Pub 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":"2025-11-03","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 : 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":"2025-10-31","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 : 2025-10-30DOI: 10.1016/j.jcss.2025.103730
Bart M.P. Jansen , Ruben F.A. Verhaegh
For an optimization problem Π on graphs whose solutions are vertex sets, a vertex v is called c-essential for Π if all solutions of size at most contain v. Recent work showed that polynomial-time algorithms to detect c-essential vertices can be used to reduce the search-space of fixed-parameter tractable algorithms solving such problems parameterized by the size k of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected n-vertex graph G in time , where ℓ is the number of vertices in an optimal solution that are not 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of c, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that ()-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.
{"title":"Search-space reduction via essential vertices revisited: Vertex multicut and cograph deletion","authors":"Bart M.P. Jansen , Ruben F.A. Verhaegh","doi":"10.1016/j.jcss.2025.103730","DOIUrl":"10.1016/j.jcss.2025.103730","url":null,"abstract":"<div><div>For an optimization problem Π on graphs whose solutions are vertex sets, a vertex <em>v</em> is called <em>c-essential</em> for Π if all solutions of size at most <figure><img></figure> contain <em>v</em>. Recent work showed that polynomial-time algorithms to detect <em>c</em>-essential vertices can be used to reduce the search-space of fixed-parameter tractable algorithms solving such problems parameterized by the size <em>k</em> of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3<span>-Essential detection for Vertex Multicut</span>, which translates into an algorithm that finds a minimum multicut of an undirected <em>n</em>-vertex graph <em>G</em> in time <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></msup><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, where <em>ℓ</em> is the number of vertices in an optimal solution that are <em>not</em> 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of <em>c</em>, the detection task becomes NP-hard assuming the <em>Unique Games Conjecture</em>. For example, we show that <span>(</span><span><math><mn>2</mn><mo>−</mo><mi>ε</mi></math></span><span>)-Essential detection for Directed Feedback Vertex Set</span> is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103730"},"PeriodicalIF":0.9,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145473537","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 : 2025-10-30DOI: 10.1016/j.jcss.2025.103729
Maximilien Gadouleau , Loïc Paulevé , Sara Riva
Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework, and we propose novel update modes, the history-based, trapping, and subcube-based modes. Building on the unified definitions, we provide a comprehensive comparison of memory-based update modes, resulting in their hierarchy by simulation and weak simulation. Finally, we highlight consequences of introducing memory on the notions of trajectory and attractors.
{"title":"Bringing memory to Boolean networks: A unifying framework","authors":"Maximilien Gadouleau , Loïc Paulevé , Sara Riva","doi":"10.1016/j.jcss.2025.103729","DOIUrl":"10.1016/j.jcss.2025.103729","url":null,"abstract":"<div><div>Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework, and we propose novel update modes, the history-based, trapping, and subcube-based modes. Building on the unified definitions, we provide a comprehensive comparison of memory-based update modes, resulting in their hierarchy by simulation and weak simulation. Finally, we highlight consequences of introducing memory on the notions of trajectory and attractors.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103729"},"PeriodicalIF":0.9,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145473539","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 : 2025-10-29DOI: 10.1016/j.jcss.2025.103726
Zohair Raza Hassan, Edith Hemaspaandra, Stanisław Radziszowski
For fixed nonnegative integers k and ℓ, the -Arrowing problem asks whether a given graph, G, has a red/blue coloring of such that there are no red copies of and no blue copies of . The problem is trivial when , but has been shown to be coNP-complete when . In this work, we show that the problem remains coNP-complete for all pairs of k and ℓ, except , and when . We define and prove the existence of special graphs that we call “transmitters.” Using transmitters, we construct gadgets for three distinct cases: 1) and , 2) , and 3) . For -Arrowing we show a polynomial-time algorithm by reducing the problem to 2SAT, thus successfully categorizing the complexity of all -Arrowing problems.
{"title":"The complexity of (Pk,Pℓ)-arrowing","authors":"Zohair Raza Hassan, Edith Hemaspaandra, Stanisław Radziszowski","doi":"10.1016/j.jcss.2025.103726","DOIUrl":"10.1016/j.jcss.2025.103726","url":null,"abstract":"<div><div>For fixed nonnegative integers <em>k</em> and <em>ℓ</em>, the <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span>-Arrowing problem asks whether a given graph, <em>G</em>, has a red/blue coloring of <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that there are no red copies of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and no blue copies of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>. The problem is trivial when <span><math><mi>max</mi><mo></mo><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo><mo>≤</mo><mn>3</mn></math></span>, but has been shown to be coNP-complete when <span><math><mi>k</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>4</mn></math></span>. In this work, we show that the problem remains coNP-complete for all pairs of <em>k</em> and <em>ℓ</em>, except <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span>, and when <span><math><mi>max</mi><mo></mo><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo><mo>≤</mo><mn>3</mn></math></span>. We define and prove the existence of special graphs that we call “transmitters.” Using transmitters, we construct gadgets for three distinct cases: 1) <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>ℓ</mi><mo>≥</mo><mn>5</mn></math></span>, 2) <span><math><mi>ℓ</mi><mo>></mo><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>, and 3) <span><math><mi>ℓ</mi><mo>=</mo><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>. For <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span>-Arrowing we show a polynomial-time algorithm by reducing the problem to 2SAT, thus successfully categorizing the complexity of all <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span>-Arrowing problems.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103726"},"PeriodicalIF":0.9,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145424690","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 : 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":"2025-10-17","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}
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":"2025-10-10","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 : 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":"2025-10-09","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 : 2025-10-09DOI: 10.1016/j.jcss.2025.103724
R. Krithika , V.K. Kutty Malu , Prafullkumar Tale
The Path Contraction and Cycle Contraction problems take as input an undirected graph G with n vertices, m edges and an integer k and determine whether one can obtain a path or a cycle, respectively, by performing at most k edge contractions in G. We revisit these NP-complete problems and prove the following results.
•
Path Contraction admits an -time algorithm. This improves over the current fastest parameterized algorithm known for the problem (Heggernes et al. (2014) [15]).
•
Cycle Contraction admits an -time algorithm where and decreases as increases.
Central to these results is an algorithm for a general variant of Path Contraction, namely Path Contraction With Constrained Ends. We also give an -time algorithm to solve the optimization version of Cycle Contraction. Next, we turn our attention to restricted graph classes and show the following results.
•
Path Contraction on planar graphs admits a polynomial-time algorithm.
•
Path Contraction on chordal graphs does not admit an -time algorithm for any , under the Orthogonal Vectors Conjecture. Here, tw is the treewidth of the input graph.
The second result complements the -time, i.e., -
路径收缩和循环收缩问题以无向图G为输入，该无向图G有n个顶点，m条边和整数k，并确定是否可以分别通过在G中执行最多k次边收缩来获得路径或循环。我们重新讨论这些np完全问题并证明了以下结果。•路径收缩允许一个O （2k）时间算法。这比目前已知的最快的参数化算法有所改进（Heggernes et al.(2014)[15]）。•循环收缩允许一个O （(2+ λ)k）时间算法，其中0<； λ≤0.5509，且λ随着λ =n−k的增加而减小。这些结果的核心是路径收缩的一般变体的算法，即带约束端点的路径收缩。我们还给出了一个O （2.5191n）时间的算法来解决循环收缩的优化版本。接下来，我们将注意力转向受限制的图类，并显示以下结果。•平面图上的路径收缩允许一个多项式时间算法。•对于任何ϵ>；0，在正交向量猜想下，弦图上的路径收缩不允许O（n2−λ·2o(tw)）时间算法。这里，tw是输入图的树宽。第二个结果补充了O（nm）时间，即O（n2⋅tw）时间，已知的问题算法（Heggernes et al.(2014)[16]）。
{"title":"Revisiting path contraction and cycle contraction","authors":"R. Krithika , V.K. Kutty Malu , Prafullkumar Tale","doi":"10.1016/j.jcss.2025.103724","DOIUrl":"10.1016/j.jcss.2025.103724","url":null,"abstract":"<div><div>The <span>Path Contraction</span> and <span>Cycle Contraction</span> problems take as input an undirected graph <em>G</em> with <em>n</em> vertices, <em>m</em> edges and an integer <em>k</em> and determine whether one can obtain a path or a cycle, respectively, by performing at most <em>k</em> edge contractions in <em>G</em>. We revisit these <span>NP</span>-complete problems and prove the following results.<ul><li><span>•</span><span><div><span>Path Contraction</span> admits an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time algorithm. This improves over the current fastest parameterized algorithm known for the problem (Heggernes et al. (2014) <span><span>[15]</span></span>).</div></span></li><li><span>•</span><span><div><span>Cycle Contraction</span> admits an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mo>(</mo><mn>2</mn><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time algorithm where <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>≤</mo><mn>0.5509</mn></math></span> and <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> decreases as <span><math><mi>ℓ</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span> increases.</div></span></li></ul> Central to these results is an algorithm for a general variant of <span>Path Contraction</span>, namely <span>Path Contraction With Constrained Ends</span>. We also give an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>2.5191</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>-time algorithm to solve the optimization version of <span>Cycle Contraction</span>. Next, we turn our attention to restricted graph classes and show the following results.<ul><li><span>•</span><span><div><span>Path Contraction</span> on planar graphs admits a polynomial-time algorithm.</div></span></li><li><span>•</span><span><div><span>Path Contraction</span> on chordal graphs does not admit an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mo>(</mo><mi>t</mi><mi>w</mi><mo>)</mo></mrow></msup><mo>)</mo></math></span>-time algorithm for any <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span>, under the <span>Orthogonal Vectors Conjecture</span>. Here, <em>tw</em> is the treewidth of the input graph.</div></span></li></ul> The second result complements the <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>m</mi><mo>)</mo></math></span>-time, i.e., <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mi>t</mi><mi>w</mi><mo>)</mo></math></span>-","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103724"},"PeriodicalIF":0.9,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333259","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 : 2025-10-08DOI: 10.1016/j.jcss.2025.103723
Hoang-Oanh Le , Van Bang Le
In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. matching cut (mc), respectively, perfect matching cut (pmc), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect matching cut. The disconnected perfect matching problem (dpm) is to decide if a graph has a perfect matching that contains a matching cut. Solving an open problem posed in [Lucke, Paulusma, Ries (ISAAC 2022, Algorithmica 2023)], we show that pmc is -complete in graphs without induced 14-vertex path . Our reduction also works simultaneously for mc and dpm, improving the previous hardness results of mc on -free graphs and of dpm on -free graphs to -free graphs for both problems. Actually, we prove a slightly stronger result: within -free 8-chordal graphs (graphs without chordless cycles of length at least 9), it is hard to distinguish between those without matching cuts (respectively, perfect matching cuts, disconnected perfect matchings) and those in which every matching cut is a perfect matching cut. Moreover, assuming the Exponential Time Hypothesis, none of these problems can be solved in time for n-vertex -free 8-chordal graphs.
On the positive side, we show that, as for mc [Moshi (JGT 1989)], dpm and pmc are polynomially solvable when restricted to 4-chordal graphs. Together with the negative results, this partly answers an open question on the complexity of pmc in k-chordal graphs asked in [Le, Telle (WG 2021, TCS 2022) & Lucke, Paulusma, Ries (MFCS 2023, TCS 2024)].
{"title":"Complexity and algorithms for matching cut problems in graphs without long induced paths and cycles","authors":"Hoang-Oanh Le , Van Bang Le","doi":"10.1016/j.jcss.2025.103723","DOIUrl":"10.1016/j.jcss.2025.103723","url":null,"abstract":"<div><div>In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. <span>matching cut</span> (<span>mc</span>), respectively, <span>perfect matching cut</span> (<span>pmc</span>), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect matching cut. The <span>disconnected perfect matching</span> problem (<span>dpm</span>) is to decide if a graph has a perfect matching that contains a matching cut. Solving an open problem posed in [Lucke, Paulusma, Ries (ISAAC 2022, Algorithmica 2023)], we show that <span>pmc</span> is <span><math><mi>NP</mi></math></span>-complete in graphs without induced 14-vertex path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>14</mn></mrow></msub></math></span>. Our reduction also works simultaneously for <span>mc</span> and <span>dpm</span>, improving the previous hardness results of <span>mc</span> on <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>15</mn></mrow></msub></math></span>-free graphs and of <span>dpm</span> on <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>19</mn></mrow></msub></math></span>-free graphs to <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>14</mn></mrow></msub></math></span>-free graphs for both problems. Actually, we prove a slightly stronger result: within <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>14</mn></mrow></msub></math></span>-free 8-chordal graphs (graphs without chordless cycles of length at least 9), it is hard to distinguish between those without matching cuts (respectively, perfect matching cuts, disconnected perfect matchings) and those in which every matching cut is a perfect matching cut. Moreover, assuming the Exponential Time Hypothesis, none of these problems can be solved in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> time for <em>n</em>-vertex <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>14</mn></mrow></msub></math></span>-free 8-chordal graphs.</div><div>On the positive side, we show that, as for <span>mc</span> [Moshi (JGT 1989)], <span>dpm</span> and <span>pmc</span> are polynomially solvable when restricted to 4-chordal graphs. Together with the negative results, this partly answers an open question on the complexity of <span>pmc</span> in <em>k</em>-chordal graphs asked in [Le, Telle (WG 2021, TCS 2022) & Lucke, Paulusma, Ries (MFCS 2023, TCS 2024)].</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103723"},"PeriodicalIF":0.9,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333256","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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