verschae%2C+J">Jos茅 Verschae</a>, <a href="/search/cs?searchtype=author&query=Wiese%2C+A">Andreas Wiese</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.08539v3-abstract-short" style="display: inline;"> The Sum of Squares (\sos{}) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of this hierarchy give integrality gaps matching the best known approximation algorithms. For many other problems, however, ad-hoc techniques give better approximation ratios than \sos{… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.08539v3-abstract-full').style.display = 'inline'; document.getElementById('1811.08539v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.08539v3-abstract-full" style="display: none;"> The Sum of Squares (\sos{}) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of this hierarchy give integrality gaps matching the best known approximation algorithms. For many other problems, however, ad-hoc techniques give better approximation ratios than \sos{} in the worst case, as shown by corresponding lower bound instances. Notably, in many cases these instances are invariant under the action of a large permutation group. This yields the question how symmetries in a formulation degrade the performance of the relaxation obtained by the \sos{} hierarchy. In this paper, we study this for the case of the minimum makespan problem on identical machines. Our first result is to show that $惟(n)$ rounds of \sos{} applied over the \emph{configuration linear program} yields an integrality gap of at least $1.0009$, where $n$ is the number of jobs. Our result is based on tools from representation theory of symmetric groups. Then, we consider the weaker \emph{assignment linear program} and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying $2^{\tilde{O}(1/\varepsilon^2)}$ rounds of the SA hierarchy to this stronger linear program reduces the integrality gap to $1+\varepsilon$, which yields a linear programming based polynomial time approximation scheme. Our results suggest that for this classical problem, symmetries were the main barrier preventing the \sos{}/ SA hierarchies to give relaxations of polynomial complexity with an integrality gap of~$1+\varepsilon$. We leave as an open question whether this phenomenon occurs for other symmetric problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.08539v3-abstract-full').style.display = 'none'; document.getElementById('1811.08539v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.01997">arXiv:1802.01997</a> <span> [<a href="https://arxiv.org/pdf/1802.01997">pdf</a>, <a href="https://arxiv.org/ps/1802.01997">ps</a>, <a href="https://arxiv.org/format/1802.01997">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Strong Algorithms for the Ordinal Matroid Secretary Problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Soto%2C+J+A">Jos茅 A. Soto</a>, <a href="/search/cs?searchtype=author&query=Turkieltaub%2C+A">Abner Turkieltaub</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1802.01997v1-abstract-short" style="display: inline;"> In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is $伪$ probability-competitive if every element from the optimum appears with probability… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.01997v1-abstract-full').style.display = 'inline'; document.getElementById('1802.01997v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.01997v1-abstract-full" style="display: none;"> In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is $伪$ probability-competitive if every element from the optimum appears with probability $1/伪$ in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on $2e$ by Korula and P谩l [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of $k$ column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a $1+O(\sqrt{\log 蟻/蟻})$ probability-competitive algorithm for uniform matroids of rank $蟻$ based on Kleinberg's $1+O(\sqrt{1/蟻})$ utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank $蟻$. We devise an $O(\log 蟻)$ probability-competitive algorithm and an $O(\log\log 蟻)$ ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the $O(\log\log 蟻)$ utility-competitive algorithm by Feldman et al.~[SODA 2015]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.01997v1-abstract-full').style.display = 'none'; document.getElementById('1802.01997v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">A preliminary version appeared at ACM-SIAM SODA 18</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.02058">arXiv:1710.02058</a> <span> [<a href="https://arxiv.org/pdf/1710.02058">pdf</a>, <a href="https://arxiv.org/format/1710.02058">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Skyline Computation with Noisy Comparisons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Groz%2C+B">Beno卯t Groz</a>, <a href="/search/cs?searchtype=author&query=Mallmann-Trenn%2C+F">Frederik Mallmann-Trenn</a>, <a href="/search/cs?searchtype=author&query=Mathieu%2C+C">Claire Mathieu</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.02058v2-abstract-short" style="display: inline;"> Given a set of $n$ points in a $d$-dimensional space, we seek to compute the skyline, i.e., those points that are not strictly dominated by any other point, using few comparisons between elements. We adopt the noisy comparison model [FRPU94] where comparisons fail with constant probability and confidence can be increased through independent repetitions of a comparison. In this model motivated by C… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.02058v2-abstract-full').style.display = 'inline'; document.getElementById('1710.02058v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.02058v2-abstract-full" style="display: none;"> Given a set of $n$ points in a $d$-dimensional space, we seek to compute the skyline, i.e., those points that are not strictly dominated by any other point, using few comparisons between elements. We adopt the noisy comparison model [FRPU94] where comparisons fail with constant probability and confidence can be increased through independent repetitions of a comparison. In this model motivated by Crowdsourcing applications, Groz & Milo [GM15] show three bounds on the query complexity for the skyline problem. We improve significantly on that state of the art and provide two output-sensitive algorithms computing the skyline with respective query complexity $O(nd\log (dk/未))$ and $O(ndk\log (k/未))$ where $k$ is the size of the skyline and $未$ the expected probability that our algorithm fails to return the correct answer. These results are tight for low dimensions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.02058v2-abstract-full').style.display = 'none'; document.getElementById('1710.02058v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.03959">arXiv:1702.03959</a> <span> [<a href="https://arxiv.org/pdf/1702.03959">pdf</a>, <a href="https://arxiv.org/format/1702.03959">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> How large is your graph? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Kanade%2C+V">Varun Kanade</a>, <a href="/search/cs?searchtype=author&query=Mallmann-Trenn%2C+F">Frederik Mallmann-Trenn</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.03959v1-abstract-short" style="display: inline;"> We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a \emph{seed} node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution $蟺$ uses… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.03959v1-abstract-full').style.display = 'inline'; document.getElementById('1702.03959v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.03959v1-abstract-full" style="display: none;"> We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a \emph{seed} node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution $蟺$ uses $O\left(\frac{1}{\|蟺\|_2} + \text{davg}\right)$ queries, we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily, the results of Katzir et al. give an upper bound that is worse by a multiplicative factor $t_\text{mix} \cdot \log(n)$. The picture becomes significantly different in the case of directed graphs. We show that without strong assumptions on the graph structure, the number of nodes cannot be predicted to within a constant multiplicative factor without using a number of queries that are at least linear in the number of nodes, in particular, rapid mixing and small diameter, properties that most real-world networks exhibit, do not suffice. The question of interest is whether any algorithm can beat breadth-first search. We introduce a new parameter, generalising the well-studied conductance, such that if a suitable bound on it exists and is known to the algorithm, the number of queries required is sublinear in the number of edges, we show that this is tight. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.03959v1-abstract-full').style.display = 'none'; document.getElementById('1702.03959v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1212.1754">arXiv:1212.1754</a> <span> [<a href="https://arxiv.org/pdf/1212.1754">pdf</a>, <a href="https://arxiv.org/ps/1212.1754">ps</a>, <a href="https://arxiv.org/format/1212.1754">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Split Scheduling with Uniform Setup Times </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Schalekamp%2C+F">Frans Schalekamp</a>, <a href="/search/cs?searchtype=author&query=Sitters%2C+R">Rene Sitters</a>, <a href="/search/cs?searchtype=author&query=van+der+Ster%2C+S">Suzanne van der Ster</a>, <a href="/search/cs?searchtype=author&query=Stougie%2C+L">Leen Stougie</a>, <a href="/search/cs?searchtype=author&query=Verdugo%2C+V">Victor Verdugo</a>, <a href="/search/cs?searchtype=author&query=van+Zuylen%2C+A">Anke van Zuylen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1212.1754v1-abstract-short" style="display: inline;"> We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and se… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1212.1754v1-abstract-full').style.display = 'inline'; document.getElementById('1212.1754v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1212.1754v1-abstract-full" style="display: none;"> We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1212.1754v1-abstract-full').style.display = 'none'; document.getElementById('1212.1754v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2012. </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul 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