Theoretical Computer Science最新文献

{"title":"The frontier of intractability for EFX with two agents","authors":"Paul W. Goldberg ,&nbsp;Kasper Høgh ,&nbsp;Alexandros Hollender","doi":"10.1016/j.tcs.2025.115367","DOIUrl":"10.1016/j.tcs.2025.115367","url":null,"abstract":"<div><div>We consider the problem of sharing a set of indivisible goods among agents in a fair manner, namely such that the allocation is envy-free up to any good (EFX). We focus on the problem of computing an EFX allocation in the two-agent case and characterize the computational complexity of the problem for most well-known valuation classes. We present a simple greedy algorithm that solves the problem when the agent valuations are weakly well-layered, a class which contains gross substitutes and budget-additive valuations. For the next largest valuation class we prove a negative result: the problem is <span>PLS</span>-complete for submodular valuations. All of our results also hold for the setting where there are many agents with identical valuations.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1052 ","pages":"Article 115367"},"PeriodicalIF":0.9,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On decidability and expressive power of fusion grammars","authors":"Tikhon Pshenitsyn","doi":"10.1016/j.tcs.2025.115420","DOIUrl":"10.1016/j.tcs.2025.115420","url":null,"abstract":"<div><div>We study algorithmic complexity and expressive power of fusion grammars, a novel formalism introduced in [Kreowski, Kuske, and Lye 2017], which extends hyperedge replacement grammars. In the first part of the work, we prove that the non-emptiness problem for fusion grammars and the membership problem for fusion grammars without markers and connectors are decidable and are in NEXPTIME. We introduce fusion grammars with bounded usage of markers and connectors and prove decidability of the membership problem for them as well. In the proofs, we develop the technique of hypergraph vertex colourings encoded in hyperedge labels and also the technique of evidence paths and their encodings.</div><div>In the second part of the work, we study the class of languages generated by connection-preserving fusion grammars. Namely, we prove Parikh's theorem for them, i.e. we show that these languages are semilinear.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1052 ","pages":"Article 115420"},"PeriodicalIF":0.9,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimizing the total completion time for a class of semi-online single machine scheduling problems","authors":"Hajar Nouinou ,&nbsp;Taha Arbaoui ,&nbsp;Alice Yaloui","doi":"10.1016/j.tcs.2025.115422","DOIUrl":"10.1016/j.tcs.2025.115422","url":null,"abstract":"<div><div>Semi-online single machine scheduling problems with information on jobs' processing times and the objective to minimize the total completion time are considered. In these problems, a set of jobs arriving over time are to be scheduled on a single machine and their characteristics become known only upon arrival. Some of the studied problems are shown to have the same competitive ratio as the online problem and online scheduling algorithms can be applied. Given some constraints on the processing times of successive jobs, new lower bounds can be achieved and a new semi-online algorithm, called <em>ϕ</em>D-SPT, is presented along with its competitive analysis.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1051 ","pages":"Article 115422"},"PeriodicalIF":0.9,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144312978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"K-balanced biclique partition: Kernelization and efficient algorithms","authors":"Yifei Li ,&nbsp;Donghua Yang ,&nbsp;Jianzhong Li","doi":"10.1016/j.tcs.2025.115410","DOIUrl":"10.1016/j.tcs.2025.115410","url":null,"abstract":"<div><div>Balanced signed biclique captures cohesive friend-foe relations in social and biological networks. We initiate the study of the k-Balanced Biclique Partition (k-BBP): given a signed bipartite graph, partition its edge set into at most k balanced signed bicliques. First, we prove that deciding whether k-BBP exists is NP-hard by a polynomial reduction from Non-negative Matrix Factorization. We present the kernelization that contracts the input graph to a kernel with no more than <span><math><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup></math></span> edges, ensuring polynomial preprocessing time. We relax the decision problem to a property-testing variant of k-BBP, designing a parameterized one-sided tester that runs in time independent of the input size. The tester accepts graphs admitting the desired partition and, with high probability, rejects graphs that are <em>ϵ</em>-far from having any such partition. Next, we study how to find an approximate k-BBP with a guaranteed error bound. We first recast the problem as a constrained discrete optimization problem and devise an alternating optimization algorithm with sub-exponential time complexity. We further relax the problem to a continuous optimization setting. Leveraging the multi-block convex objective, we design a linear-time approximation algorithm.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1052 ","pages":"Article 115410"},"PeriodicalIF":0.9,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unit refutability of horn constraint systems - certification and parallel complexity","authors":"P. Wojciechowski,&nbsp;K. Subramani","doi":"10.1016/j.tcs.2025.115419","DOIUrl":"10.1016/j.tcs.2025.115419","url":null,"abstract":"<div><div>This paper is concerned with algorithmic procedures for checking if a Horn Constraint System (HCS) has unit refutations of linear feasibility. A Horn constraint is a linear constraint of the form <span><math><mi>a</mi><mo>⋅</mo><mi>x</mi><mo>≥</mo><mi>b</mi></math></span>, where each <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span> and at most one of the <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s is positive. Horn constraints generalize Horn clauses and find applications in a number of practical domains such as program verification and econometrics. There exist quite a few algorithms in the literature for checking linear (and hence integer) feasibility in HCSs. We are concerned with the complementary problem, viz., that of finding refutations. In particular, we focus on unit refutations, i.e., refutations in which a unit variable constraint <strong>must</strong> be used in each inference step. Unit refutations are an analogue of unit resolution, which is a widely used refutation system in Boolean constraint solving. Whereas unit resolution is <strong>complete</strong> for Horn clausal systems, unit refutation is <strong>incomplete</strong> for HCSs. Although incomplete, unit refutations are very useful from the perspective of establishing a certain type of infeasibility. Our main contribution is showing that the problem of checking if an infeasible HCS has a unit refutation is in <strong>P</strong>. Towards this end, we provide two algorithms: a non-certifying algorithm which runs in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time and a certifying algorithm which runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time, where <em>m</em> and <em>n</em> are the number of constraints and number of variables in the input HCS, respectively. Our algorithms are based on a combination of new and existing insights regarding Horn constraints. What is particularly interesting about our approach is that we are able to efficiently certify the existence of a unit refutation, although the actual refutation itself could be <em>exponential</em> in the size of the input. From the perspective of parallel algorithms, we show that the unit refutation problem in HCSs is <strong>P-complete</strong>.<span><span>²</span></span> </div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1053 ","pages":"Article 115419"},"PeriodicalIF":0.9,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some Results on Critical (P5,H)-free Graphs","authors":"Wen Xia ,&nbsp;Jorik Jooken ,&nbsp;Jan Goedgebeur ,&nbsp;Shenwei Huang","doi":"10.1016/j.tcs.2025.115411","DOIUrl":"10.1016/j.tcs.2025.115411","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Given two graphs &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, a graph is &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-free if it contains no induced subgraph isomorphic to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. A graph &lt;em&gt;G&lt;/em&gt; is &lt;em&gt;k&lt;/em&gt;-vertex-critical if every proper induced subgraph of &lt;em&gt;G&lt;/em&gt; has chromatic number less than &lt;em&gt;k&lt;/em&gt;, but &lt;em&gt;G&lt;/em&gt; has chromatic number &lt;em&gt;k&lt;/em&gt;. The study of &lt;em&gt;k&lt;/em&gt;-vertex-critical graphs for specific graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there exists a polynomial-time certifying algorithm to decide the &lt;em&gt;k&lt;/em&gt;-colorability of a graph in the class.&lt;/div&gt;&lt;div&gt;In this paper, we show that: (1) for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, there are finitely many &lt;em&gt;k&lt;/em&gt;-vertex-critical &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-free graphs; (2) for &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, there are finitely many 5-vertex-critical &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-free graphs; (3) for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, there are finitely many &lt;em&gt;k&lt;/em&gt;-vertex-critical &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-free graphs. Moreover, we characterize all 5-vertex-critical &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-free graphs where &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/m","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1051 ","pages":"Article 115411"},"PeriodicalIF":0.9,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simple random sampling of binary forests with fixed number of nodes and trees","authors":"Stoyan Dimitrov","doi":"10.1016/j.tcs.2025.115418","DOIUrl":"10.1016/j.tcs.2025.115418","url":null,"abstract":"<div><div>We generalize the classical algorithm of Rémy for random sampling of full binary trees with a given number of leaves. As a result, we give a simple linear time algorithm for random generation of full binary forests with given number of trees and leaves. The algorithm is obtained from a bijection that we construct in order to give a direct proof of the well-known fact that these forests are enumerated by the <em>k</em>-th fold self-convolution of the Catalan numbers. Via some well-known bijections, the given algorithm can be used to sample random objects from several other classes enumerated by self-convolutions of the Catalan numbers, e.g., binary forests with a given number of trees, lists of a given number of balanced strings and others.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1051 ","pages":"Article 115418"},"PeriodicalIF":0.9,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144312976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An approximation algorithm for the parity-constrained k-supplier problem","authors":"Xinlan Xia ,&nbsp;Lu Han ,&nbsp;Lili Mei","doi":"10.1016/j.tcs.2025.115413","DOIUrl":"10.1016/j.tcs.2025.115413","url":null,"abstract":"<div><div>This paper studies the parity-constrained <em>k</em>-supplier (PAR <em>k</em>-supplier) problem, which extends the well-known <em>k</em>-supplier problem. In the PAR <em>k</em>-supplier problem, we are given a set of vertices in a metric space with distances and an integer <em>k</em>. The vertex set is partitioned into a facility set and a client set. Each facility has an odd or even parity requirement. The objective is to select at most <em>k</em> facilities to open and assign each client to an open facility, ensuring that the number of clients assigned to each open facility matches its parity requirement, while also minimizing the maximum distance of any client to its assigned facility.</div><div>As our main contribution, we design the first constant-factor 9-approximation algorithm for the parity-constrained <em>k</em>-supplier problem. The algorithm is divided into two main phases. In the first phase, we determine the initial set of open facilities with a maximum cardinality of <em>k</em> and the assignment of all the clients. There may include some so-called invalid facilities that do not meet their parity requirements under the initial assignment. In the second phase, we find a matching based on the invalid facilities and reassign some clients accordingly to obtain a feasible solution.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1051 ","pages":"Article 115413"},"PeriodicalIF":0.9,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144297455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parameterized algorithms for multi-label periodic temporal graph realization","authors":"Thomas Erlebach ,&nbsp;Nils Morawietz ,&nbsp;Petra Wolf","doi":"10.1016/j.tcs.2025.115412","DOIUrl":"10.1016/j.tcs.2025.115412","url":null,"abstract":"<div><div>In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <em>D</em> of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between any pair of vertices matches the one specified by <em>D</em>. We generalize the problem from simple temporal graphs to temporal graphs where each edge can appear up to <em>ℓ</em> times in each period, for some given integer <em>ℓ</em>. For the resulting problem <span>Multi-Label Periodic TGR</span>, we show that it is fixed-parameter tractable for parameter <em>n</em> and for parameter <span><math><mrow><mi>vc</mi></mrow><mo>+</mo><mi>Δ</mi></math></span>, where vc is the vertex cover number of the underlying graph. We also show the existence of a polynomial kernel for parameter <span><math><mrow><mi>nu</mi></mrow><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span>, where nu is the number of non-universal vertices of the underlying graph and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span> is the largest entry of <em>D</em>. Furthermore, we show that the problem is <em>NP</em>-hard for each <span><math><mi>ℓ</mi><mo>≥</mo><mn>5</mn></math></span>, even if the underlying graph is a tree, a case that was known to be solvable in polynomial time if the task is to construct a simple periodic temporal graph, that is, if <span><math><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1051 ","pages":"Article 115412"},"PeriodicalIF":0.9,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Robust matroid bandit optimization: Near-optimal rates under adversarial contamination","authors":"Youming Tao ,&nbsp;Xiuzhen Cheng ,&nbsp;Falko Dressler ,&nbsp;Zhipeng Cai ,&nbsp;Dongxiao Yu","doi":"10.1016/j.tcs.2025.115416","DOIUrl":"10.1016/j.tcs.2025.115416","url":null,"abstract":"<div><div>We study the matroid bandit optimization problem, a fundamental and broadly applicable framework for combinatorial multi-armed bandits where the action space is constrained by a matroid. In particular, we address the challenge of designing algorithms that remain effective under adversarial contamination of feedback rewards, which may severely degrade performance or even mislead existing methods. Our main contribution is an efficient and robust algorithm named <span>ROMM</span>, which builds upon the principle of optimistic matroid maximization and leverages robust statistical estimators to assess base arm quality in polynomial time. Under the <em>ϵ</em>-contamination model, we establish lower bounds and prove that <span>ROMM</span> achieves near-optimal regret guarantees up to polylogarithmic factors. Our analysis further reveals a sharp phase transition between the low and high contamination regimes. Notably, <span>ROMM</span> can tolerate up to a universal constant fraction of corrupted feedback, which is optimal under mild conditions. Finally, we validate our theoretical findings with numerical experiments that demonstrate the effectiveness of the proposed method.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1051 ","pages":"Article 115416"},"PeriodicalIF":0.9,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}