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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Aardal, K"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <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/2411.18321">arXiv:2411.18321</a> <span> [<a href="https://arxiv.org/pdf/2411.18321">pdf</a>, <a href="https://arxiv.org/format/2411.18321">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="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Software">cs.MS</span> </div> </div> <p class="title is-5 mathjax"> Learning optimal objective values for MILP </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a>, <a href="/search/cs?searchtype=author&query=Yorke-Smith%2C+N">Neil Yorke-Smith</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="2411.18321v1-abstract-short" style="display: inline;"> Modern Mixed Integer Linear Programming (MILP) solvers use the Branch-and-Bound algorithm together with a plethora of auxiliary components that speed up the search. In recent years, there has been an explosive development in the use of machine learning for enhancing and supporting these algorithmic components. Within this line, we propose a methodology for predicting the optimal objective value, o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.18321v1-abstract-full').style.display = 'inline'; document.getElementById('2411.18321v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.18321v1-abstract-full" style="display: none;"> Modern Mixed Integer Linear Programming (MILP) solvers use the Branch-and-Bound algorithm together with a plethora of auxiliary components that speed up the search. In recent years, there has been an explosive development in the use of machine learning for enhancing and supporting these algorithmic components. Within this line, we propose a methodology for predicting the optimal objective value, or, equivalently, predicting if the current incumbent is optimal. For this task, we introduce a predictor based on a graph neural network (GNN) architecture, together with a set of dynamic features. Experimental results on diverse benchmarks demonstrate the efficacy of our approach, achieving high accuracy in the prediction task and outperforming existing methods. These findings suggest new opportunities for integrating ML-driven predictions into MILP solvers, enabling smarter decision-making and improved performance. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.18321v1-abstract-full').style.display = 'none'; document.getElementById('2411.18321v1-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> 27 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.05501">arXiv:2402.05501</a> <span> [<a href="https://arxiv.org/pdf/2402.05501">pdf</a>, <a href="https://arxiv.org/format/2402.05501">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="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Machine Learning Augmented Branch and Bound for Mixed Integer Linear Programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a>, <a href="/search/cs?searchtype=author&query=Lodi%2C+A">Andrea Lodi</a>, <a href="/search/cs?searchtype=author&query=Yorke-Smith%2C+N">Neil Yorke-Smith</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="2402.05501v1-abstract-short" style="display: inline;"> Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving MILPs, and many commercial and academic software packages exist. Nevertheless, the availability of data, both from problem instances and from solvers, and the desir… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.05501v1-abstract-full').style.display = 'inline'; document.getElementById('2402.05501v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.05501v1-abstract-full" style="display: none;"> Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving MILPs, and many commercial and academic software packages exist. Nevertheless, the availability of data, both from problem instances and from solvers, and the desire to solve new problems and larger (real-life) instances, trigger the need for continuing algorithmic development. MILP solvers use branch and bound as their main component. In recent years, there has been an explosive development in the use of machine learning algorithms for enhancing all main tasks involved in the branch-and-bound algorithm, such as primal heuristics, branching, cutting planes, node selection and solver configuration decisions. This paper presents a survey of such approaches, addressing the vision of integration of machine learning and mathematical optimization as complementary technologies, and how this integration can benefit MILP solving. In particular, we give detailed attention to machine learning algorithms that automatically optimize some metric of branch-and-bound efficiency. We also address how to represent MILPs in the context of applying learning algorithms, MILP benchmarks and software. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.05501v1-abstract-full').style.display = 'none'; document.getElementById('2402.05501v1-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> 8 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.11107">arXiv:2205.11107</a> <span> [<a href="https://arxiv.org/pdf/2205.11107">pdf</a>, <a href="https://arxiv.org/format/2205.11107">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</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"> Learning to branch with Tree MDPs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Chen%2C+F+Y">Feng Yang Chen</a>, <a href="/search/cs?searchtype=author&query=Ch%C3%A9telat%2C+D">Didier Ch茅telat</a>, <a href="/search/cs?searchtype=author&query=Gasse%2C+M">Maxime Gasse</a>, <a href="/search/cs?searchtype=author&query=Lodi%2C+A">Andrea Lodi</a>, <a href="/search/cs?searchtype=author&query=Yorke-Smith%2C+N">Neil Yorke-Smith</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</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="2205.11107v3-abstract-short" style="display: inline;"> State-of-the-art Mixed Integer Linear Program (MILP) solvers combine systematic tree search with a plethora of hard-coded heuristics, such as the branching rule. The idea of learning branching rules from data has received increasing attention recently, and promising results have been obtained by learning fast approximations of the strong branching expert. In this work, we instead propose to learn… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.11107v3-abstract-full').style.display = 'inline'; document.getElementById('2205.11107v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.11107v3-abstract-full" style="display: none;"> State-of-the-art Mixed Integer Linear Program (MILP) solvers combine systematic tree search with a plethora of hard-coded heuristics, such as the branching rule. The idea of learning branching rules from data has received increasing attention recently, and promising results have been obtained by learning fast approximations of the strong branching expert. In this work, we instead propose to learn branching rules from scratch via Reinforcement Learning (RL). We revisit the work of Etheve et al. (2020) and propose tree Markov Decision Processes, or tree MDPs, a generalization of temporal MDPs that provides a more suitable framework for learning to branch. We derive a tree policy gradient theorem, which exhibits a better credit assignment compared to its temporal counterpart. We demonstrate through computational experiments that tree MDPs improve the learning convergence, and offer a promising framework for tackling the learning-to-branch problem in MILPs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.11107v3-abstract-full').style.display = 'none'; document.getElementById('2205.11107v3-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 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </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">10 pages, 2 figures, plus supplementary material</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2009.08748">arXiv:2009.08748</a> <span> [<a href="https://arxiv.org/pdf/2009.08748">pdf</a>, <a href="https://arxiv.org/ps/2009.08748">ps</a>, <a href="https://arxiv.org/format/2009.08748">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</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="Combinatorics">math.CO</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"> A polynomial size model with implicit SWAP gate counting for exact qubit reordering </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Mulderij%2C+J">Jesse Mulderij</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K+I">Karen I. Aardal</a>, <a href="/search/cs?searchtype=author&query=Chiscop%2C+I">Irina Chiscop</a>, <a href="/search/cs?searchtype=author&query=Phillipson%2C+F">Frank Phillipson</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="2009.08748v1-abstract-short" style="display: inline;"> Due to the physics behind quantum computing, quantum circuit designers must adhere to the constraints posed by the limited interaction distance of qubits. Existing circuits need therefore to be modified via the insertion of SWAP gates, which alter the qubit order by interchanging the location of two qubits' quantum states. We consider the Nearest Neighbor Compliance problem on a linear array, wher… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.08748v1-abstract-full').style.display = 'inline'; document.getElementById('2009.08748v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2009.08748v1-abstract-full" style="display: none;"> Due to the physics behind quantum computing, quantum circuit designers must adhere to the constraints posed by the limited interaction distance of qubits. Existing circuits need therefore to be modified via the insertion of SWAP gates, which alter the qubit order by interchanging the location of two qubits' quantum states. We consider the Nearest Neighbor Compliance problem on a linear array, where the number of required SWAP gates is to be minimized. We introduce an Integer Linear Programming model of the problem of which the size scales polynomially in the number of qubits and gates. Furthermore, we solve $131$ benchmark instances to optimality using the commercial solver CPLEX. The benchmark instances are substantially larger in comparison to those evaluated with exact methods before. The largest circuits contain up to $18$ qubits or over $100$ quantum gates. This formulation also seems to be suitable for developing heuristic methods since (near) optimal solutions are discovered quickly in the search process. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.08748v1-abstract-full').style.display = 'none'; document.getElementById('2009.08748v1-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 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2020. </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">18 pages, 3 figures, 4 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94C30; 68Q06 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> B.6.3 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1410.1409">arXiv:1410.1409</a> <span> [<a href="https://arxiv.org/pdf/1410.1409">pdf</a>, <a href="https://arxiv.org/format/1410.1409">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 class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.orl.2014.09.008">10.1016/j.orl.2014.09.008 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Approximation algorithms for the Transportation Problem with Market Choice and related models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a>, <a href="/search/cs?searchtype=author&query=Bodic%2C+P+L">Pierre Le Bodic</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="1410.1409v1-abstract-short" style="display: inline;"> Given facilities with capacities and clients with penalties and demands, the transportation problem with market choice consists in finding the minimum-cost way to partition the clients into unserved clients, paying the penalties, and into served clients, paying the transportation cost to serve them. We give polynomial-time reductions from this problem and variants to the (un)capacitated facility l… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.1409v1-abstract-full').style.display = 'inline'; document.getElementById('1410.1409v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1410.1409v1-abstract-full" style="display: none;"> Given facilities with capacities and clients with penalties and demands, the transportation problem with market choice consists in finding the minimum-cost way to partition the clients into unserved clients, paying the penalties, and into served clients, paying the transportation cost to serve them. We give polynomial-time reductions from this problem and variants to the (un)capacitated facility location problem, directly yielding approximation algorithms, two with constant factors in the metric case, one with a logarithmic factor in the general case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.1409v1-abstract-full').style.display = 'none'; document.getElementById('1410.1409v1-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 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2014. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1311.4759">arXiv:1311.4759</a> <span> [<a href="https://arxiv.org/pdf/1311.4759">pdf</a>, <a href="https://arxiv.org/format/1311.4759">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="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Approximation Algorithms for Hard Capacitated $k$-facility Location Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a>, <a href="/search/cs?searchtype=author&query=Berg%2C+P+v+d">Pieter van den Berg</a>, <a href="/search/cs?searchtype=author&query=Gijswijt%2C+D">Dion Gijswijt</a>, <a href="/search/cs?searchtype=author&query=Li%2C+S">Shanfei Li</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="1311.4759v4-abstract-short" style="display: inline;"> We study the capacitated $k$-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number $k$. It costs $f_i$ to open facility $i$, and $c_{ij}$ for facility $i$ to serve one unit of demand from client $j$. The objective is to open at most $k$ facilities serving all the demands and satisfying the capacity constraints whil… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1311.4759v4-abstract-full').style.display = 'inline'; document.getElementById('1311.4759v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1311.4759v4-abstract-full" style="display: none;"> We study the capacitated $k$-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number $k$. It costs $f_i$ to open facility $i$, and $c_{ij}$ for facility $i$ to serve one unit of demand from client $j$. The objective is to open at most $k$ facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs. In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated $k$-facility location problem. Then, we show that the capacitated $k$-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated $k$-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening cost. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1311.4759v4-abstract-full').style.display = 'none'; document.getElementById('1311.4759v4-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> 12 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 November, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2013. </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">We add new results obtained with Karen Aardal and Pieter van den Berg to the previous version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90B80 (primary); 68W25 (secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/cs/0703010">arXiv:cs/0703010</a> <span> [<a href="https://arxiv.org/pdf/cs/0703010">pdf</a>, <a href="https://arxiv.org/ps/cs/0703010">ps</a>, <a href="https://arxiv.org/format/cs/0703010">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"> An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Byrka%2C+J">Jaroslaw Byrka</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</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="cs/0703010v2-abstract-short" style="display: inline;"> We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463. An algorithm is a {\em ($位_f$,$位_c$)-approximation algorithm} if the solution it produces has total cost at most… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('cs/0703010v2-abstract-full').style.display = 'inline'; document.getElementById('cs/0703010v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="cs/0703010v2-abstract-full" style="display: none;"> We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463. An algorithm is a {\em ($位_f$,$位_c$)-approximation algorithm} if the solution it produces has total cost at most $位_f \cdot F^* + 位_c \cdot C^*$, where $F^*$ and $C^*$ are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the $(1+2/e)$-approximation algorithm of Chudak and Shmoys, is a (1.6774,1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve $(纬_f, 1+2e^{-纬_f})$ established by Jain, Mahdian and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11,1.7764)-approximation algorithm proposed by Jain et al., and later analyzed by Mahdian et al., we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('cs/0703010v2-abstract-full').style.display = 'none'; document.getElementById('cs/0703010v2-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 February, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 March, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2007. </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 journal version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.2.2 </p> </li> </ol> <div class="is-hidden-tablet">  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