Pub Date : 2026-01-03DOI: 10.1016/j.tcs.2025.115720
Jonathan Wagner, Reshef Meir
We present a strategy-proof public goods budgeting mechanism where agents determine both the total volume of expanses and the specific allocation. It is constructed as a modification of VCG to a non-typical environment, namely where we do not assume quasi-linear utilities nor direct revelation. We further show that under plausible assumptions it satisfies strategyproofness in strictly dominant strategies, and consequently implements the social optimum as a Coalition-Proof Nash Equilibrium. A primary (albeit not an exclusive) motivation of our model is Participatory Budgeting, where members of a community collectively decide the spending policy of public tax dollars. While incentives alignment in our mechanism, as in classic VCG, is achieved via individual payments we charge from agents, in a PB context that seems unreasonable. Our second main result thus provides that, under further specifications relevant in that context, these payments will vanish in large populations. In the last section we expand the mechanism’s definition to a class of mechanisms in which the designer can prioritize certain outcomes she sees as desirable. In particular we give the example of favoring equitable/egalitarian allocations.
{"title":"Strategy-proof budgeting via a VCG-like mechanism","authors":"Jonathan Wagner, Reshef Meir","doi":"10.1016/j.tcs.2025.115720","DOIUrl":"10.1016/j.tcs.2025.115720","url":null,"abstract":"<div><div>We present a strategy-proof public goods budgeting mechanism where agents determine both the total volume of expanses and the specific allocation. It is constructed as a modification of VCG to a non-typical environment, namely where we do not assume quasi-linear utilities nor direct revelation. We further show that under plausible assumptions it satisfies strategyproofness in strictly dominant strategies, and consequently implements the social optimum as a Coalition-Proof Nash Equilibrium. A primary (albeit not an exclusive) motivation of our model is Participatory Budgeting, where members of a community collectively decide the spending policy of public tax dollars. While incentives alignment in our mechanism, as in classic VCG, is achieved via individual payments we charge from agents, in a PB context that seems unreasonable. Our second main result thus provides that, under further specifications relevant in that context, these payments will vanish in large populations. In the last section we expand the mechanism’s definition to a class of mechanisms in which the designer can prioritize certain outcomes she sees as desirable. In particular we give the example of favoring equitable/egalitarian allocations.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115720"},"PeriodicalIF":1.0,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039653","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 : 2026-01-02DOI: 10.1016/j.tcs.2025.115725
Xiangping Chu, Qingguo Li
In this paper, we investigate the product of H-posets and their function spaces. Our main result establishes that the function space [X → Y], consisting of all Scott continuous functions from X to Y, is an H-poset when X is a continuous poset and Y is an H-poset with a smallest element. This finding generalizes earlier results reported by Kou, Liu, and Luo. Building on these insights, we derive a cartesian closed full subcategory of posets and Scott continuous functions. Additionally, we prove that the finite product of meet continuous H-posets remains a meet continuous H-poset.
{"title":"The category of H-posets","authors":"Xiangping Chu, Qingguo Li","doi":"10.1016/j.tcs.2025.115725","DOIUrl":"10.1016/j.tcs.2025.115725","url":null,"abstract":"<div><div>In this paper, we investigate the product of <em>H</em>-posets and their function spaces. Our main result establishes that the function space [<em>X</em> → <em>Y</em>], consisting of all Scott continuous functions from <em>X</em> to <em>Y</em>, is an <em>H</em>-poset when <em>X</em> is a continuous poset and <em>Y</em> is an <em>H</em>-poset with a smallest element. This finding generalizes earlier results reported by Kou, Liu, and Luo. Building on these insights, we derive a cartesian closed full subcategory of posets and Scott continuous functions. Additionally, we prove that the finite product of meet continuous <em>H</em>-posets remains a meet continuous <em>H</em>-poset.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115725"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145908870","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 : 2026-01-02DOI: 10.1016/j.tcs.2025.115730
Marcos E. González Laffitte , René González-Martínez , Amanda Montejano
We study local and global amoebas, which are graphs that have unique properties with respect to interpolation techniques in graphs. Our work includes a proof that almost every graph is not an amoeba, and the identification of a special type of edge-replacement called weird-edge-replacements. Additionally, we provide an infinite family of trees that are both weird local and global amoebas. Our contributions extend to the development and implementation of several algorithms for detecting local and global amoebas, which are made available in a public repository along with multiple examples.
{"title":"On the detection of local and global amoebas: Theoretical insights and practical algorithms","authors":"Marcos E. González Laffitte , René González-Martínez , Amanda Montejano","doi":"10.1016/j.tcs.2025.115730","DOIUrl":"10.1016/j.tcs.2025.115730","url":null,"abstract":"<div><div>We study local and global amoebas, which are graphs that have unique properties with respect to interpolation techniques in graphs. Our work includes a proof that almost every graph is not an amoeba, and the identification of a special type of edge-replacement called weird-edge-replacements. Additionally, we provide an infinite family of trees that are both weird local and global amoebas. Our contributions extend to the development and implementation of several algorithms for detecting local and global amoebas, which are made available in a public repository along with multiple examples.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115730"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928840","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 : 2026-01-02DOI: 10.1016/j.tcs.2025.115741
Hao Wang, Yan Wang, Baolei Cheng, Jianxi Fan
Spanning trees are completely independent spanning trees (CISTs) if and only if they are edge-disjoint and each node is internal in at most one tree. CISTs have a wide range of applications in routing protection, data transmission, etc., and can improve reliability, fault tolerance, and information security. Line graphs have received increasing attention in recent years for the construction of multiple CISTs, as they are more likely to satisfy the structural conditions required for their existence. This paper introduces an algorithm for constructing multiple CISTs in the line graph of the complete tripartite graph (denoted by ), using multiple two-dimensional matrices to guide the construction process. Furthermore, it presents a method for constructing multiple CISTs in the line graph of the complete multipartite graph (denoted by , when φ ≥ 4), utilizing edge-disjoint Hamiltonian cycles of the complete graph. In our simulation study, we employed multiple CISTs as transmission paths and compared their performance with shortest-path routing in terms of transmission latency and resilience to node failures.
{"title":"Completely independent spanning trees in the line graph of complete multipartite graphs","authors":"Hao Wang, Yan Wang, Baolei Cheng, Jianxi Fan","doi":"10.1016/j.tcs.2025.115741","DOIUrl":"10.1016/j.tcs.2025.115741","url":null,"abstract":"<div><div>Spanning trees <span><math><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>T</mi><mi>t</mi></msub></mrow></math></span> are completely independent spanning trees (CISTs) if and only if they are edge-disjoint and each node is internal in at most one tree. CISTs have a wide range of applications in routing protection, data transmission, etc., and can improve reliability, fault tolerance, and information security. Line graphs have received increasing attention in recent years for the construction of multiple CISTs, as they are more likely to satisfy the structural conditions required for their existence. This paper introduces an algorithm for constructing multiple CISTs in the line graph of the complete tripartite graph <span><math><msub><mi>K</mi><mrow><msub><mi>n</mi><mn>3</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub></math></span> (denoted by <span><math><mrow><mi>L</mi><mo>(</mo><msub><mi>K</mi><mrow><msub><mi>n</mi><mn>3</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub><mo>)</mo></mrow></math></span>), using multiple two-dimensional matrices to guide the construction process. Furthermore, it presents a method for constructing multiple CISTs in the line graph of the complete multipartite graph <span><math><msub><mi>K</mi><mrow><msub><mi>n</mi><mi>φ</mi></msub><mo>,</mo><msub><mi>n</mi><mrow><mi>φ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub></math></span> (denoted by <span><math><mrow><mi>L</mi><mo>(</mo><msub><mi>K</mi><mrow><msub><mi>n</mi><mi>φ</mi></msub><mo>,</mo><msub><mi>n</mi><mrow><mi>φ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub><mo>)</mo></mrow></math></span>, when φ ≥ 4), utilizing edge-disjoint Hamiltonian cycles of the complete graph. In our simulation study, we employed multiple CISTs as transmission paths and compared their performance with shortest-path routing in terms of transmission latency and resilience to node failures.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115741"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928842","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 : 2026-01-02DOI: 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}
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