Pub Date : 2026-01-02DOI: 10.1016/j.tcs.2025.115740
Dariusz R. Kowalski , Miguel A. Mosteiro
In seminal work on Adversarial Dynamic Networks, Kuhn, Lynch and Oshman (STOC 2010) [19] studied dynamic networks in which links are selected by an adversary and the number of network nodes is initially unknown. In such networks, they showed upper and lower bounds for computing the size of the network and any computable function of the nodes initial inputs. In this work, we address the same question in dynamic networks which additionally are: anonymous, possibly disconnected, and where internal memory and links’ bandwith are logarithmically limited. In the above framework, we study a fundamental communication principle – the All-to-all problem: each node has an input message to be delivered to all other nodes. (Once a node receives all inputs, any function can be computed locally.) Because of anonymity, each node needs to receive only a set of all input messages, each accompanied by a number of their initiating nodes (message multiplicity). We prove that this can be done deterministically in time proportional to the total number of messages’ bits multiplied by a small polynomial in networks’ parameters – namely, in the (initially unknown) number of nodes n and in the lower bound on the isoperimetric numbers of dynamically evolving graphs imin. Our results prove that a polynomial bit-throughput is possible in adversarial and anonymous dynamic networks with logarithmically limited bandwidth and internal memory.
{"title":"Anonymous adversarial dynamic networks with logarithmic memory and communication","authors":"Dariusz R. Kowalski , Miguel A. Mosteiro","doi":"10.1016/j.tcs.2025.115740","DOIUrl":"10.1016/j.tcs.2025.115740","url":null,"abstract":"<div><div>In seminal work on Adversarial Dynamic Networks, Kuhn, Lynch and Oshman (STOC 2010) [19] studied dynamic networks in which links are selected by an adversary and the number of network nodes is initially unknown. In such networks, they showed upper and lower bounds for computing the size of the network and any computable function of the nodes initial inputs. In this work, we address the same question in dynamic networks which additionally are: anonymous, possibly disconnected, and where internal memory and links’ bandwith are logarithmically limited. In the above framework, we study a fundamental communication principle – the All-to-all problem: each node has an input message to be delivered to all other nodes. (Once a node receives all inputs, any function can be computed locally.) Because of anonymity, each node needs to receive only a set of all input messages, each accompanied by a number of their initiating nodes (message multiplicity). We prove that this can be done deterministically in time proportional to the total number of messages’ bits multiplied by a small polynomial in networks’ parameters – namely, in the (initially unknown) number of nodes <em>n</em> and in the lower bound on the isoperimetric numbers of dynamically evolving graphs <em>i</em><sub>min</sub>. Our results prove that a polynomial bit-throughput is possible in adversarial and anonymous dynamic networks with logarithmically limited bandwidth and internal memory.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115740"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145908869","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 : 2026-01-01DOI: 10.1016/j.tcs.2025.115732
Gwenaël Richomme
The binomial notation represents the number of occurrences of the word u as a (scattered) subword in w. We first introduce and study possible uses of a geometrical interpretation of and when a and b are distinct letters. We then study the structure of the 2-binomial equivalence class of a binary word w (two words are 2-binomially equivalent if they have the same binomial coefficients, that is, the same numbers of occurrences, for each word of length at most 2). Especially we prove the existence of an isomorphism between the graph of the 2-binomial equivalence class of w with respect to a particular rewriting rule and the lattice of partitions of the integer with parts and greatest part bounded by . Finally we study binary fair words, the words over {a, b} having the same numbers of occurrences of ab and ba as subwords (). In particular, we prove a recent conjecture related to a special case of the least square approximation.
{"title":"On some 2-binomial coefficients of binary words: geometrical interpretation, partitions of integers, and fair words","authors":"Gwenaël Richomme","doi":"10.1016/j.tcs.2025.115732","DOIUrl":"10.1016/j.tcs.2025.115732","url":null,"abstract":"<div><div>The binomial notation <span><math><mrow><mo>(</mo><mfrac><mi>w</mi><mi>u</mi></mfrac><mo>)</mo></mrow></math></span> represents the number of occurrences of the word <em>u</em> as a (scattered) subword in <em>w</em>. We first introduce and study possible uses of a geometrical interpretation of <span><math><mrow><mo>(</mo><mfrac><mi>w</mi><mrow><mi>a</mi><mi>b</mi></mrow></mfrac><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><mfrac><mi>w</mi><mrow><mi>b</mi><mi>a</mi></mrow></mfrac><mo>)</mo></mrow></math></span> when <em>a</em> and <em>b</em> are distinct letters. We then study the structure of the 2-binomial equivalence class of a binary word <em>w</em> (two words are 2-binomially equivalent if they have the same binomial coefficients, that is, the same numbers of occurrences, for each word of length at most 2). Especially we prove the existence of an isomorphism between the graph of the 2-binomial equivalence class of <em>w</em> with respect to a particular rewriting rule and the lattice of partitions of the integer <span><math><mrow><mo>(</mo><mfrac><mi>w</mi><mrow><mi>a</mi><mi>b</mi></mrow></mfrac><mo>)</mo></mrow></math></span> with <span><math><mrow><mo>(</mo><mfrac><mi>w</mi><mi>a</mi></mfrac><mo>)</mo></mrow></math></span> parts and greatest part bounded by <span><math><mrow><mo>(</mo><mfrac><mi>w</mi><mi>b</mi></mfrac><mo>)</mo></mrow></math></span>. Finally we study binary fair words, the words over {<em>a, b</em>} having the same numbers of occurrences of <em>ab</em> and <em>ba</em> as subwords (<span><math><mrow><mrow><mo>(</mo><mfrac><mi>w</mi><mrow><mi>a</mi><mi>b</mi></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mfrac><mi>w</mi><mrow><mi>b</mi><mi>a</mi></mrow></mfrac><mo>)</mo></mrow></mrow></math></span>). In particular, we prove a recent conjecture related to a special case of the least square approximation.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115732"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145908868","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 : 2025-12-31DOI: 10.1016/j.tcs.2025.115731
Bojan Bašić , Danijela Popović
We consider the game Hackenforb, which has been introduced recently and for which it has been shown that it is capable of mimicking a vast spectrum of impartial combinatorial games. We show that it is capable of mimicking any game that has the property that from every position there exists a move to an ending position; for misère play, this amounts to any game in which the players are entitled to resign the game on any move.
{"title":"Hackenforb the chameleon: A game capable of mimicking (practically) any misère game","authors":"Bojan Bašić , Danijela Popović","doi":"10.1016/j.tcs.2025.115731","DOIUrl":"10.1016/j.tcs.2025.115731","url":null,"abstract":"<div><div>We consider the game Hackenforb, which has been introduced recently and for which it has been shown that it is capable of mimicking a vast spectrum of impartial combinatorial games. We show that it is capable of mimicking any game that has the property that from every position there exists a move to an ending position; for misère play, this amounts to any game in which the players are entitled to resign the game on any move.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115731"},"PeriodicalIF":1.0,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980989","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 : 2025-12-31DOI: 10.1016/j.tcs.2025.115727
Ramin Javadi, Hossein Shokouhi
Given a bipartite graph , a left-perfect many-to-one matching is a subset M⊆E such that each vertex in U is incident with exactly one edge in M. If U is partitioned into some groups, the matching is called fair if for every v ∈ V, the difference between the number of vertices matched with v in any two groups does not exceed a given threshold. In this paper, we investigate parameterized complexity of the fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of U, combined. Also, it is W[1]-hard with respect to the path-width, the number of groups and the maximum degree of U, combined. On the positive side, we prove that the problem is FPT with respect to the treewidth and the maximum degree of V. Also, it is FPT with respect to the neighborhood diversity of the input graph (which implies being FPT with respect to the vertex cover number and modular-width). Finally, we prove that the problem is FPT with respect to the tree-depth and the number of groups.
{"title":"Parameterized complexity of fair many-to-one matchings","authors":"Ramin Javadi, Hossein Shokouhi","doi":"10.1016/j.tcs.2025.115727","DOIUrl":"10.1016/j.tcs.2025.115727","url":null,"abstract":"<div><div>Given a bipartite graph <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>U</mi><mo>∪</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span>, a left-perfect many-to-one matching is a subset <em>M</em>⊆<em>E</em> such that each vertex in <em>U</em> is incident with exactly one edge in <em>M</em>. If <em>U</em> is partitioned into some groups, the matching is called fair if for every <em>v</em> ∈ <em>V</em>, the difference between the number of vertices matched with <em>v</em> in any two groups does not exceed a given threshold. In this paper, we investigate parameterized complexity of the fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of <em>U</em>, combined. Also, it is W[1]-hard with respect to the path-width, the number of groups and the maximum degree of <em>U</em>, combined. On the positive side, we prove that the problem is FPT with respect to the treewidth and the maximum degree of <em>V</em>. Also, it is FPT with respect to the neighborhood diversity of the input graph (which implies being FPT with respect to the vertex cover number and modular-width). Finally, we prove that the problem is FPT with respect to the tree-depth and the number of groups.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115727"},"PeriodicalIF":1.0,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145908867","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}
In Two-Sets Cut-Uncut, we are given an undirected graph and two terminal sets S and T. The task is to find a minimum cut C in G (if there is any) separating S from T under the following “uncut” condition. In the graph (V, E∖C), the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum s-t-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters.
On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP coNP/poly).
{"title":"The parameterized complexity landscape of two-sets cut-uncut","authors":"Matthias Bentert , Fedor V. Fomin , Fanny Hauser , Saket Saurabh","doi":"10.1016/j.tcs.2025.115726","DOIUrl":"10.1016/j.tcs.2025.115726","url":null,"abstract":"<div><div>In <span>Two-Sets Cut-Uncut</span>, we are given an undirected graph <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span> and two terminal sets <em>S</em> and <em>T</em>. The task is to find a minimum cut <em>C</em> in <em>G</em> (if there is any) separating <em>S</em> from <em>T</em> under the following “uncut” condition. In the graph (<em>V, E</em>∖<em>C</em>), the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem <span>Minimum s-t-Cut</span>, <span>Two-Sets Cut-Uncut</span> is computationally challenging. In particular, even deciding whether such a cut of <em>any size</em> exists, is already NP-complete. We initiate a systematic study of <span>Two-Sets Cut-Uncut</span> within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters.</div><div>On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP <span><math><mrow><mo>¬</mo><mo>⊆</mo></mrow></math></span> coNP/poly).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1065 ","pages":"Article 115726"},"PeriodicalIF":1.0,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927310","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 : 2025-12-29DOI: 10.1016/j.tcs.2025.115722
Mingchao Zhou, Zhao Zhang
<div><div>We propose the <em>fair stochastic maximum coverage</em> (FSMC) problem. Given an element set <em>U</em> and a collection of sets <span><math><mi>S</mi></math></span> partitioned into ℓ disjoint groups <span><math><mrow><msub><mi>S</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>S</mi><mi>ℓ</mi></msub></mrow></math></span>, each set <em>S</em> has a cost <em>c<sub>S</sub></em>. For a set of demand scenarios Ω, each occurring with probability <em>p<sub>ω</sub></em>, let <em>r</em><sub><em>e,ω</em></sub> ∈ {0, 1} indicate whether element <em>e</em> is demanded in scenario <em>ω</em>. The objective is to choose a subcollection <span><math><mrow><mi>F</mi><mo>⊆</mo><mi>S</mi></mrow></math></span> maximizing the expected number of demanded elements covered, subject to a global budget <span><math><mrow><msub><mo>∑</mo><mrow><mi>S</mi><mo>∈</mo><mi>F</mi></mrow></msub><msub><mi>c</mi><mi>S</mi></msub><mo>≤</mo><mi>B</mi></mrow></math></span> and fairness constraints <span><math><mrow><msub><mi>L</mi><mi>i</mi></msub><mo>≤</mo><msub><mo>∑</mo><mrow><mi>S</mi><mo>∈</mo><mi>F</mi><mo>∩</mo><msub><mi>S</mi><mi>i</mi></msub></mrow></msub><msub><mi>c</mi><mi>S</mi></msub><mo>≤</mo><msub><mi>B</mi><mi>i</mi></msub></mrow></math></span> for each group. FSMC strictly generalizes the maximum group set cover (MGSC) problem by incorporating both stochasticity and lower-bound fairness constraints. Under two mild assumptions, we design a deterministic <span><math><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>)</mo></mrow><mi>f</mi></msup><mo>)</mo></mrow></mrow></math></span>-approximation algorithm running in <span><math><mrow><mi>O</mi><mo>(</mo><mi>T</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></math></span> time, where <em>f</em> is the maximum frequency of any element, <em>T</em> is the LP-solving time, and <span><math><mrow><mi>n</mi><mo>=</mo><mo>|</mo><mi>S</mi><mo>|</mo></mrow></math></span>. Relaxing one assumption yields a bi-criteria algorithm with the same ratio and total budget at most 2<em>B</em>. For the cardinality version (FCSMC), our algorithm achieves an unconditional <span><math><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>)</mo></mrow><mi>f</mi></msup><mo>)</mo></mrow></math></span>-approximation. When <em>L<sub>i</sub></em> ≡ 0, FSMC reduces to the stochastic MGSC problem, where we obtain approximation ratio <span><math><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>)</mo></mrow><mi>f</mi></msup><mo>)</mo></mrow><mo>></mo><mn>0.316</mn></mrow></math></span> under the relaxed assumption, and <span><math><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1<
{"title":"Approximation algorithm for fair stochastic maximum coverage problem","authors":"Mingchao Zhou, Zhao Zhang","doi":"10.1016/j.tcs.2025.115722","DOIUrl":"10.1016/j.tcs.2025.115722","url":null,"abstract":"<div><div>We propose the <em>fair stochastic maximum coverage</em> (FSMC) problem. Given an element set <em>U</em> and a collection of sets <span><math><mi>S</mi></math></span> partitioned into ℓ disjoint groups <span><math><mrow><msub><mi>S</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>S</mi><mi>ℓ</mi></msub></mrow></math></span>, each set <em>S</em> has a cost <em>c<sub>S</sub></em>. For a set of demand scenarios Ω, each occurring with probability <em>p<sub>ω</sub></em>, let <em>r</em><sub><em>e,ω</em></sub> ∈ {0, 1} indicate whether element <em>e</em> is demanded in scenario <em>ω</em>. The objective is to choose a subcollection <span><math><mrow><mi>F</mi><mo>⊆</mo><mi>S</mi></mrow></math></span> maximizing the expected number of demanded elements covered, subject to a global budget <span><math><mrow><msub><mo>∑</mo><mrow><mi>S</mi><mo>∈</mo><mi>F</mi></mrow></msub><msub><mi>c</mi><mi>S</mi></msub><mo>≤</mo><mi>B</mi></mrow></math></span> and fairness constraints <span><math><mrow><msub><mi>L</mi><mi>i</mi></msub><mo>≤</mo><msub><mo>∑</mo><mrow><mi>S</mi><mo>∈</mo><mi>F</mi><mo>∩</mo><msub><mi>S</mi><mi>i</mi></msub></mrow></msub><msub><mi>c</mi><mi>S</mi></msub><mo>≤</mo><msub><mi>B</mi><mi>i</mi></msub></mrow></math></span> for each group. FSMC strictly generalizes the maximum group set cover (MGSC) problem by incorporating both stochasticity and lower-bound fairness constraints. Under two mild assumptions, we design a deterministic <span><math><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>)</mo></mrow><mi>f</mi></msup><mo>)</mo></mrow></mrow></math></span>-approximation algorithm running in <span><math><mrow><mi>O</mi><mo>(</mo><mi>T</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></math></span> time, where <em>f</em> is the maximum frequency of any element, <em>T</em> is the LP-solving time, and <span><math><mrow><mi>n</mi><mo>=</mo><mo>|</mo><mi>S</mi><mo>|</mo></mrow></math></span>. Relaxing one assumption yields a bi-criteria algorithm with the same ratio and total budget at most 2<em>B</em>. For the cardinality version (FCSMC), our algorithm achieves an unconditional <span><math><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>)</mo></mrow><mi>f</mi></msup><mo>)</mo></mrow></math></span>-approximation. When <em>L<sub>i</sub></em> ≡ 0, FSMC reduces to the stochastic MGSC problem, where we obtain approximation ratio <span><math><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>f</mi></mfrac><mo>)</mo></mrow><mi>f</mi></msup><mo>)</mo></mrow><mo>></mo><mn>0.316</mn></mrow></math></span> under the relaxed assumption, and <span><math><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mn>1<","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1065 ","pages":"Article 115722"},"PeriodicalIF":1.0,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927412","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 : 2025-12-27DOI: 10.1016/j.tcs.2025.115729
Petra Wiederhold , Tonatiuh Matos-Wiederhold
The paper presents a border tracing algorithm for sets of polygonal tiles which are identified with the nodes of an oriented adjacency graph. This algorithm is the first one proposed for objects in an arbitrary edge-to-edge tiling of convex polygons equipped with an adjacency relation, although it adapts a known method to determine so-called border meshes of subgraphs of oriented graphs. We employ several adjacencies to study the corresponding border meshes and contours resulting from the algorithm, and compare them to another type of boundary. Special attention is given to border tracing for connected objects in rectangular, triangular or hexagonal tilings, using diverse adjacency based connectivity types. The latter could be useful, for example, to analyse digital images made of square, rectangular, triangular or hexagonal pixels.
{"title":"Border tracing in oriented adjacency graphs of polygonal tilings","authors":"Petra Wiederhold , Tonatiuh Matos-Wiederhold","doi":"10.1016/j.tcs.2025.115729","DOIUrl":"10.1016/j.tcs.2025.115729","url":null,"abstract":"<div><div>The paper presents a border tracing algorithm for sets of polygonal tiles which are identified with the nodes of an oriented adjacency graph. This algorithm is the first one proposed for objects in an arbitrary edge-to-edge tiling of convex polygons equipped with an adjacency relation, although it adapts a known method to determine so-called border meshes of subgraphs of oriented graphs. We employ several adjacencies to study the corresponding border meshes and contours resulting from the algorithm, and compare them to another type of boundary. Special attention is given to border tracing for connected objects in rectangular, triangular or hexagonal tilings, using diverse adjacency based connectivity types. The latter could be useful, for example, to analyse digital images made of square, rectangular, triangular or hexagonal pixels.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1065 ","pages":"Article 115729"},"PeriodicalIF":1.0,"publicationDate":"2025-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927309","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 : 2025-12-26DOI: 10.1016/j.tcs.2025.115728
Guozhen Zhang , Tao Wen , Dajin Wang
<div><div>A network’s connectivity is a crucial indicator for its reliability. There are various ways to measure the connectivity, and the <em>extra connectivity</em> and the <em>structure connectivity</em> are two variants of the classic, original connectivity. In this paper, we incorporate the two to study the <em><strong>extra structure connectivity</strong></em> for the <em>modified bubble-sort network MB<sub>n</sub></em>, which is one of the proposed models for the interconnection network of multiprocessor systems. Let <em>H</em> be a connected subgraph of a graph <em>G</em>, and let <span><math><mrow><mi>F</mi><mo>=</mo><mo>{</mo><msub><mi>H</mi><mn>1</mn></msub><mo>,</mo><msub><mi>H</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>H</mi><mi>j</mi></msub><mo>}</mo></mrow></math></span> be a set of subgraphs of <em>G</em>, such that 1) each <em>H<sub>i</sub></em> is isomorphic to <em>H</em>; 2) <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> is disconnected; and 3) each component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> has at least <span><math><mrow><mi>g</mi><mo>+</mo><mn>1</mn></mrow></math></span> nodes. The minimum <em>j</em> for such an <em>F</em> is called the <em>g</em>-extra <em>H</em>-structure connectivity of <em>G</em>, denoted <em>κ<sub>g</sub></em>(<em>G</em>; <em>H</em>). Let <span><math><mrow><mi>F</mi><mo>=</mo><mo>{</mo><msub><mi>J</mi><mn>1</mn></msub><mo>,</mo><msub><mi>J</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>J</mi><mi>k</mi></msub><mo>}</mo></mrow></math></span> be a set of subgraphs of <em>G</em>, such that 1) each <em>J<sub>i</sub></em> is isomorphic to a subgraph of <em>H</em>; 2) <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> is disconnected; and 3) each component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> has at least <span><math><mrow><mi>g</mi><mo>+</mo><mn>1</mn></mrow></math></span> nodes. The minimum <em>k</em> for such an <em>F</em> is called the <em>g</em>-extra <em>H</em>-substructure connectivity of <em>G</em>, denoted <span><math><mrow><msubsup><mi>κ</mi><mrow><mi>g</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. We will prove that for <em>P</em><sub>3<em>l</em></sub>, a path on 3<em>l</em> nodes, <span><math><mstyle><mrow><msub><mi>κ</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>M</mi><mspace></mspace><msub><mi>B</mi><mi>n</mi></msub><mo>;</mo><msub><mi>P</mi><mrow><mn>3</mn><mi>l</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><msubsup><mi>κ</mi><mrow><mn>1</mn></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>M</mi><mspace></mspace><msub><mi>B</mi><mi>n</mi></msub><mo>;</mo><msub><mi>P</mi><mrow><mn>3</mn><mi>l</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mi>l</mi></mfrac><mo>⌉</mo></mrow></mrow></mstyle></math></span> for <em>n</em> ≥ 9 and
{"title":"Extra path-structure connectivity of modified bubble-sort networks","authors":"Guozhen Zhang , Tao Wen , Dajin Wang","doi":"10.1016/j.tcs.2025.115728","DOIUrl":"10.1016/j.tcs.2025.115728","url":null,"abstract":"<div><div>A network’s connectivity is a crucial indicator for its reliability. There are various ways to measure the connectivity, and the <em>extra connectivity</em> and the <em>structure connectivity</em> are two variants of the classic, original connectivity. In this paper, we incorporate the two to study the <em><strong>extra structure connectivity</strong></em> for the <em>modified bubble-sort network MB<sub>n</sub></em>, which is one of the proposed models for the interconnection network of multiprocessor systems. Let <em>H</em> be a connected subgraph of a graph <em>G</em>, and let <span><math><mrow><mi>F</mi><mo>=</mo><mo>{</mo><msub><mi>H</mi><mn>1</mn></msub><mo>,</mo><msub><mi>H</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>H</mi><mi>j</mi></msub><mo>}</mo></mrow></math></span> be a set of subgraphs of <em>G</em>, such that 1) each <em>H<sub>i</sub></em> is isomorphic to <em>H</em>; 2) <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> is disconnected; and 3) each component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> has at least <span><math><mrow><mi>g</mi><mo>+</mo><mn>1</mn></mrow></math></span> nodes. The minimum <em>j</em> for such an <em>F</em> is called the <em>g</em>-extra <em>H</em>-structure connectivity of <em>G</em>, denoted <em>κ<sub>g</sub></em>(<em>G</em>; <em>H</em>). Let <span><math><mrow><mi>F</mi><mo>=</mo><mo>{</mo><msub><mi>J</mi><mn>1</mn></msub><mo>,</mo><msub><mi>J</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>J</mi><mi>k</mi></msub><mo>}</mo></mrow></math></span> be a set of subgraphs of <em>G</em>, such that 1) each <em>J<sub>i</sub></em> is isomorphic to a subgraph of <em>H</em>; 2) <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> is disconnected; and 3) each component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> has at least <span><math><mrow><mi>g</mi><mo>+</mo><mn>1</mn></mrow></math></span> nodes. The minimum <em>k</em> for such an <em>F</em> is called the <em>g</em>-extra <em>H</em>-substructure connectivity of <em>G</em>, denoted <span><math><mrow><msubsup><mi>κ</mi><mrow><mi>g</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. We will prove that for <em>P</em><sub>3<em>l</em></sub>, a path on 3<em>l</em> nodes, <span><math><mstyle><mrow><msub><mi>κ</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>M</mi><mspace></mspace><msub><mi>B</mi><mi>n</mi></msub><mo>;</mo><msub><mi>P</mi><mrow><mn>3</mn><mi>l</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><msubsup><mi>κ</mi><mrow><mn>1</mn></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>M</mi><mspace></mspace><msub><mi>B</mi><mi>n</mi></msub><mo>;</mo><msub><mi>P</mi><mrow><mn>3</mn><mi>l</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mi>l</mi></mfrac><mo>⌉</mo></mrow></mrow></mstyle></math></span> for <em>n</em> ≥ 9 and","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1065 ","pages":"Article 115728"},"PeriodicalIF":1.0,"publicationDate":"2025-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927308","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 : 2025-12-24DOI: 10.1016/j.tcs.2025.115723
Bar Menashe, Meirav Zehavi
We study the optimization version of the classic Disjoint Paths problem, known as Min-Sum Disjoint Paths, as well as its restriction to shortest paths, known as Disjoint Shortest Paths. Both problems are notoriously hard in the sense that very few positive results are known in their context even when confined to grids, in contrast to the classic Disjoint Paths problem, despite significant research efforts in recent years. In light of this, we focus on restricted graph classes, being subclasses of chordal graphs: specifically, we consider the classes of split graphs, well-partitioned chordal graphs, and threshold graphs. For each of the two problems and each of these graph classes, we provide either a polynomial-time algorithm or a fixed-parameter algorithm (when a polynomial-time algorithm is unlikely to exist).
{"title":"Min-Sum disjoint paths on subclasses of chordal graphs","authors":"Bar Menashe, Meirav Zehavi","doi":"10.1016/j.tcs.2025.115723","DOIUrl":"10.1016/j.tcs.2025.115723","url":null,"abstract":"<div><div>We study the optimization version of the classic <span>Disjoint Paths</span> problem, known as <span>Min-Sum Disjoint Paths</span>, as well as its restriction to shortest paths, known as <span>Disjoint Shortest Paths</span>. Both problems are notoriously hard in the sense that very few positive results are known in their context even when confined to grids, in contrast to the classic <span>Disjoint Paths</span> problem, despite significant research efforts in recent years. In light of this, we focus on restricted graph classes, being subclasses of chordal graphs: specifically, we consider the classes of split graphs, well-partitioned chordal graphs, and threshold graphs. For each of the two problems and each of these graph classes, we provide either a polynomial-time algorithm or a fixed-parameter algorithm (when a polynomial-time algorithm is unlikely to exist).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115723"},"PeriodicalIF":1.0,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885130","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 : 2025-12-24DOI: 10.1016/j.tcs.2025.115724
Zhidan Feng , Henning Fernau , Kevin Mann , Xingqin Qi
Signed graphs have been introduced to enrich graph structures expressing relationships between persons or general social entities, introducing edge signs to reflect the nature of the relationship, e.g., friendship or enmity. Independently, offensive alliances have been defined and studied for undirected, unsigned graphs. We join both lines of research and define offensive alliances in signed graphs, hence considering the nature of relationships. Apart from some combinatorial results, mainly on k-balanced and k-anti-balanced signed graphs (where the latter is a newly introduced family of signed graphs), we focus on the algorithmic complexity of finding smallest offensive alliances, looking at a number of parameterizations. While the parameter solution size leads to an FPTresult for unsigned graphs, we obtain -completeness for the signed setting. We introduce new parameters for signed graphs, e.g., distance to weakly balanced signed graphs, that could be of independent interest. We show that these parameters yield FPTresults. Here, we make use of the recently introduced parameter neighborhood diversity for signed graphs.
{"title":"Offensive alliances in signed graphs","authors":"Zhidan Feng , Henning Fernau , Kevin Mann , Xingqin Qi","doi":"10.1016/j.tcs.2025.115724","DOIUrl":"10.1016/j.tcs.2025.115724","url":null,"abstract":"<div><div>Signed graphs have been introduced to enrich graph structures expressing relationships between persons or general social entities, introducing edge signs to reflect the nature of the relationship, e.g., friendship or enmity. Independently, offensive alliances have been defined and studied for undirected, unsigned graphs. We join both lines of research and define offensive alliances in signed graphs, hence considering the nature of relationships. Apart from some combinatorial results, mainly on <em>k</em>-balanced and <em>k</em>-anti-balanced signed graphs (where the latter is a newly introduced family of signed graphs), we focus on the algorithmic complexity of finding smallest offensive alliances, looking at a number of parameterizations. While the parameter solution size leads to an <span>FPT</span>result for unsigned graphs, we obtain <span><math><mrow><mi>W</mi><mo>[</mo><mn>2</mn><mo>]</mo></mrow></math></span>-completeness for the signed setting. We introduce new parameters for signed graphs, e.g., distance to weakly balanced signed graphs, that could be of independent interest. We show that these parameters yield <span>FPT</span>results. Here, we make use of the recently introduced parameter neighborhood diversity for signed graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1065 ","pages":"Article 115724"},"PeriodicalIF":1.0,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886131","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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