Pub Date : 2026-05-01Epub Date: 2025-11-03DOI: 10.1016/j.jcss.2025.103732
Jun Le Goh , Joseph S. Miller , Mariya I. Soskova , Linda Westrick
Let denote the set of infinite sequences of effective dimension t. Greenberg, Miller, Shen, and Westrick [6] studied both how near and how far an infinite sequence of dimension s can be from the closest sequence of dimension t, where distance in is measured using the Besicovitch pseudometric. They found and for all , except for the supremum when . This case is made difficult by the fact that the information in a dimension s sequence can be coded redundantly, so it is not clear what density of changes is needed to erase enough of that information. We completely solve the dimension reduction problem. We also identify classes of sequences for which these infima and suprema are realized as minima and maxima. When , we find is minimized when X is a Bernoulli -random, and maximized when X belongs to a class of infinite sequences that we call s-codewords. When , the situation is reversed. Finally, we prove that all distances between the extrema are realized.
{"title":"Redundancy of information: Lowering effective dimension","authors":"Jun Le Goh , Joseph S. Miller , Mariya I. Soskova , Linda Westrick","doi":"10.1016/j.jcss.2025.103732","DOIUrl":"10.1016/j.jcss.2025.103732","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msup></math></span> denote the set of infinite sequences of effective dimension <em>t</em>. Greenberg, Miller, Shen, and Westrick <span><span>[6]</span></span> studied both how near and how far an infinite sequence of dimension <em>s</em> can be from the closest sequence of dimension <em>t</em>, where distance in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is measured using the Besicovitch pseudometric. They found <span><math><msub><mrow><mi>inf</mi></mrow><mrow><mi>X</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></msub><mo></mo><mi>d</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>sup</mi></mrow><mrow><mi>X</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></msub><mo></mo><mi>d</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></math></span> for all <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, except for the supremum when <span><math><mi>t</mi><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn></math></span>. This case is made difficult by the fact that the information in a dimension <em>s</em> sequence can be coded redundantly, so it is not clear what density of changes is needed to erase enough of that information. We completely solve the dimension reduction problem. We also identify classes of sequences for which these infima and suprema are realized as minima and maxima. When <span><math><mi>t</mi><mo><</mo><mi>s</mi></math></span>, we find <span><math><mi>d</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></math></span> is minimized when <em>X</em> is a Bernoulli <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>s</mi><mo>)</mo></math></span>-random, and maximized when <em>X</em> belongs to a class of infinite sequences that we call <em>s</em>-codewords. When <span><math><mi>s</mi><mo><</mo><mi>t</mi></math></span>, the situation is reversed. Finally, we prove that all distances between the extrema are realized.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"157 ","pages":"Article 103732"},"PeriodicalIF":0.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145521312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-12-11DOI: 10.1016/j.jcss.2025.103747
Michael R. Fellows , Mario Grobler , Nicole Megow , Amer E. Mouawad , Vijayaragunathan Ramamoorthi , Frances A. Rosamond , Daniel Schmand , Sebastian Siebertz
The dynamics of real-world applications and systems require efficient methods for improving infeasible solutions or restoring corrupted ones by making modifications to the current state of a system in a restricted way. We propose a new framework of solution discovery via reconfiguration for constructing a feasible solution for a given problem by executing a sequence of small modifications starting from a given state or configuration. Our framework integrates and formalizes different aspects of classical local search, reoptimization, and combinatorial reconfiguration. We exemplify our framework on a multitude of fundamental combinatorial problems, namely Vertex Cover, Independent Set, Dominating Set, and Coloring. We study the classical as well as the parameterized complexity of the solution discovery variants of those problems and explore the boundary between tractable and intractable instances.
{"title":"On solution discovery via reconfiguration","authors":"Michael R. Fellows , Mario Grobler , Nicole Megow , Amer E. Mouawad , Vijayaragunathan Ramamoorthi , Frances A. Rosamond , Daniel Schmand , Sebastian Siebertz","doi":"10.1016/j.jcss.2025.103747","DOIUrl":"10.1016/j.jcss.2025.103747","url":null,"abstract":"<div><div>The dynamics of real-world applications and systems require efficient methods for improving infeasible solutions or restoring corrupted ones by making modifications to the current state of a system in a restricted way. We propose a new framework of <em>solution discovery via reconfiguration</em> for constructing a feasible solution for a given problem by executing a sequence of small modifications starting from a given state or configuration. Our framework integrates and formalizes different aspects of classical local search, reoptimization, and combinatorial reconfiguration. We exemplify our framework on a multitude of fundamental combinatorial problems, namely <span>Vertex Cover</span>, <span>Independent Set</span>, <span>Dominating Set</span>, and <span>Coloring</span>. We study the classical as well as the parameterized complexity of the solution discovery variants of those problems and explore the boundary between tractable and intractable instances.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"157 ","pages":"Article 103747"},"PeriodicalIF":0.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus and recovers the known tight bound for the planar case ().
{"title":"Exploration of graphs with excluded minors","authors":"Júlia Baligács, Yann Disser, Irene Heinrich, Pascal Schweitzer","doi":"10.1016/j.jcss.2025.103725","DOIUrl":"10.1016/j.jcss.2025.103725","url":null,"abstract":"<div><div>We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm <figure><img></figure> and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus <span><math><mi>g</mi><mo>≥</mo><mn>1</mn></math></span> and recovers the known tight bound for the planar case (<span><math><mi>g</mi><mo>=</mo><mn>0</mn></math></span>).</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103725"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-01DOI: 10.1016/j.jcss.2025.103720
Eva-Maria C. Hols , Stefan Kratsch, Astrid Pieterse
We extend the notion of lossy kernelization, introduced by Lokshtanov et al. (2017) [19], to approximate Turing kernelization. An α-approximate Turing kernelization for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs c-approximate solutions in time, computes an -approximate solution to the considered problem, using calls to the oracle of size at most for some function f that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth ℓ has a -approximate Turing kernelization with vertices, answering an open question posed by Lokshtanov et al. (2017) [19]. Furthermore, we give -approximate Turing kernelizations for the following graph problems parameterized by treewidth: Vertex Cover, Edge Clique Cover, Edge-Disjoint Triangle Packing, and Connected Vertex Cover. We generalize the result for Independent Set and Vertex Cover by showing that all graph problems that we will call friendly admit -approximate Turing kernelizations of polynomial size when parameterized by treewidth. We use this to establish approximate Turing kernelizations for Vertex-DisjointH-packing for connected graphs H, Clique Cover, Feedback Vertex Set, and Edge Dominating Set.
{"title":"Approximate Turing kernelization for problems parameterized by treewidth","authors":"Eva-Maria C. Hols , Stefan Kratsch, Astrid Pieterse","doi":"10.1016/j.jcss.2025.103720","DOIUrl":"10.1016/j.jcss.2025.103720","url":null,"abstract":"<div><div>We extend the notion of lossy kernelization, introduced by Lokshtanov et al. (2017) <span><span>[19]</span></span>, to approximate Turing kernelization. An <em>α</em>-approximate Turing kernelization for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs <em>c</em>-approximate solutions in <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> time, computes an <span><math><mi>α</mi><mo>⋅</mo><mi>c</mi></math></span>-approximate solution to the considered problem, using calls to the oracle of size at most <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> for some function <em>f</em> that only depends on the parameter. Using this definition, we show that <span>Independent Set</span> parameterized by treewidth <em>ℓ</em> has a <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximate Turing kernelization with <span><math><mi>O</mi><mo>(</mo><mfrac><mrow><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>ε</mi></mrow></mfrac><mo>)</mo></math></span> vertices, answering an open question posed by Lokshtanov et al. (2017) <span><span>[19]</span></span>. Furthermore, we give <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximate Turing kernelizations for the following graph problems parameterized by treewidth: <span>Vertex Cover</span>, <span>Edge Clique Cover</span>, <span>Edge-Disjoint Triangle Packing</span>, and <span>Connected Vertex Cover</span>. We generalize the result for <span>Independent Set</span> and <span>Vertex Cover</span> by showing that all graph problems that we will call <em>friendly</em> admit <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximate Turing kernelizations of polynomial size when parameterized by treewidth. We use this to establish approximate Turing kernelizations for <span>Vertex-Disjoint</span> <em>H</em><span>-packing</span> for connected graphs <em>H</em>, <span>Clique Cover</span>, <span>Feedback Vertex Set</span>, and <span>Edge Dominating Set</span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103720"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-31DOI: 10.1016/j.jcss.2025.103731
Hsu-Chun Yen , Di-De Yen
Finite transducers are finite automata with outputs. A transducer is finite-valued if the number of different outputs for any input string is bounded by a constant, and is single-valued if the constant is one. It is known that finite-valued one-way finite transducers enjoy a nice property that they can be decomposed into finitely many single-valued ones. In this paper, we develop an analytical technique for finite-valued 2-way finite transducers, capable of not only showing the decomposability result but also revealing the decomposition complexity. In particular, we show that every finite-valued two-way finite transducer can be effectively decomposed into a finite collection of single-valued two-way finite transducers. The number of such single-valued transducers is bounded by a tower of three exponentials in the size of the original transducer, while the size of each single-valued transducer is bounded by a tower of five exponentials. For special classes of 2-way transducers such as sweeping transducers and reversal-bounded transducers, lower decomposition complexity can be achieved by simplifying certain steps in the decomposition procedure. Finally, our decomposition analysis also allows us to derive complexity bounds for the equivalence problem for various classes of finite-valued 2-way finite transducers.
{"title":"Decomposing finite-valued two-way finite transducers","authors":"Hsu-Chun Yen , Di-De Yen","doi":"10.1016/j.jcss.2025.103731","DOIUrl":"10.1016/j.jcss.2025.103731","url":null,"abstract":"<div><div>Finite transducers are finite automata with outputs. A transducer is finite-valued if the number of different outputs for any input string is bounded by a constant, and is single-valued if the constant is one. It is known that finite-valued one-way finite transducers enjoy a nice property that they can be decomposed into finitely many single-valued ones. In this paper, we develop an analytical technique for finite-valued 2-way finite transducers, capable of not only showing the decomposability result but also revealing the decomposition complexity. In particular, we show that every finite-valued two-way finite transducer can be effectively decomposed into a finite collection of single-valued two-way finite transducers. The number of such single-valued transducers is bounded by a tower of three exponentials in the size of the original transducer, while the size of each single-valued transducer is bounded by a tower of five exponentials. For special classes of 2-way transducers such as sweeping transducers and reversal-bounded transducers, lower decomposition complexity can be achieved by simplifying certain steps in the decomposition procedure. Finally, our decomposition analysis also allows us to derive complexity bounds for the equivalence problem for various classes of finite-valued 2-way finite transducers.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103731"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145473538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-17DOI: 10.1016/j.jcss.2025.103714
Jan Bok , Jiří Fiala , Petr Hliněný , Nikola Jedličková , Jan Kratochvíl
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello, Fellows, and Stillwell asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs).
{"title":"Computational complexity of covering multigraphs with semi-edges: Small cases","authors":"Jan Bok , Jiří Fiala , Petr Hliněný , Nikola Jedličková , Jan Kratochvíl","doi":"10.1016/j.jcss.2025.103714","DOIUrl":"10.1016/j.jcss.2025.103714","url":null,"abstract":"<div><div>We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for <em>graphs with semi-edges</em>. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello, Fellows, and Stillwell asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding <em>k</em>-tuples of pairwise disjoint perfect matchings in regular graphs).</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103714"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-09-24DOI: 10.1016/j.jcss.2025.103716
Tesshu Hanaka , Hironori Kiya , Michael Lampis , Hirotaka Ono , Kanae Yoshiwatari
Arc Kayles and Colored Arc Kayles are generalized versions of well-studied combinatorial games Cram and Domineering, respectively. In Arc Kayles, two players alternately choose an edge to remove with its adjacent edges, and the player who cannot move is the loser. Colored Arc Kayles is similarly played on a graph with edges colored in black, white, or gray, in which the black (resp., white) player can choose only a gray or black (resp., white) edge. For Arc Kayles, the vertex cover number τ (i.e., the minimum size of a vertex cover) is an essential invariant because it is known that twice the vertex cover number upper bounds the number of turns of Arc Kayles, and for the winner determination of (Colored) Arc Kayles, -time algorithms are known, where n is the number of vertices. In this paper, we first give a polynomial kernel for Colored Arc Kayles parameterized by τ, which leads to a faster -time algorithm for Colored Arc Kayles. We then focus on Arc Kayles on trees, and propose a -time algorithm. Furthermore, we show that determining the winner of Arc Kayles on a tree can be done in time, which improves the best-known running time of . Finally, we show that Colored Arc Kayles is NP-hard, the first hardness result in the family of the above games.
{"title":"Faster winner determination algorithms for (Colored) Arc Kayles","authors":"Tesshu Hanaka , Hironori Kiya , Michael Lampis , Hirotaka Ono , Kanae Yoshiwatari","doi":"10.1016/j.jcss.2025.103716","DOIUrl":"10.1016/j.jcss.2025.103716","url":null,"abstract":"<div><div><span>Arc Kayles</span> and <span>Colored Arc Kayles</span> are generalized versions of well-studied combinatorial games <span>Cram</span> and <span>Domineering</span>, respectively. In <span>Arc Kayles</span>, two players alternately choose an edge to remove with its adjacent edges, and the player who cannot move is the loser. <span>Colored Arc Kayles</span> is similarly played on a graph with edges colored in black, white, or gray, in which the black (resp., white) player can choose only a gray or black (resp., white) edge. For <span>Arc Kayles</span>, the vertex cover number <em>τ</em> (i.e., the minimum size of a vertex cover) is an essential invariant because it is known that twice the vertex cover number upper bounds the number of turns of <span>Arc Kayles</span>, and for the winner determination of <span>(Colored) Arc Kayles</span>, <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>-time algorithms are known, where <em>n</em> is the number of vertices. In this paper, we first give a polynomial kernel for <span>Colored Arc Kayles</span> parameterized by <em>τ</em>, which leads to a faster <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>τ</mi><mi>log</mi><mo></mo><mi>τ</mi><mo>)</mo></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>-time algorithm for <span>Colored Arc Kayles</span>. We then focus on <span>Arc Kayles</span> on trees, and propose a <span><math><msup><mrow><mn>2.2361</mn></mrow><mrow><mi>τ</mi></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>-time algorithm. Furthermore, we show that determining the winner of <span>Arc Kayles</span> on a tree can be done in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>1.3831</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> time, which improves the best-known running time of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>1.4143</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. Finally, we show that <span>Colored Arc Kayles</span> is NP-hard, the first hardness result in the family of the above games.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103716"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-10-09DOI: 10.1016/j.jcss.2025.103722
Lars Jaffke , Laure Morelle , Ignasi Sau , Dimitrios M. Thilikos
We revisit a graph width parameter that we dub bipartite treewidth (btw). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one “bipartite” vertex, while the width of such decomposition measures the number of “non-bipartite” vertices in a bag. We provide para-NP-completeness results and develop dynamic programming techniques to solve problems on graphs of small btw. In particular, we show that -Subgraph-Cover, Weighted Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are parameterized by btw. We also provide the following dichotomy when H is a 2-connected graph: if H is bipartite, then H-{Subgraph/Induced-Subgraph/Odd-Minor/Scattered}-Packing is para-NP-complete parameterized by btw while, if H is non-bipartite, then the problem is solvable in XP-time.
{"title":"Dynamic programming on bipartite tree decompositions","authors":"Lars Jaffke , Laure Morelle , Ignasi Sau , Dimitrios M. Thilikos","doi":"10.1016/j.jcss.2025.103722","DOIUrl":"10.1016/j.jcss.2025.103722","url":null,"abstract":"<div><div>We revisit a graph width parameter that we dub <em>bipartite treewidth</em> (<span>btw</span>). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a <em>bipartite tree decomposition</em> is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one “bipartite” vertex, while the width of such decomposition measures the number of “non-bipartite” vertices in a bag. We provide <span>para-NP</span>-completeness results and develop dynamic programming techniques to solve problems on graphs of small <span>btw</span>. In particular, we show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span>-Subgraph-Cover</span>, <span>Weighted Independent Set</span>, <span>Odd Cycle Transversal</span>, and <span>Maximum Weighted Cut</span> are <span><math><mtext>FPT</mtext></math></span> parameterized by <span>btw</span>. We also provide the following dichotomy when <em>H</em> is a 2-connected graph: if <em>H</em> is bipartite, then <em>H</em><span>-{Subgraph/Induced-Subgraph/Odd-Minor/Scattered}-Packing</span> is <span>para-NP</span>-complete parameterized by <span>btw</span> while, if <em>H</em> is non-bipartite, then the problem is solvable in <span>XP</span>-time.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103722"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-03-01Epub Date: 2025-09-24DOI: 10.1016/j.jcss.2025.103718
Karen Frilya Celine , Ziyuan Gao , Sanjay Jain , Ryan Lou , Frank Stephan , Guohua Wu
This paper studies the recursion- and automata-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We first investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings α and β such that α is strictly quasi-isometrically reducible to β, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka. Furthermore, we also study automatic quasi-isometric reductions between automatic structures, and show that automatic quasi-isometry may be separable from general quasi-isometry depending on the growth of the automatic domain.
{"title":"Quasi-isometric reductions between infinite strings","authors":"Karen Frilya Celine , Ziyuan Gao , Sanjay Jain , Ryan Lou , Frank Stephan , Guohua Wu","doi":"10.1016/j.jcss.2025.103718","DOIUrl":"10.1016/j.jcss.2025.103718","url":null,"abstract":"<div><div>This paper studies the recursion- and automata-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We first investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings <em>α</em> and <em>β</em> such that <em>α</em> is strictly quasi-isometrically reducible to <em>β</em>, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka. Furthermore, we also study automatic quasi-isometric reductions between automatic structures, and show that automatic quasi-isometry may be separable from general quasi-isometry depending on the growth of the automatic domain.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103718"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A connected graph has a -cover if each of its edges is contained in at least ℓ cliques of order k. Motivated by recent advances in extremal combinatorics and the literature on edge modification problems, we study the algorithmic version of the -cover problem. Given a connected graph G, the -cover problem is to identify the smallest subset of non-edges of G such that their addition to G results in a graph with a -cover. For every constant , we show that the -cover problem is -complete for general graphs. Moreover, we show that for every constant , the -cover problem admits no polynomial-time constant-factor approximation algorithm unless . However, we show that the -cover problem can be solved in polynomial time when the input graph is chordal. For the class of trees and general values of k, we show that the -cover problem is -hard even for spiders. However, we show that for every , the -cover and the -cover problems are constant-factor approximable when the input graph is a tree.
{"title":"Algorithms and hardness results for the (k,ℓ)-cover problem","authors":"Amirali Madani , Anil Maheshwari , Babak Miraftab , Bodhayan Roy","doi":"10.1016/j.jcss.2025.103727","DOIUrl":"10.1016/j.jcss.2025.103727","url":null,"abstract":"<div><div>A connected graph has a <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>-cover if each of its edges is contained in at least <em>ℓ</em> cliques of order <em>k</em>. Motivated by recent advances in extremal combinatorics and the literature on edge modification problems, we study the algorithmic version of the <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>-cover problem. Given a connected graph <em>G</em>, the <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>-cover problem is to identify the smallest subset of non-edges of <em>G</em> such that their addition to <em>G</em> results in a graph with a <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>-cover. For every constant <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, we show that the <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-cover problem is <span><math><mi>NP</mi></math></span>-complete for general graphs. Moreover, we show that for every constant <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, the <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-cover problem admits no polynomial-time constant-factor approximation algorithm unless <span><math><mi>P</mi><mo>=</mo><mrow><mi>NP</mi></mrow></math></span>. However, we show that the <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-cover problem can be solved in polynomial time when the input graph is chordal. For the class of trees and general values of <em>k</em>, we show that the <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-cover problem is <span><math><mi>NP</mi></math></span>-hard even for spiders. However, we show that for every <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>, the <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>-cover and the <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-cover problems are constant-factor approximable when the input graph is a tree.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103727"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145473540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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