Pub Date : 2026-02-06DOI: 10.1016/j.tcs.2026.115802
Vacharapat Mettanant
This paper studies a special case of the Maximum Interval Multi-Cover (MaxIMC) problem, called the Interval Maximum Coverage Problem. Given a set of points P on the real line, a collection of intervals I, and a budget K, the goal is to select up to K intervals that maximize the number of covered points. While the computational complexity of the general MaxIMC problem with arbitrary coverage requirements remains open, this special case admits efficient polynomial-time solutions. We develop an exact algorithm that improves computational efficiency when the number of intervals is extremely large, and a near-linear-time approximation algorithm for the case where each interval covers exactly r points. We provide formal proofs of correctness, detailed complexity analysis, and experimental results demonstrating the practical efficiency and effectiveness of the proposed algorithms.
{"title":"Efficient algorithms for the interval maximum coverage problem","authors":"Vacharapat Mettanant","doi":"10.1016/j.tcs.2026.115802","DOIUrl":"10.1016/j.tcs.2026.115802","url":null,"abstract":"<div><div>This paper studies a special case of the Maximum Interval Multi-Cover (MaxIMC) problem, called the <em>Interval Maximum Coverage Problem</em>. Given a set of points <em>P</em> on the real line, a collection of intervals <em>I</em>, and a budget <em>K</em>, the goal is to select up to <em>K</em> intervals that maximize the number of covered points. While the computational complexity of the general MaxIMC problem with arbitrary coverage requirements remains open, this special case admits efficient polynomial-time solutions. We develop an exact algorithm that improves computational efficiency when the number of intervals is extremely large, and a near-linear-time approximation algorithm for the case where each interval covers exactly <em>r</em> points. We provide formal proofs of correctness, detailed complexity analysis, and experimental results demonstrating the practical efficiency and effectiveness of the proposed algorithms.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115802"},"PeriodicalIF":1.0,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-05DOI: 10.1016/j.tcs.2026.115798
Hortensia Galeana-Sánchez , Carlos Vilchis-Alfaro
Let G be a multigraph. A transition in G is a pair of adjacent edges. For each vertex x of G, a set T(x) of allowed transition with respect to x is a set of unordered pairs of edges incident to x. A transition system T is a set {T(x): x ∈ V(G)}, where T(x) is a fixed set of allowed transition with respect to x. Given a multigraph G and a transition system T, for each x ∈ V(G), the transition graph of x, denoted by Gx, is a graph such that its vertex set is the set of edges incident to x; and two vertices e and g of Gx are adjacent whenever eg ∈ T(x). A trail in G is T-compatible if for every , .
In this paper we deal with the problem of finding T-compatible trails between s and t two given vertices in a multigraph with transition system T. First, we prove that finding a T-compatible trail between two given edges can be done in polynomial time. Consequently, finding a T-compatible trail can be done in polynomial time. Moreover, it can be found a shortest T-compatible trail and a closed T-compatible trail containing a given vertex in polynomial time. Finally, we study multigraphs with transition systems such that their transition graph is connected or complete multipartite graph. The properly colored setting is a particular case of transition systems where all its transition graphs are complete multipartite graphs.
设G是一个多重图。G中的一个过渡是一对相邻的边。每个顶点x (G, T (x)允许过渡对x是一组无序对边缘入射x。过渡系统T是一个集{T (x): x ∈ V (G)},其中T (x)是一组固定的允许过渡对x。鉴于油印G和T转换系统,为每个x ∈ V (G), x的过渡图,用Gx,就是这样一个图表,其顶点集的边缘入射x;当eg ∈ T(x)时，两个顶点e与g （Gx）相邻。小道P = (x0, e0 x1,…,xk−1,埃克−1,xk)在G T-compatible如果每我∈{0…k−2},eiei T + 1∈(xi + 1)。本文研究了具有过渡系统t的多重图中s和t两个给定顶点之间的t -相容轨迹问题，首先证明了在多项式时间内找到两个给定边之间的t -相容轨迹是可行的。因此，找到t兼容的s - t轨迹可以在多项式时间内完成。此外，在多项式时间内可以找到一条最短的t -兼容s - t轨迹和一条包含给定顶点的t -兼容闭合轨迹。最后，我们研究了具有过渡系统的多图，其过渡图是连通或完全多部图。适当的着色设置是过渡系统的特殊情况，其中所有的过渡图都是完全多部图。
{"title":"Finding trails in multigraphs with restricted transitions","authors":"Hortensia Galeana-Sánchez , Carlos Vilchis-Alfaro","doi":"10.1016/j.tcs.2026.115798","DOIUrl":"10.1016/j.tcs.2026.115798","url":null,"abstract":"<div><div>Let <em>G</em> be a multigraph. A transition in <em>G</em> is a pair of adjacent edges. For each vertex <em>x</em> of <em>G</em>, a set <em>T</em>(<em>x</em>) of allowed transition with respect to <em>x</em> is a set of unordered pairs of edges incident to <em>x</em>. A transition system <em>T</em> is a set {<em>T</em>(<em>x</em>): <em>x</em> ∈ <em>V</em>(<em>G</em>)}, where <em>T</em>(<em>x</em>) is a fixed set of allowed transition with respect to <em>x</em>. Given a multigraph <em>G</em> and a transition system <em>T</em>, for each <em>x</em> ∈ <em>V</em>(<em>G</em>), the transition graph of <em>x</em>, denoted by <em>G<sub>x</sub></em>, is a graph such that its vertex set is the set of edges incident to <em>x</em>; and two vertices <em>e</em> and <em>g</em> of <em>G<sub>x</sub></em> are adjacent whenever <em>eg</em> ∈ <em>T</em>(<em>x</em>). A trail <span><math><mrow><mi>P</mi><mo>=</mo><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>e</mi><mn>0</mn></msub><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>e</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi><mi>k</mi></msub><mo>)</mo></mrow></math></span> in <em>G</em> is <em>T</em>-compatible if for every <span><math><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>}</mo></mrow></math></span>, <span><math><mrow><msub><mi>e</mi><mi>i</mi></msub><msub><mi>e</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><mi>T</mi><mrow><mo>(</mo><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>.</div><div>In this paper we deal with the problem of finding <em>T</em>-compatible trails between <em>s</em> and <em>t</em> two given vertices in a multigraph with transition system <em>T</em>. First, we prove that finding a <em>T</em>-compatible trail between two given edges can be done in polynomial time. Consequently, finding a <em>T</em>-compatible <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> trail can be done in polynomial time. Moreover, it can be found a shortest <em>T</em>-compatible <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> trail and a closed <em>T</em>-compatible trail containing a given vertex in polynomial time. Finally, we study multigraphs with transition systems such that their transition graph is connected or complete multipartite graph. The properly colored setting is a particular case of transition systems where all its transition graphs are complete multipartite graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115798"},"PeriodicalIF":1.0,"publicationDate":"2026-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-02DOI: 10.1016/j.tcs.2026.115797
Guillaume Blin , Adrian Miclăuş , Sebastian Ordyniak , Alexandru Popa
In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.
{"title":"SOBRA - Shielding Optimization for BRAchytherapy","authors":"Guillaume Blin , Adrian Miclăuş , Sebastian Ordyniak , Alexandru Popa","doi":"10.1016/j.tcs.2026.115797","DOIUrl":"10.1016/j.tcs.2026.115797","url":null,"abstract":"<div><div>In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115797"},"PeriodicalIF":1.0,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ring signatures, a novel cryptographic primitive, were introduced by Rivest, Shamir, and Tauman at ASIACRYPT 2001. It allows a member of a group to sign a message on behalf of an ad-hoc group of users, known as a ring, without revealing its identity. Ring signatures, initially introduced for applications such as whistle-blowing and confidential disclosures, have since gained significant relevance and widespread adoption across various domains, including blockchain technologies, ad-hoc networks, anonymous transactions, location-based services, cloud computing platforms, and outsourced computation. However, traditional ring signatures are vulnerable if the signers’ keys are compromised. To resolve this vulnerability, Liu and Wong introduced the notion of forward-secure ring signatures at IJNS 2008. Forward-security ensures that even if a signer’s secret key is compromised, signatures generated in prior time periods remain secure. It significantly reduces the damage caused by secret key exposure, making it crucial for long-term security in applications such as whistle-blowing or anonymous authentication. However, most existing forward-secure ring signature schemes are based on number-theoretic assumptions, making them vulnerable to quantum adversaries. Yu and Wang proposed a quantum-safe forward-secure ring signature scheme at ICICS 2023, and Yu et al. further extended this work in Computer Networks (2025), retaining the same ring signature scheme with forward security. But the scheme suffers from large signatures that grow linearly with the ring size. In this paper, we propose a lattice-based forward-secure ring signature scheme in the quantum random oracle model (QROM) that simultaneously achieves security against quantum adversaries and compact signature size. Notably, the signature size is logarithmic in the number of members in the ring, enabling significantly better scalability compared to prior constructions.
{"title":"Lattice-based logarithmic-size forward-secure ring signatures in QROM","authors":"Priyanka Dutta, Willy Susilo, Fuchun Guo, Dung Hoang Duong","doi":"10.1016/j.tcs.2026.115787","DOIUrl":"10.1016/j.tcs.2026.115787","url":null,"abstract":"<div><div>Ring signatures, a novel cryptographic primitive, were introduced by Rivest, Shamir, and Tauman at ASIACRYPT 2001. It allows a member of a group to sign a message on behalf of an ad-hoc group of users, known as a ring, without revealing its identity. Ring signatures, initially introduced for applications such as whistle-blowing and confidential disclosures, have since gained significant relevance and widespread adoption across various domains, including blockchain technologies, ad-hoc networks, anonymous transactions, location-based services, cloud computing platforms, and outsourced computation. However, traditional ring signatures are vulnerable if the signers’ keys are compromised. To resolve this vulnerability, Liu and Wong introduced the notion of forward-secure ring signatures at IJNS 2008. Forward-security ensures that even if a signer’s secret key is compromised, signatures generated in prior time periods remain secure. It significantly reduces the damage caused by secret key exposure, making it crucial for long-term security in applications such as whistle-blowing or anonymous authentication. However, most existing forward-secure ring signature schemes are based on number-theoretic assumptions, making them vulnerable to quantum adversaries. Yu and Wang proposed a quantum-safe forward-secure ring signature scheme at ICICS 2023, and Yu et al. further extended this work in Computer Networks (2025), retaining the same ring signature scheme with forward security. But the scheme suffers from large signatures that grow linearly with the ring size. In this paper, we propose a lattice-based forward-secure ring signature scheme in the quantum random oracle model (<span>QROM</span>) that simultaneously achieves security against quantum adversaries and compact signature size. Notably, the signature size is logarithmic in the number of members in the ring, enabling significantly better scalability compared to prior constructions.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115787"},"PeriodicalIF":1.0,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-24DOI: 10.1016/j.tcs.2026.115785
Chenzheng Feng, Wen Liu, Gengsheng Zhang, Bo Hou
In this paper, we consider the k-product uncapacitated facility location problem with submodular penalties. In this problem, we are given a set of demand points where clients are located, a set of potential locations where unlimited capacity facilities can be opened and a set of k different kinds of products. Each open facility can only supply one kind of product, and its open cost is determined by the product it supplies. There is a service cost between each pair of locations. Assume these costs of service are metric. Each client is either supplied k different kinds of products by a set of k different open facilities or completely rejected and a rejection cost has to be paid, which is determined by a submodular function. The objective is to minimize the total cost, including the cost of opening facilities at sites, the service cost for providing products to clients from the open facilities, and the penalty cost of the set of the rejected clients. Based on the LP rounding technique, we propose a -approximation algorithm for this problem.
{"title":"Approximation algorithm for k-Product uncapacitated facility location problem with submodular penalties","authors":"Chenzheng Feng, Wen Liu, Gengsheng Zhang, Bo Hou","doi":"10.1016/j.tcs.2026.115785","DOIUrl":"10.1016/j.tcs.2026.115785","url":null,"abstract":"<div><div>In this paper, we consider the <em>k</em>-product uncapacitated facility location problem with submodular penalties. In this problem, we are given a set of demand points where clients are located, a set of potential locations where unlimited capacity facilities can be opened and a set of <em>k</em> different kinds of products. Each open facility can only supply one kind of product, and its open cost is determined by the product it supplies. There is a service cost between each pair of locations. Assume these costs of service are metric. Each client is either supplied <em>k</em> different kinds of products by a set of <em>k</em> different open facilities or completely rejected and a rejection cost has to be paid, which is determined by a submodular function. The objective is to minimize the total cost, including the cost of opening facilities at sites, the service cost for providing products to clients from the open facilities, and the penalty cost of the set of the rejected clients. Based on the LP rounding technique, we propose a <span><math><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></math></span>-approximation algorithm for this problem.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115785"},"PeriodicalIF":1.0,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-24DOI: 10.1016/j.tcs.2026.115767
Kohei Nomura, Koichi Yamazaki
Studying minimal separators in graph theory is important for both practical and theoretical reasons. Some graphs have polynomially many minimal separators, while others have exponentially many. Recently, there has been increasing interest in understanding which graph structures lead to exponentially many minimal separators. One such structure is a k-creature. It has been observed that graphs containing a k-creature as an induced subgraph have exponentially many minimal separators. Because of this, it was initially conjectured that forbidding k-creatures would be sufficient to characterize graph classes with only a polynomial number of minimal separators.
However, this conjecture was disproven by Gartland and Lokshtanov, who introduced a graph class known as k-twisted ladders. Although these graphs are free of k-creatures, they still have exponentially many minimal separators. This counterexample motivated the introduction of a new graph structure called the k-critter, which generalizes the k-twisted ladder.
This paper aims to clarify the fundamental differences between k-creatures and k-critters by analyzing their associated lattice structures. We focus on two representative graph types: k-ladders, which exemplify k-creatures, and k-twisted ladders, which represent the k-critter structure. By comparing the lattices formed by the minimal separators of each graph, we demonstrate that, despite their similar graph structures, their lattice structures differ significantly. Our results show that lattice-theoretical methods provide useful insights for studying minimal a, b-separators.
{"title":"Comparison of k-creature and t-critter","authors":"Kohei Nomura, Koichi Yamazaki","doi":"10.1016/j.tcs.2026.115767","DOIUrl":"10.1016/j.tcs.2026.115767","url":null,"abstract":"<div><div>Studying minimal separators in graph theory is important for both practical and theoretical reasons. Some graphs have polynomially many minimal separators, while others have exponentially many. Recently, there has been increasing interest in understanding which graph structures lead to exponentially many minimal separators. One such structure is a <em>k</em>-creature. It has been observed that graphs containing a <em>k</em>-creature as an induced subgraph have exponentially many minimal separators. Because of this, it was initially conjectured that forbidding <em>k</em>-creatures would be sufficient to characterize graph classes with only a polynomial number of minimal separators.</div><div>However, this conjecture was disproven by Gartland and Lokshtanov, who introduced a graph class known as <em>k</em>-twisted ladders. Although these graphs are free of <em>k</em>-creatures, they still have exponentially many minimal separators. This counterexample motivated the introduction of a new graph structure called the <em>k</em>-critter, which generalizes the <em>k</em>-twisted ladder.</div><div>This paper aims to clarify the fundamental differences between <em>k</em>-creatures and <em>k</em>-critters by analyzing their associated lattice structures. We focus on two representative graph types: <em>k</em>-ladders, which exemplify <em>k</em>-creatures, and <em>k</em>-twisted ladders, which represent the <em>k</em>-critter structure. By comparing the lattices formed by the minimal separators of each graph, we demonstrate that, despite their similar graph structures, their lattice structures differ significantly. Our results show that lattice-theoretical methods provide useful insights for studying minimal <em>a, b</em>-separators.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115767"},"PeriodicalIF":1.0,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-24DOI: 10.1016/j.tcs.2026.115784
Nesrine Abbas
In this paper we show that convex bipartite graphs can be decomposed into induced subgraphs consisting of biconvex connected components with no neighbourhood set containment. We use this result to enumerate minimum weight blue dominating sets in convex bipartite graphs. In the k blue (red) domination problem for a vertex-weighted bipartite graph where w is a function that assigns positive integer weights to vertices, we seek a subset D⊆Y (respectively D⊆X) of total vertex weight at most k that dominates vertices of X (respectively Y). Because convex bipartite graphs are asymmetrical with regards to the neighbourhood sets of vertices in X and Y, an algorithm for red domination does not necessarily apply to blue domination for this subclass of bipartite graphs. The decision version of red (blue) domination is -complete for general bipartite graphs. We strengthen those results by showing that it remains -complete for perfect elimination bipartite graphs. We present a tight upper bound on the number of such sets in bipartite graphs. We present linear space linear delay enumeration algorithms for both the minimum weight red and blue dominating sets in convex bipartite graphs, that need linear and quadratic preprocessing time, respectively. We show that the number of minimum weight red and blue dominating sets in convex bipartite graphs can be computed in linear and quadratic time, respectively. We show that our results can be used to compute minimum weight red and blue dominating sets in convex bipartite graphs in polynomial time. Thus, they extend results in the literature for computing minimum cardinality red and blue dominating sets in unweighted convex bipartite graphs.
{"title":"Decomposing convex bipartite graphs into biconvex graphs and enumerating minimum weight red blue dominating sets","authors":"Nesrine Abbas","doi":"10.1016/j.tcs.2026.115784","DOIUrl":"10.1016/j.tcs.2026.115784","url":null,"abstract":"<div><div>In this paper we show that convex bipartite graphs can be decomposed into induced subgraphs consisting of biconvex connected components with no neighbourhood set containment. We use this result to enumerate minimum weight blue dominating sets in convex bipartite graphs. In the <em>k</em> blue (red) domination problem for a vertex-weighted bipartite graph <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo></mrow></math></span> where <em>w</em> is a function that assigns positive integer weights to vertices, we seek a subset <em>D</em>⊆<em>Y</em> (respectively <em>D</em>⊆<em>X</em>) of total vertex weight at most <em>k</em> that dominates vertices of <em>X</em> (respectively <em>Y</em>). Because convex bipartite graphs are asymmetrical with regards to the neighbourhood sets of vertices in <em>X</em> and <em>Y</em>, an algorithm for red domination does not necessarily apply to blue domination for this subclass of bipartite graphs. The decision version of red (blue) domination is <span><math><mtext>NP</mtext></math></span>-complete for general bipartite graphs. We strengthen those results by showing that it remains <span><math><mtext>NP</mtext></math></span>-complete for perfect elimination bipartite graphs. We present a tight upper bound on the number of such sets in bipartite graphs. We present linear space linear delay enumeration algorithms for both the minimum weight red and blue dominating sets in convex bipartite graphs, that need linear and quadratic preprocessing time, respectively. We show that the number of minimum weight red and blue dominating sets in convex bipartite graphs can be computed in linear and quadratic time, respectively. We show that our results can be used to compute minimum weight red and blue dominating sets in convex bipartite graphs in polynomial time. Thus, they extend results in the literature for computing minimum cardinality red and blue dominating sets in unweighted convex bipartite graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115784"},"PeriodicalIF":1.0,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-23DOI: 10.1016/j.tcs.2026.115775
Irena Rusu
The problems Cluster Vertex Deletion (or Cluster-VD) and its generalization s-Club Cluster Vertex Deletion (or s-Club-VD, for any integer s ≥ 1), have been introduced with the aim of detecting highly-connected parts in complex systems. Their NP-completeness has been established for several classes of graphs, but remains open for smaller classes, including subcubic planar bipartite graphs and cubic graphs. In this paper, we show that Cluster-VD and more generally s-Club-VD are NP-complete for cubic planar bipartite graphs, as well as for subcubic planar bipartite graphs with unbounded girth (when s is odd) and with girth exactly (when s is even). We also deduce new results for the related k-Path Vertex Cover problem (or k-PVC), namely 3-PVC is NP-complete for cubic planar bipartite graphs, whereas k-PVC with k ≥ 4 is NP-complete for subcubic planar (and bipartite, when k is odd) graphs of arbitrarily large girth.
针对复杂系统中高连通部分的检测问题，提出了聚类顶点删除（Cluster- vd）及其推广的s- club聚类顶点删除（s- club - vd，适用于任意整数s ≥ 1）。它们的np完备性已被建立在若干类图上，但对于更小的类，包括次三次平面二部图和三次图，仍然是开放的。在本文中，我们证明了Cluster-VD和更一般的s- club - vd对于三次平面二部图以及具有无界周长（当s为奇数）和周长正好为s+2（当s为偶数）的次三次平面二部图是np完全的。对于相关的k- path顶点覆盖问题（或k- pvc），我们也推导出新的结果，即对于三次平面二部图，3-PVC是np完全的，而对于任意大周长的次三次平面（和二部图，当k为奇数时）图，k- pvc （k ≥ 4）是np完全的。
{"title":"Cluster vertex deletion problems on cubic graphs","authors":"Irena Rusu","doi":"10.1016/j.tcs.2026.115775","DOIUrl":"10.1016/j.tcs.2026.115775","url":null,"abstract":"<div><div>The problems <span>Cluster Vertex Deletion</span> (or <span>Cluster-VD</span>) and its generalization <em>s</em>-<span>Club Cluster Vertex Deletion</span> (or <em>s</em>-<span>Club-VD</span>, for any integer <em>s</em> ≥ 1), have been introduced with the aim of detecting highly-connected parts in complex systems. Their NP-completeness has been established for several classes of graphs, but remains open for smaller classes, including subcubic planar bipartite graphs and cubic graphs. In this paper, we show that <span>Cluster-VD</span> and more generally <em>s</em>-<span>Club-VD</span> are NP-complete for cubic planar bipartite graphs, as well as for subcubic planar bipartite graphs with unbounded girth (when <em>s</em> is odd) and with girth exactly <span><math><mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></math></span> (when <em>s</em> is even). We also deduce new results for the related <em>k</em>-<span>Path Vertex Cover</span> problem (or <em>k</em>-<span>PVC</span>), namely <span>3</span>-PVC is NP-complete for cubic planar bipartite graphs, whereas <em>k</em>-<span>PVC</span> with <em>k</em> ≥ 4 is NP-complete for subcubic planar (and bipartite, when <em>k</em> is odd) graphs of arbitrarily large girth.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115775"},"PeriodicalIF":1.0,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-23DOI: 10.1016/j.tcs.2026.115771
Paweł Gawrychowski , Samah Ghazawi , Gad M. Landau
Given two indeterminate equal-length strings p and t with a set of r characters per position in both strings, we obtain a determinate string pw from p and a determinate string tw from t by choosing one character per position. Then, we say that p and t match when pw and tw match for some choice of characters. We systematically study the complexity of string matching for indeterminate equal-length strings, for different notions of matching: parameterized matching, order-preserving matching, and Cartesian tree matching. We use n to denote the length of both strings and r the upper bound on the number of characters per position. First, we provide an algorithm for the Cartesian tree version that runs in time using space, for any constant r. Second, we establish NP-hardness of the order-preserving version for , thus solving a question explicitly stated by Henriques et al. [CPM 2018], who showed hardness for . Third, we establish NP-hardness of the parameterized version for . As both parameterized and order-preserving indeterminate matching reduce to the standard determinate matching for , this provides a complete classification for these three variants.
{"title":"On the complexity of indeterminate strings matching","authors":"Paweł Gawrychowski , Samah Ghazawi , Gad M. Landau","doi":"10.1016/j.tcs.2026.115771","DOIUrl":"10.1016/j.tcs.2026.115771","url":null,"abstract":"<div><div>Given two indeterminate equal-length strings <em>p</em> and <em>t</em> with a set of <em>r</em> characters per position in both strings, we obtain a determinate string <em>p<sub>w</sub></em> from <em>p</em> and a determinate string <em>t<sub>w</sub></em> from <em>t</em> by choosing one character per position. Then, we say that <em>p</em> and <em>t</em> match when <em>p<sub>w</sub></em> and <em>t<sub>w</sub></em> match for some choice of characters. We systematically study the complexity of string matching for indeterminate equal-length strings, for different notions of matching: parameterized matching, order-preserving matching, and Cartesian tree matching. We use <em>n</em> to denote the length of both strings and <em>r</em> the upper bound on the number of characters per position. First, we provide an algorithm for the Cartesian tree version that runs in <span><math><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mi>n</mi><mi>log</mi><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> time using <span><math><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span> space, for any constant <em>r</em>. Second, we establish NP-hardness of the order-preserving version for <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span>, thus solving a question explicitly stated by Henriques et al. [CPM 2018], who showed hardness for <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span>. Third, we establish NP-hardness of the parameterized version for <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span>. As both parameterized and order-preserving indeterminate matching reduce to the standard determinate matching for <span><math><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></math></span>, this provides a complete classification for these three variants.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115771"},"PeriodicalIF":1.0,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-22DOI: 10.1016/j.tcs.2026.115774
János Balogh , József Békési , György Dósa , Leah Epstein , Asaf Levin
We revisit online bin packing with cardinality constraints. In this problem, a set of items of positive sizes not larger than 1 and an integer parameter k ≥ 2 are given. The goal is to partition the items into the minimum number of valid bins, where a valid bin is a set of at most k items whose total size is at most 1. We provide better bounds on the asymptotic competitive ratio for cardinality constrained bin packing for , showcasing current methods for designing algorithms for bin packing problems. We extend the lower bound construction for for other values of k, improving all known lower bounds on the best possible asymptotic competitive ratio for small k ≥ 3.
{"title":"More on online cardinality constrained bin packing with small cardinality bounds","authors":"János Balogh , József Békési , György Dósa , Leah Epstein , Asaf Levin","doi":"10.1016/j.tcs.2026.115774","DOIUrl":"10.1016/j.tcs.2026.115774","url":null,"abstract":"<div><div>We revisit online bin packing with cardinality constraints. In this problem, a set of items of positive sizes not larger than 1 and an integer parameter <em>k</em> ≥ 2 are given. The goal is to partition the items into the minimum number of valid bins, where a valid bin is a set of at most <em>k</em> items whose total size is at most 1. We provide better bounds on the asymptotic competitive ratio for cardinality constrained bin packing for <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></math></span>, showcasing current methods for designing algorithms for bin packing problems. We extend the lower bound construction for <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></math></span> for other values of <em>k</em>, improving all known lower bounds on the best possible asymptotic competitive ratio for small <em>k</em> ≥ 3.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115774"},"PeriodicalIF":1.0,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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