Pub Date : 2026-08-01Epub Date: 2026-02-05DOI: 10.1016/j.jcss.2026.103780
Konrad K. Dabrowski , Peter Jonsson , Sebastian Ordyniak , George Osipov
Difference Logic (DL) is a fragment of linear arithmetic where atoms are constraints for variables (ranging over or ) and integer k. We study the complexity of deciding the truth of existential DL sentences. This problem appears in many contexts: examples include verification, bioinformatics, telecommunications, and spatio-temporal reasoning in AI. We begin by considering sentences in CNF with rational-valued variables. We restrict the allowed clauses via two natural parameters: arity and coefficient bounds. The problem is NP-hard for most choices of these parameters. As a response to this, we refine our understanding by analysing the time complexity and the parameterized complexity (with respect to well-studied parameters such as primal and incidence treewidth). We obtain a comprehensive picture of the complexity landscape in both cases. Finally, we generalise our results to integer domains and sentences that are not in CNF.
{"title":"Algorithms and complexity of difference logic","authors":"Konrad K. Dabrowski , Peter Jonsson , Sebastian Ordyniak , George Osipov","doi":"10.1016/j.jcss.2026.103780","DOIUrl":"10.1016/j.jcss.2026.103780","url":null,"abstract":"<div><div><em>Difference Logic</em> (DL) is a fragment of linear arithmetic where atoms are constraints <span><math><mi>x</mi><mo>+</mo><mi>k</mi><mo>≤</mo><mi>y</mi></math></span> for variables <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> (ranging over <span><math><mi>Q</mi></math></span> or <span><math><mi>Z</mi></math></span>) and integer <em>k</em>. We study the complexity of deciding the truth of existential DL sentences. This problem appears in many contexts: examples include verification, bioinformatics, telecommunications, and spatio-temporal reasoning in AI. We begin by considering sentences in CNF with rational-valued variables. We restrict the allowed clauses via two natural parameters: <em>arity</em> and <em>coefficient bounds</em>. The problem is <span>NP</span>-hard for most choices of these parameters. As a response to this, we refine our understanding by analysing the time complexity and the parameterized complexity (with respect to well-studied parameters such as primal and incidence treewidth). We obtain a comprehensive picture of the complexity landscape in both cases. Finally, we generalise our results to integer domains and sentences that are not in CNF.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"159 ","pages":"Article 103780"},"PeriodicalIF":0.9,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154235","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 : 2026-08-01Epub Date: 2026-02-03DOI: 10.1016/j.jcss.2026.103782
Jungho Ahn , Jinha Kim , O-joung Kwon
Let be a family of graphs and be an integer. For a graph G and an integer k, -Covering asks whether there is a set of size at most k such that every induced subgraph of G isomorphic to a graph in is at distance at most r from D. -Packing asks whether G has k induced subgraphs such that each is isomorphic to a graph in and the distance between distinct and in G is more than r. We show that for every fixed nonempty finite family of connected graphs and , -Covering and -Packing admit almost linear kernels on every nowhere dense class of graphs, parameterized by the solution size k. As corollaries, we prove that Distance-rVertex Cover, Distance-rMatching, -Free Vertex Deletion, and Induced--Packing for any fixed finite family of connected graphs admit almost linear kernels on every nowhere dense class of graphs. Our results extend the results for Distance-rDominating Set by Drange et al. (2016) [17] and Eickmeyer et al. (2017) [20] and for Distance-rIndependent Set by Pilipczuk and Siebertz (2021) [41].
设F为图族，r≥0为整数。图G和整数k (r、F)覆盖询问是否有一组D⊆最多k大小的V (G),这样每一个G的诱导子图同构图形在距离最多r F D . (r, F)包装要求是否G k诱导子图H1,…,香港,这样每个嗨同构的图F和之间的距离不同V(嗨)和V (Hj) G比r。我们表明,对于每一个固定的连接图和非空的有限的家庭F r≥0,（r，F）-覆盖和（r，F）-填充在每一个无处密集的图类上都承认几乎线性核，用解大小k参数化。作为推论，我们证明了对于任何固定有限族的连通图F，距离-r顶点覆盖、距离-r匹配、F-自由顶点删除和诱导-F-填充在每一个无处密集的图类上都承认几乎线性核。我们的结果扩展了Distance-r支配集(Drange et al.(2016)[17]和Eickmeyer et al.(2017)[20]以及Pilipczuk和Siebertz(2021)[41]的Distance-r独立集的结果。
{"title":"Unified almost linear kernels for generalized covering and packing problems on nowhere dense classes","authors":"Jungho Ahn , Jinha Kim , O-joung Kwon","doi":"10.1016/j.jcss.2026.103782","DOIUrl":"10.1016/j.jcss.2026.103782","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a family of graphs and <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span> be an integer. For a graph <em>G</em> and an integer <em>k</em>, <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span><span>-Covering</span> asks whether there is a set <span><math><mi>D</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of size at most <em>k</em> such that every induced subgraph of <em>G</em> isomorphic to a graph in <span><math><mi>F</mi></math></span> is at distance at most <em>r</em> from <em>D</em>. <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span><span>-Packing</span> asks whether <em>G</em> has <em>k</em> induced subgraphs <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that each <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is isomorphic to a graph in <span><math><mi>F</mi></math></span> and the distance between distinct <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span> in <em>G</em> is more than <em>r</em>. We show that for every fixed nonempty finite family <span><math><mi>F</mi></math></span> of connected graphs and <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span><span>-Covering</span> and <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span><span>-Packing</span> admit almost linear kernels on every nowhere dense class of graphs, parameterized by the solution size <em>k</em>. As corollaries, we prove that <span>Distance-</span><em>r</em> <span>Vertex Cover</span>, <span>Distance-</span><em>r</em> <span>Matching</span>, <span><math><mi>F</mi></math></span><span>-Free Vertex Deletion</span>, and <span>Induced-</span><span><math><mi>F</mi></math></span><span>-Packing</span> for any fixed finite family <span><math><mi>F</mi></math></span> of connected graphs admit almost linear kernels on every nowhere dense class of graphs. Our results extend the results for <span>Distance-</span><em>r</em> <span>Dominating Set</span> by Drange et al. (2016) <span><span>[17]</span></span> and Eickmeyer et al. (2017) <span><span>[20]</span></span> and for <span>Distance-</span><em>r</em> <span>Independent Set</span> by Pilipczuk and Siebertz (2021) <span><span>[41]</span></span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"159 ","pages":"Article 103782"},"PeriodicalIF":0.9,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154238","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 : 2026-08-01Epub Date: 2026-02-06DOI: 10.1016/j.jcss.2026.103773
Mengmeng He , Ralf Klasing , Yaping Mao , Xiaoyan Zhang
<div><div>Let <span><math><mi>k</mi><mspace></mspace><mo>(</mo><mi>k</mi><mo>≥</mo><mn>2</mn><mo>)</mo></math></span> be a positive integer, and let <em>G</em> be a simple graph with the vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. A Roman <em>k</em>-dominating function (Rk-DF) on <em>G</em> is a function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span> such that every vertex <em>u</em> with <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> is adjacent to at least <em>k</em> vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></math></span>. The minimum Roman <em>k</em>-dominating set problem aims to compute a Roman <em>k</em>-dominating function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub></math></span> that minimizes the total weight <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span>. The minimum Connected Roman <em>k</em>-dominating set problem (MinCR<em>k</em>DS) seeks to find a minimum-weight Roman <em>k</em>-dominating function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub></math></span> such that the subgraph of <em>G</em> induced by <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>|</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>1</mn><mtext> or </mtext><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo>}</mo></math></span> is connected. As far as we know, this paper is the first paper to solve MinCR<em>k</em>DS in general graphs. We present a greedy algorithm for MinCR<em>k</em>DS with an approximation ratio <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mo>(</mo><mn>2</mn><mo>+</mo><mi>ln</mi><mo></mo><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>Δ</mi><mo>)</mo><mo>)</mo></math></span> for any <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, wher
{"title":"Approximation algorithm for connected Roman k-dominating set","authors":"Mengmeng He , Ralf Klasing , Yaping Mao , Xiaoyan Zhang","doi":"10.1016/j.jcss.2026.103773","DOIUrl":"10.1016/j.jcss.2026.103773","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mspace></mspace><mo>(</mo><mi>k</mi><mo>≥</mo><mn>2</mn><mo>)</mo></math></span> be a positive integer, and let <em>G</em> be a simple graph with the vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. A Roman <em>k</em>-dominating function (Rk-DF) on <em>G</em> is a function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span> such that every vertex <em>u</em> with <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> is adjacent to at least <em>k</em> vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></math></span>. The minimum Roman <em>k</em>-dominating set problem aims to compute a Roman <em>k</em>-dominating function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub></math></span> that minimizes the total weight <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span>. The minimum Connected Roman <em>k</em>-dominating set problem (MinCR<em>k</em>DS) seeks to find a minimum-weight Roman <em>k</em>-dominating function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub></math></span> such that the subgraph of <em>G</em> induced by <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>|</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>1</mn><mtext> or </mtext><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi><mi>R</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo>}</mo></math></span> is connected. As far as we know, this paper is the first paper to solve MinCR<em>k</em>DS in general graphs. We present a greedy algorithm for MinCR<em>k</em>DS with an approximation ratio <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mo>(</mo><mn>2</mn><mo>+</mo><mi>ln</mi><mo></mo><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>Δ</mi><mo>)</mo><mo>)</mo></math></span> for any <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, wher","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"159 ","pages":"Article 103773"},"PeriodicalIF":0.9,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154237","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 : 2026-08-01Epub Date: 2026-02-06DOI: 10.1016/j.jcss.2026.103772
Zhao Wang , Yaping Mao , Sun-Yuan Hsieh , Ralf Klasing
In 1996, Fàbrega and Fiol introduced the g-extra connectivity of G as an important parameter for the fault tolerance of an interconnection network. A subset of vertices S is said to be a cutset if is not connected. A cutset S is called an -cutset, where g is a non-negative integer, if every component of has at least vertices. If G has at least one -cutset, the g-extra connectivity of G, denoted by , is then defined as the minimum cardinality over all -cutsets of G. In this paper, we obtain the exact values of the g-extra connectivity of some special graph classes, and show that for , and graphs with and trees with are characterized, respectively. We also derive three extremal results for the g-extra connectivity.
{"title":"On the g-extra connectivity of graphs","authors":"Zhao Wang , Yaping Mao , Sun-Yuan Hsieh , Ralf Klasing","doi":"10.1016/j.jcss.2026.103772","DOIUrl":"10.1016/j.jcss.2026.103772","url":null,"abstract":"<div><div>In 1996, Fàbrega and Fiol introduced the <em>g</em>-extra connectivity of <em>G</em> as an important parameter for the fault tolerance of an interconnection network. A subset of vertices <em>S</em> is said to be a <em>cutset</em> if <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> is not connected. A cutset <em>S</em> is called an <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span><em>-cutset</em>, where <em>g</em> is a non-negative integer, if every component of <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> has at least <span><math><mi>g</mi><mo>+</mo><mn>1</mn></math></span> vertices. If <em>G</em> has at least one <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>-cutset, the <em>g-extra connectivity</em> of <em>G</em>, denoted by <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is then defined as the minimum cardinality over all <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>-cutsets of <em>G</em>. In this paper, we obtain the exact values of the <em>g</em>-extra connectivity of some special graph classes, and show that <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn><mi>g</mi><mo>−</mo><mn>2</mn></math></span> for <span><math><mn>0</mn><mo>≤</mo><mi>g</mi><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></math></span>, and graphs with <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span> and trees with <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mn>2</mn><mi>g</mi><mo>−</mo><mn>2</mn></math></span> are characterized, respectively. We also derive three extremal results for the <em>g</em>-extra connectivity.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"159 ","pages":"Article 103772"},"PeriodicalIF":0.9,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154234","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 : 2026-06-01Epub Date: 2026-01-22DOI: 10.1016/j.jcss.2026.103760
Guilherme C.M. Gomes , Bruno P. Masquio , Paulo E.D. Pinto , Dieter Rautenbach , Vinicius F. dos Santos , Jayme L. Szwarcfiter , Florian Werner
A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.
{"title":"Complexity of deciding the equality of matching numbers","authors":"Guilherme C.M. Gomes , Bruno P. Masquio , Paulo E.D. Pinto , Dieter Rautenbach , Vinicius F. dos Santos , Jayme L. Szwarcfiter , Florian Werner","doi":"10.1016/j.jcss.2026.103760","DOIUrl":"10.1016/j.jcss.2026.103760","url":null,"abstract":"<div><div>A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"158 ","pages":"Article 103760"},"PeriodicalIF":0.9,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081672","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 : 2026-06-01Epub Date: 2026-02-03DOI: 10.1016/j.jcss.2026.103775
Walter Didimo , Michael Kaufmann , Giuseppe Liotta , Giacomo Ortali , Maurizio Patrignani
A rectilinear-upward planar drawing of a digraph G is a crossing-free drawing of G where each edge is either a horizontal or a vertical segment, and such that no directed edge points downward. Rectilinear-Upward Planarity Testing is the problem of deciding whether a digraph G admits a rectilinear-upward planar drawing. We study the complexity of Rectilinear-Upward Planarity Testing and provide several algorithmic results. Precisely, we prove that: (i) the problem is NP-complete, even if G is biconnected; it can be solved in linear time when an upward planar embedding of G is fixed; the problem is polynomial-time solvable for biconnected digraphs of treewidth at most two, i.e., for digraphs whose underlying undirected graph is a series-parallel graph; the problem is fixed-parameter tractable (namely, fixed-parameter linear) for all biconnected graphs, when parameterized by the number of sources and sinks in the digraph.
{"title":"Rectilinear-upward planarity testing of digraphs","authors":"Walter Didimo , Michael Kaufmann , Giuseppe Liotta , Giacomo Ortali , Maurizio Patrignani","doi":"10.1016/j.jcss.2026.103775","DOIUrl":"10.1016/j.jcss.2026.103775","url":null,"abstract":"<div><div>A <em>rectilinear-upward planar drawing</em> of a digraph <em>G</em> is a crossing-free drawing of <em>G</em> where each edge is either a horizontal or a vertical segment, and such that no directed edge points downward. <span>Rectilinear-Upward Planarity Testing</span> is the problem of deciding whether a digraph <em>G</em> admits a rectilinear-upward planar drawing. We study the complexity of <span>Rectilinear-Upward Planarity Testing</span> and provide several algorithmic results. Precisely, we prove that: (<em>i</em>) the problem is NP-complete, even if <em>G</em> is biconnected; <span><math><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></math></span> it can be solved in linear time when an upward planar embedding of <em>G</em> is fixed; <span><math><mo>(</mo><mi>i</mi><mi>i</mi><mi>i</mi><mo>)</mo></math></span> the problem is polynomial-time solvable for biconnected digraphs of treewidth at most two, i.e., for digraphs whose underlying undirected graph is a series-parallel graph; <span><math><mo>(</mo><mi>i</mi><mi>v</mi><mo>)</mo></math></span> the problem is fixed-parameter tractable (namely, fixed-parameter linear) for all biconnected graphs, when parameterized by the number of sources and sinks in the digraph.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"158 ","pages":"Article 103775"},"PeriodicalIF":0.9,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190711","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}
Consider , the free monoid generated by the finite alphabet A with the concatenation operation. Two words have the same commutative image when one is a permutation of the symbols of the other. The commutative closure of a language is the set of words whose commutative image coincides with that of some word in L. We provide an algorithm that, given a regular language L, constructs a finite state automaton that accepts the commutative closure , in all the cases where regular. The problem of deciding whether is regular was solved by Ginsburg and Spanier in 1966 using the decidability of Presburger sentences, and by Gohon in 1985 via formal power series. Some recent work constructs the finite state automaton accepting the commutative closure of permutation languages; however, to date there had been no general algorithm that handles all the cases where is regular. This algorithm is the main contribution of this work.
{"title":"Automata for the commutative closure of regular languages","authors":"Verónica Becher, Simón Lew Deveali, Ignacio Mollo Cunningham","doi":"10.1016/j.jcss.2026.103762","DOIUrl":"10.1016/j.jcss.2026.103762","url":null,"abstract":"<div><div>Consider <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the free monoid generated by the finite alphabet <em>A</em> with the concatenation operation. Two words have the same commutative image when one is a permutation of the symbols of the other. The commutative closure of a language <span><math><mi>L</mi><mo>⊆</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is the set <span><math><mi>C</mi><mo>(</mo><mi>L</mi><mo>)</mo><mo>⊆</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> of words whose commutative image coincides with that of some word in <em>L</em>. We provide an algorithm that, given a regular language <em>L</em>, constructs a finite state automaton that accepts the commutative closure <span><math><mi>C</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span>, in all the cases where <span><math><mi>C</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span> regular. The problem of deciding whether <span><math><mi>C</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span> is regular was solved by Ginsburg and Spanier in 1966 using the decidability of Presburger sentences, and by Gohon in 1985 via formal power series. Some recent work constructs the finite state automaton accepting the commutative closure of permutation languages; however, to date there had been no general algorithm that handles all the cases where <span><math><mi>C</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span> is regular. This algorithm is the main contribution of this work.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"158 ","pages":"Article 103762"},"PeriodicalIF":0.9,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146045202","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 : 2026-06-01Epub Date: 2026-01-23DOI: 10.1016/j.jcss.2026.103759
Ruben Becker , Davide Cenzato , Sung-Hwan Kim , Tomasz Kociumaka , Bojana Kodric , Alberto Policriti , Nicola Prezza
Co-lex partial orders (Cotumaccio and Prezza (2021) [9] and Cotumaccio et al. (2023) [8]) are a powerful tool to index finite automata generalizing Wheeler orders (Gagie et al. (2017) [14]). The co-lex width p of an automaton measures how sortable its states are w.r.t. the co-lexicographic order among its accepted strings. Automata of co-lex width p can be compressed to bits per edge and admit regular expression matching in time proportional to per matched character. The deterministic co-lex width of a regular language is the smallest width of such a co-lex order, among all DFAs recognizing . Since languages of small co-lex width admit efficient solutions to hard computational problems, computing the co-lex width is relevant in applications. Previous work showed that the deterministic co-lex width p of a language can be computed in for a DFA with m transitions accepting . For constant p (in particular Wheeler languages, where ), the constant in the exponent is large and the exact complexity remains unknown. In this work, we show that one can decide in if the deterministic co-lex width of the language recognized by a given minimum DFA is strictly smaller than . We complement this with a matching conditional lower bound based on the Strong Exponential Time Hypothesis. Hence, our paper essentially settles the complexity of the problem.
Co-lex偏阶(Cotumaccio and Prezza（2021)[9]和Cotumaccio et al.(2023)[8]）是索引有限自动机泛化Wheeler阶（Gagie et al.(2017)[14]）的强大工具。自动机的协词法宽度p衡量其状态的可排序程度，而不是其接受的字符串之间的协词法顺序。协环宽度为p的自动机可以被压缩到每条边O（log (p)）比特，并且允许正则表达式匹配在时间上与每个匹配字符p2成正比。在所有识别L的dfa中，正则语言L的确定性协lex宽度是该协lex阶的最小宽度。由于较小协lex宽度的语言可以有效地解决难计算问题，因此计算协lex宽度在应用中是相关的。先前的工作表明，对于接受L的m个转换的DFA a，语言L的确定性协lex宽度p可以用mO(p)来计算。对于常数p（特别是惠勒语言，其中p=1），指数中的常数很大，确切的复杂性仍然未知。在这项工作中，我们证明了如果给定最小DFA识别的语言的确定性协lex宽度严格小于p≥2，则可以在O（mp）中确定。我们补充了一个基于强指数时间假设的匹配条件下界。因此，我们的论文基本上解决了这个问题的复杂性。
{"title":"On the complexity of computing the co-lexicographic width of a regular language","authors":"Ruben Becker , Davide Cenzato , Sung-Hwan Kim , Tomasz Kociumaka , Bojana Kodric , Alberto Policriti , Nicola Prezza","doi":"10.1016/j.jcss.2026.103759","DOIUrl":"10.1016/j.jcss.2026.103759","url":null,"abstract":"<div><div>Co-lex partial orders (Cotumaccio and Prezza (2021) <span><span>[9]</span></span> and Cotumaccio et al. (2023) <span><span>[8]</span></span>) are a powerful tool to index finite automata generalizing Wheeler orders (Gagie et al. (2017) <span><span>[14]</span></span>). The co-lex width <em>p</em> of an automaton measures how sortable its states are w.r.t. the co-lexicographic order among its accepted strings. Automata of co-lex width <em>p</em> can be compressed to <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>p</mi><mo>)</mo></math></span> bits per edge and admit regular expression matching in time proportional to <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> per matched character. The deterministic co-lex width of a regular language <span><math><mi>L</mi></math></span> is the smallest width of such a co-lex order, among all DFAs recognizing <span><math><mi>L</mi></math></span>. Since languages of small co-lex width admit efficient solutions to hard computational problems, computing the co-lex width is relevant in applications. Previous work showed that the deterministic co-lex width <em>p</em> of a language <span><math><mi>L</mi></math></span> can be computed in <span><math><msup><mrow><mi>m</mi></mrow><mrow><mi>O</mi><mo>(</mo><mi>p</mi><mo>)</mo></mrow></msup></math></span> for a DFA <span><math><mi>A</mi></math></span> with <em>m</em> transitions accepting <span><math><mi>L</mi></math></span>. For constant <em>p</em> (in particular Wheeler languages, where <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span>), the constant in the exponent is large and the exact complexity remains unknown. In this work, we show that one can decide in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo></math></span> if the deterministic co-lex width of the language recognized by a given minimum DFA is strictly smaller than <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>. We complement this with a matching conditional lower bound based on the Strong Exponential Time Hypothesis. Hence, our paper essentially settles the complexity of the problem.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"158 ","pages":"Article 103759"},"PeriodicalIF":0.9,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081578","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 : 2026-06-01Epub Date: 2026-01-26DOI: 10.1016/j.jcss.2026.103761
Yue Sun , Donglei Du , Longkun Guo , Dachuan Xu
In the k edge-disjoint restricted shortest path (kRSP) problem, we are given a directed graph with a specified source vertex s and a sink vertex t, where each edge has a non-negative cost and delay. For a given delay bound , the kRSP problem aims to find k edge-disjoint directed paths from s to t, such that the total delay of the edges on the k paths does not exceed D, while minimizing the total cost. The problem is a variant of the well-known disjoint path problem and is NP-hard even when . In this paper, we present an optimal absolute approximation algorithm based on the LP-rounding technique, which guarantees finding at least edge-disjoint paths that strictly satisfy the delay constraint with a total cost no greater than that of an optimal solution, where is an optimal approximation of k. The key observation behind our approach is that, in any basic optimal solution of the linear programming relaxation for kRSP, the underlying graph formed by edges with fractional values is exactly a cycle. We show that this cycle always includes a subset of edges that, together with the integral edges from the LP solution, can be used to construct the desired set of paths. Finally, we present a novel method for rounding the edges from this cycle to obtain the desired solution.
{"title":"An optimal absolute approximation algorithm for computing k disjoint restricted shortest paths","authors":"Yue Sun , Donglei Du , Longkun Guo , Dachuan Xu","doi":"10.1016/j.jcss.2026.103761","DOIUrl":"10.1016/j.jcss.2026.103761","url":null,"abstract":"<div><div>In the <em>k</em> edge-disjoint restricted shortest path (<em>k</em>RSP) problem, we are given a directed graph with a specified source vertex <em>s</em> and a sink vertex <em>t</em>, where each edge has a non-negative cost and delay. For a given delay bound <span><math><mi>D</mi><mo>∈</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>, the <em>k</em>RSP problem aims to find <em>k</em> edge-disjoint directed paths from <em>s</em> to <em>t</em>, such that the total delay of the edges on the <em>k</em> paths does not exceed <em>D</em>, while minimizing the total cost. The problem is a variant of the well-known disjoint path problem and is NP-hard even when <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>. In this paper, we present an <em>optimal absolute approximation algorithm</em> based on the LP-rounding technique, which guarantees finding at least <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> edge-disjoint paths that strictly satisfy the delay constraint with a total cost no greater than that of an optimal solution, where <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> is an optimal approximation of <em>k</em>. The key observation behind our approach is that, in any basic optimal solution of the linear programming relaxation for <em>k</em>RSP, the underlying graph formed by edges with fractional values is exactly a cycle. We show that this cycle always includes a subset of edges that, together with the integral edges from the LP solution, can be used to construct the desired set of paths. Finally, we present a novel method for rounding the edges from this cycle to obtain the desired solution.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"158 ","pages":"Article 103761"},"PeriodicalIF":0.9,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190713","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 : 2026-06-01Epub Date: 2026-02-03DOI: 10.1016/j.jcss.2026.103776
Antoine Dailly , Florent Foucaud , Anni Hakanen
In the Metric Dimension problem, one asks for a minimum-size set R of vertices such that for any pair of vertices of the graph, there is a vertex from R whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (non-trivially) extends a previous algorithm for oriented trees. We then extend the method to orientations of unicyclic graphs. We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that Metric Dimension is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.
{"title":"Algorithms and hardness for Metric Dimension on digraphs","authors":"Antoine Dailly , Florent Foucaud , Anni Hakanen","doi":"10.1016/j.jcss.2026.103776","DOIUrl":"10.1016/j.jcss.2026.103776","url":null,"abstract":"<div><div>In the <span>Metric Dimension</span> problem, one asks for a minimum-size set <em>R</em> of vertices such that for any pair of vertices of the graph, there is a vertex from <em>R</em> whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (non-trivially) extends a previous algorithm for oriented trees. We then extend the method to orientations of unicyclic graphs. We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that <span>Metric Dimension</span> is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"158 ","pages":"Article 103776"},"PeriodicalIF":0.9,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190712","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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