Pub Date : 2026-03-01Epub 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":"2026-03-01","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}
Pub Date : 2026-03-01Epub Date: 2025-10-06DOI: 10.1016/j.jcss.2025.103719
Ivo F.D. Oliveira , Ricardo H.C. Takahashi
We find a searching method on ordered lists that surprisingly outperforms binary searching with respect to average query complexity while retaining minmax optimality. The method is shown to require queries on average while never exceeding queries in the worst case, i.e. the minmax bound of binary searching. Our average results assume a uniform distribution hypothesis similar to those of previous authors under which the expected query complexity of interpolation search of is known to be optimal. Hence our method turns out to be optimal with respect to both minmax and average performance. We further provide robustness guarantees and perform several numerical experiments with both artificial and real data. Our results suggest that time savings range roughly from a constant factor of 10% to 50% to a logarithmic factor spanning orders of magnitude when different metrics are considered.
{"title":"Minmax optimal list searching with log2log2n average cost","authors":"Ivo F.D. Oliveira , Ricardo H.C. Takahashi","doi":"10.1016/j.jcss.2025.103719","DOIUrl":"10.1016/j.jcss.2025.103719","url":null,"abstract":"<div><div>We find a searching method on ordered lists that surprisingly outperforms binary searching with respect to average query complexity while retaining minmax optimality. The method is shown to require <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mi>n</mi><mo>)</mo></math></span> queries on average while never exceeding <span><math><mo>⌈</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mi>n</mi><mo>⌉</mo></math></span> queries in the worst case, i.e. the minmax bound of binary searching. Our average results assume a uniform distribution hypothesis similar to those of previous authors under which the expected query complexity of interpolation search of <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mi>n</mi><mo>)</mo></math></span> is known to be optimal. Hence our method turns out to be optimal with respect to both minmax and average performance. We further provide robustness guarantees and perform several numerical experiments with both artificial and real data. Our results suggest that time savings range roughly from a constant factor of 10% to 50% to a logarithmic factor spanning orders of magnitude when different metrics are considered.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103719"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333257","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: 2025-10-29DOI: 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":"2026-03-01","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}
We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and—in addition to a global budget limit, there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to be hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.
{"title":"Participatory budgeting with project groups","authors":"Pallavi Jain , Krzysztof Sornat , Nimrod Talmon , Meirav Zehavi","doi":"10.1016/j.jcss.2025.103702","DOIUrl":"10.1016/j.jcss.2025.103702","url":null,"abstract":"<div><div>We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and—in addition to a global budget limit, there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to be hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103702"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060902","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: 2025-09-23DOI: 10.1016/j.jcss.2025.103715
Juhi Chaudhary , Harmender Gahlawat , Michal Wlodarczyk , Meirav Zehavi
Given an undirected graph G and a multiset of k terminal pairs , the Vertex-Disjoint Paths () and Edge-Disjoint Paths () problems ask whether G has k pairwise internally vertex-disjoint paths and k pairwise edge-disjoint paths, respectively, connecting every terminal pair in . In this paper, we study the kernelization complexity of and on subclasses of chordal graphs. For , we design a 4k vertex kernel on split graphs and an vertex kernel on well-partitioned chordal graphs. We also show that the problem becomes polynomial-time solvable on threshold graphs. For EDP, we first prove that the problem is -complete on complete graphs. Then, we design an vertex kernel for on split graphs, and improve it to a vertex kernel on threshold graphs. Lastly, we provide an vertex kernel for on block graphs and a vertex kernel for clique paths. Our contributions improve upon several results in the literature, as well as resolve an open question by Heggernes et al. (2015) [27].
{"title":"Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs","authors":"Juhi Chaudhary , Harmender Gahlawat , Michal Wlodarczyk , Meirav Zehavi","doi":"10.1016/j.jcss.2025.103715","DOIUrl":"10.1016/j.jcss.2025.103715","url":null,"abstract":"<div><div>Given an undirected graph <em>G</em> and a multiset of <em>k</em> terminal pairs <span><math><mi>X</mi></math></span>, the <span>Vertex-Disjoint Paths</span> (<figure><img></figure>) and <span>Edge-Disjoint Paths</span> (<figure><img></figure>) problems ask whether <em>G</em> has <em>k</em> pairwise internally vertex-disjoint paths and <em>k</em> pairwise edge-disjoint paths, respectively, connecting every terminal pair in <span><math><mi>X</mi></math></span>. In this paper, we study the kernelization complexity of <figure><img></figure> and <figure><img></figure> on subclasses of chordal graphs. For <figure><img></figure>, we design a 4<em>k</em> vertex kernel on split graphs and an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> vertex kernel on well-partitioned chordal graphs. We also show that the problem becomes polynomial-time solvable on threshold graphs. For <span>EDP</span>, we first prove that the problem is <span><math><mi>NP</mi></math></span>-complete on complete graphs. Then, we design an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2.75</mn></mrow></msup><mo>)</mo></math></span> vertex kernel for <figure><img></figure> on split graphs, and improve it to a <span><math><mn>7</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span> vertex kernel on threshold graphs. Lastly, we provide an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> vertex kernel for <figure><img></figure> on block graphs and a <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span> vertex kernel for clique paths. Our contributions improve upon several results in the literature, as well as resolve an open question by Heggernes et al. (2015) <span><span>[27]</span></span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103715"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227417","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: 2025-09-03DOI: 10.1016/j.jcss.2025.103704
Mingyang Gong , Zhi-Zhong Chen , Guohui Lin , Lusheng Wang
This paper studies , which is to cover as many vertices as possible in a given graph by vertex-disjoint -paths (i.e., paths each with at least five vertices). is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of and runs in time. In this paper, we present a new approximation algorithm for which achieves a ratio of 2.511 and runs in time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion.
{"title":"Approximately covering vertices by order-5 or longer paths","authors":"Mingyang Gong , Zhi-Zhong Chen , Guohui Lin , Lusheng Wang","doi":"10.1016/j.jcss.2025.103704","DOIUrl":"10.1016/j.jcss.2025.103704","url":null,"abstract":"<div><div>This paper studies <span><math><mi>M</mi><mi>P</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>5</mn><mo>+</mo></mrow></msubsup></math></span>, which is to cover as many vertices as possible in a given graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> by vertex-disjoint <span><math><msup><mrow><mn>5</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span>-paths (i.e., paths each with at least five vertices). <span><math><mi>M</mi><mi>P</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>5</mn><mo>+</mo></mrow></msubsup></math></span> is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of <span><math><mfrac><mrow><mn>19</mn></mrow><mrow><mn>7</mn></mrow></mfrac><mo>≈</mo><mn>2.714</mn></math></span> and runs in <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span> time. In this paper, we present a new approximation algorithm for <span><math><mi>M</mi><mi>P</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>5</mn><mo>+</mo></mrow></msubsup></math></span> which achieves a ratio of 2.511 and runs in <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2.5</mn></mrow></msup><mo>|</mo><mi>E</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103704"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997693","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.2025.103724
R. Krithika , V.K. Kutty Malu , Prafullkumar Tale
<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>-
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":"2026-03-01","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 : 2026-03-01Epub Date: 2025-09-26DOI: 10.1016/j.jcss.2025.103717
Florian Bridoux, Aymeric Picard Marchetto, Adrien Richard
An automata network with n components over a finite alphabet Q of size q is a discrete dynamical system described by the successive iterations of a function . In most applications, the main parameter is the interaction graph of f: the digraph with vertex set that contains an arc from j to i if depends on input j. What can be said on the set of the interaction graphs of the automata networks isomorphic to f? It seems that this simple question has never been studied. In a previous paper, we prove that the complete digraph , with arcs, is universal in that whenever f is not constant nor the identity (and ). In this paper, taking the opposite direction, we prove that there exist universal automata networks f, in that contains all the digraphs on , excepted the empty one. Actually, we prove that the presence of only three specific digraphs in implies the universality of f, and we prove that this forces the alphabet size q to have at least n prime factors (with multiplicity). However, we prove that for any fixed , there exists almost universal functions, that is, functions such that the probability that a random digraph belongs to tends to 1 as . We do not know if this holds in the binary case , providing only partial results.
{"title":"Interaction graphs of isomorphic automata networks II: Universal dynamics","authors":"Florian Bridoux, Aymeric Picard Marchetto, Adrien Richard","doi":"10.1016/j.jcss.2025.103717","DOIUrl":"10.1016/j.jcss.2025.103717","url":null,"abstract":"<div><div>An automata network with <em>n</em> components over a finite alphabet <em>Q</em> of size <em>q</em> is a discrete dynamical system described by the successive iterations of a function <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. In most applications, the main parameter is the interaction graph of <em>f</em>: the digraph with vertex set <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> that contains an arc from <em>j</em> to <em>i</em> if <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> depends on input <em>j</em>. What can be said on the set <span><math><mi>G</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of the interaction graphs of the automata networks isomorphic to <em>f</em>? It seems that this simple question has never been studied. In a previous paper, we prove that the complete digraph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, with <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> arcs, is universal in that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi>G</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> whenever <em>f</em> is not constant nor the identity (and <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>). In this paper, taking the opposite direction, we prove that there exist universal automata networks <em>f</em>, in that <span><math><mi>G</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> contains all the digraphs on <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, excepted the empty one. Actually, we prove that the presence of only three specific digraphs in <span><math><mi>G</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> implies the universality of <em>f</em>, and we prove that this forces the alphabet size <em>q</em> to have at least <em>n</em> prime factors (with multiplicity). However, we prove that for any fixed <span><math><mi>q</mi><mo>≥</mo><mn>3</mn></math></span>, there exists almost universal functions, that is, functions <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that the probability that a random digraph belongs to <span><math><mi>G</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> tends to 1 as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. We do not know if this holds in the binary case <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>, providing only partial results.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103717"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227408","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: 2025-10-03DOI: 10.1016/j.jcss.2025.103721
Ismaël Jecker , Nicolas Mazzocchi , Petra Wolf
A deterministic finite automaton, (DFA) is composite if its language can be expressed as the intersection of languages from smaller DFAs; otherwise, it is prime. This concept, introduced by Kupferman and Mosheiff in 2013, remains computationally challenging, with a doubly-exponential gap between the upper and lower bounds. This work focuses on permutation DFAs. We present an NP algorithm to decide compositionality and show that the difficulty stems from the number of non-accepting states. A fixed-parameter tractable algorithm is provided, using the count of rejecting states as the parameter. We further explore commutative permutation DFAs, whose structure enables decision procedures in NL and even LOGSPACE when the alphabet size is fixed. Despite this low complexity, intricate behavior persists: we provide a family of composite DFAs requiring polynomially many factors relative to their size. Additionally, we examine the k-factor composite variant — whether a DFA can be decomposed into k smaller DFAs. For commutative permutation DFAs, limiting the number of factors increases complexity, making the problem NP-complete. More generally, determining k-factor compositionality lies in PSPACE, and in LOGSPACE for DFAs over a singleton alphabet.
{"title":"Decomposing permutation automata","authors":"Ismaël Jecker , Nicolas Mazzocchi , Petra Wolf","doi":"10.1016/j.jcss.2025.103721","DOIUrl":"10.1016/j.jcss.2025.103721","url":null,"abstract":"<div><div>A deterministic finite automaton, (<span>DFA</span>) is composite if its language can be expressed as the intersection of languages from smaller <span>DFAs</span>; otherwise, it is prime. This concept, introduced by Kupferman and Mosheiff in 2013, remains computationally challenging, with a doubly-exponential gap between the upper and lower bounds. This work focuses on permutation <span>DFAs</span>. We present an <span>NP</span> algorithm to decide compositionality and show that the difficulty stems from the number of non-accepting states. A fixed-parameter tractable algorithm is provided, using the count of rejecting states as the parameter. We further explore commutative permutation <span>DFAs</span>, whose structure enables decision procedures in NL and even <span>LOGSPACE</span> when the alphabet size is fixed. Despite this low complexity, intricate behavior persists: we provide a family of composite <span>DFAs</span> requiring polynomially many factors relative to their size. Additionally, we examine the <em>k</em>-factor composite variant — whether a <span>DFA</span> can be decomposed into <em>k</em> smaller <span>DFAs</span>. For commutative permutation <span>DFAs</span>, limiting the number of factors increases complexity, making the problem NP-complete. More generally, determining <em>k</em>-factor compositionality lies in <span>PSPACE</span>, and in <span>LOGSPACE</span> for <span>DFAs</span> over a singleton alphabet.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"156 ","pages":"Article 103721"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333249","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: 2025-10-30DOI: 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":"2026-03-01","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}
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