Computational Complexity

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rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 2 of 2 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2503.15771" title="Abstract" id="2503.15771"> arXiv:2503.15771 </a> [<a href="/pdf/2503.15771" title="Download PDF" id="pdf-2503.15771" aria-labelledby="pdf-2503.15771">pdf</a>, <a href="https://arxiv.org/html/2503.15771v1" title="View HTML" id="html-2503.15771" aria-labelledby="html-2503.15771" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.15771" title="Other formats" id="oth-2503.15771" aria-labelledby="oth-2503.15771">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Recognizing and Realizing Temporal Reachability Graphs </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Erlebach,+T">Thomas Erlebach</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Michail,+O">Othon Michail</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Morawietz,+N">Nils Morawietz</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Computational Complexity (cs.CC)</span> </div> <p class='mathjax'> A temporal graph $\mathcal{G}=(G,\lambda)$ can be represented by an underlying graph $G=(V,E)$ together with a function $\lambda$ that assigns to each edge $e\in E$ the set of time steps during which $e$ is present. The reachability graph of $\mathcal{G}$ is the directed graph $D=(V,A)$ with $(u,v)\in A$ if only if there is a temporal path from $u$ to $v$. We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph $D=(V,A)$ is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on $\lambda$ (proper, simple, or happy). Answering an open question posed by Casteigts et al. (Theoretical Computer Science 991 (2024)), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is polynomial-time solvable if the solid graph is a tree and fixed-parameter tractable with respect to the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set or treedepth, since the problem is W[2]-hard with respect to these parameters. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2503.16089" title="Abstract" id="2503.16089"> arXiv:2503.16089 </a> [<a href="/pdf/2503.16089" title="Download PDF" id="pdf-2503.16089" aria-labelledby="pdf-2503.16089">pdf</a>, <a href="https://arxiv.org/html/2503.16089v1" title="View HTML" id="html-2503.16089" aria-labelledby="html-2503.16089" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.16089" title="Other formats" id="oth-2503.16089" aria-labelledby="oth-2503.16089">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Query-Efficient Fixpoints of $\ell_p$-Contractions </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Haslebacher,+S">Sebastian Haslebacher</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Lill,+J">Jonas Lill</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Schnider,+P">Patrick Schnider</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Weber,+S">Simon Weber</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 33 pages, 4 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Computational Complexity (cs.CC)</span>; Computational Geometry (cs.CG) </div> <p class='mathjax'> We prove that an $\epsilon$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}{\epsilon} + \log\frac{1}{1-\lambda}))$ queries to $f$ if $f$ is $\lambda$-contracting with respect to an $\ell_p$-metric for some $p\in [1,\infty)\cup\{\infty\}$. This generalizes a recent result of Chen, Li, and Yannakakis [STOC'24] from the $\ell_\infty$-case to all $\ell_p$-metrics. Previously, all query upper bounds for $p\in [1,\infty) \setminus \{2\}$ were either exponential in $d$, $\log\frac{1}{\epsilon}$, or $\log\frac{1}{1-\lambda}$. <br>Chen, Li, and Yannakakis also show how to ensure that all queries to $f$ lie on a discrete grid of limited granularity in the $\ell_\infty$-case. We provide such a rounding for the $\ell_1$-case, placing an appropriately defined version of the $\ell_1$-case in $\textsf{FP}^{dt}$. <br>To prove our results, we introduce the notion of $\ell_p$-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any $p \in [1, \infty) \cup \{\infty\}$ and any mass distribution (or point set), we prove that there exists a centerpoint $c$ such that every $\ell_p$-halfspace defined by $c$ and a normal vector contains at least a $\frac{1}{d+1}$-fraction of the mass (or points). </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 2 of 2 entries)</h3> <dt> <a name='item3'>[3]</a> <a href ="/abs/2502.02393" title="Abstract" id="2502.02393"> arXiv:2502.02393 </a> (replaced) [<a href="/pdf/2502.02393" title="Download PDF" id="pdf-2502.02393" aria-labelledby="pdf-2502.02393">pdf</a>, <a href="https://arxiv.org/html/2502.02393v2" title="View HTML" id="html-2502.02393" aria-labelledby="html-2502.02393" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.02393" title="Other formats" id="oth-2502.02393" aria-labelledby="oth-2502.02393">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Amiri,+A">Alireza Amiri</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Huang,+X">Xinting Huang</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Rofin,+M">Mark Rofin</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Hahn,+M">Michael Hahn</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Machine Learning (cs.LG)</span>; Computational Complexity (cs.CC) </div> <p class='mathjax'> Chain-of-thought reasoning and scratchpads have emerged as critical tools for enhancing the computational capabilities of transformers. While theoretical results show that polynomial-length scratchpads can extend transformers' expressivity from $TC^0$ to $PTIME$, their required length remains poorly understood. Empirical evidence even suggests that transformers need scratchpads even for many problems in $TC^0$, such as Parity or Multiplication, challenging optimistic bounds derived from circuit complexity. In this work, we initiate the study of systematic lower bounds for the number of CoT steps across different algorithmic problems, in the hard-attention regime. We study a variety of algorithmic problems, and provide bounds that are tight up to logarithmic factors. Overall, these results contribute to emerging understanding of the power and limitations of chain-of-thought reasoning. </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2503.15242" title="Abstract" id="2503.15242"> arXiv:2503.15242 </a> (replaced) [<a href="/pdf/2503.15242" title="Download PDF" id="pdf-2503.15242" aria-labelledby="pdf-2503.15242">pdf</a>, <a href="https://arxiv.org/html/2503.15242v2" title="View HTML" id="html-2503.15242" aria-labelledby="html-2503.15242" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.15242" title="Other formats" id="oth-2503.15242" aria-labelledby="oth-2503.15242">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> BigO(Bench) -- Can LLMs Generate Code with Controlled Time and Space Complexity? </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Chambon,+P">Pierre Chambon</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Roziere,+B">Baptiste Roziere</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Sagot,+B">Benoit Sagot</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Synnaeve,+G">Gabriel Synnaeve</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Computation and Language (cs.CL)</span>; Artificial Intelligence (cs.AI); Computational Complexity (cs.CC) </div> <p class='mathjax'> We introduce BigO(Bench), a novel coding benchmark designed to evaluate the capabilities of generative language models in understanding and generating code with specified time and space complexities. This benchmark addresses the gap in current evaluations that often overlook the ability of models to comprehend and produce code constrained by computational complexity. BigO(Bench) includes tooling to infer the algorithmic complexity of any Python function from profiling measurements, including human- or LLM-generated solutions. BigO(Bench) also includes of set of 3,105 coding problems and 1,190,250 solutions from Code Contests annotated with inferred (synthetic) time and space complexity labels from the complexity framework, as well as corresponding runtime and memory footprint values for a large set of input sizes. We present results from evaluating multiple state-of-the-art language models on this benchmark, highlighting their strengths and weaknesses in handling complexity requirements. In particular, token-space reasoning models are unrivaled in code generation but not in complexity understanding, hinting that they may not generalize well to tasks for which no reward was given at training time. </p> </div> </dd> </dl> <div class='paging'>Total of 4 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/cs.CC/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em">  <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; 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Computational Complexity