Pub Date : 2025-12-14DOI: 10.1016/j.tcs.2025.115708
Byung-Cheon Choi , Myoung-Ju Park
We investigate a family of single-machine scheduling problems that combine convex resource consumption with job rejection, where the performance measure is the total completion time of the accepted jobs. Across four natural variants-distinguished by whether the resource consumption cost and rejection cost appear in the objective or as budget constraints-we analyze their computational complexity and develop efficient algorithms. We show that one variant is polynomially solvable, two variants are weakly NP-hard but admit fully polynomial-time approximation schemes (FPTASs), and the remaining variant admits an FPTAS while its exact complexity remains open.
{"title":"Minimizing total completion time in single-machine scheduling with convex resource consumption and job rejection","authors":"Byung-Cheon Choi , Myoung-Ju Park","doi":"10.1016/j.tcs.2025.115708","DOIUrl":"10.1016/j.tcs.2025.115708","url":null,"abstract":"<div><div>We investigate a family of single-machine scheduling problems that combine convex resource consumption with job rejection, where the performance measure is the total completion time of the accepted jobs. Across four natural variants-distinguished by whether the resource consumption cost and rejection cost appear in the objective or as budget constraints-we analyze their computational complexity and develop efficient algorithms. We show that one variant is polynomially solvable, two variants are weakly NP-hard but admit fully polynomial-time approximation schemes (FPTASs), and the remaining variant admits an FPTAS while its exact complexity remains open.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115708"},"PeriodicalIF":1.0,"publicationDate":"2025-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841934","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 : 2025-12-13DOI: 10.1016/j.tcs.2025.115682
Nina Hammer, Frank Kammer, Johannes Meintrup
We present a novel space-efficient graph coarsening technique for n-vertex planar graphs G, called cloud partition, which partitions the vertices V(G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/log n) vertices. The combination of is referred to as a cloud decomposition.
For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition constructed for a planar graph G we are able to find a balanced separator of G in O(n/log n) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to H-minor-free graphs for any fixed graph H. This allows us to construct the succinct encoding scheme for H-minor-free graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of Θ(log n).
As an additional application of our cloud decomposition we show that, for H-minor-free graphs, a tree decomposition of width for any ϵ > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width for graphs of treewidth in sublinear space and polynomial time.
{"title":"Space-efficient graph coarsening with applications to succinct planar encodings","authors":"Nina Hammer, Frank Kammer, Johannes Meintrup","doi":"10.1016/j.tcs.2025.115682","DOIUrl":"10.1016/j.tcs.2025.115682","url":null,"abstract":"<div><div>We present a novel space-efficient graph coarsening technique for <em>n</em>-vertex planar graphs <em>G</em>, called <em>cloud partition</em>, which partitions the vertices <em>V</em>(<em>G</em>) into disjoint sets <em>C</em> of size <em>O</em>(log <em>n</em>) such that each <em>C</em> induces a connected subgraph of <em>G</em>. Using this partition <span><math><mi>P</mi></math></span> we construct a so-called <em>structure-maintaining minor F</em> of <em>G</em> via specific contractions within the disjoint sets such that <em>F</em> has <em>O</em>(<em>n</em>/log <em>n</em>) vertices. The combination of <span><math><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>P</mi><mo>)</mo></mrow></math></span> is referred to as a <em>cloud decomposition</em>.</div><div>For planar graphs we show that a cloud decomposition can be constructed in <em>O</em>(<em>n</em>) time and using <em>O</em>(<em>n</em>) bits. Given a cloud decomposition <span><math><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>P</mi><mo>)</mo></mrow></math></span> constructed for a planar graph <em>G</em> we are able to find a balanced separator of <em>G</em> in <em>O</em>(<em>n</em>/log <em>n</em>) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to <em>H</em>-minor-free graphs for any fixed graph <em>H</em>. This allows us to construct the succinct encoding scheme for <em>H</em>-minor-free graphs due to Blelloch and Farzan (CPM 2010) in <em>O</em>(<em>n</em>) time and <em>O</em>(<em>n</em>) bits improving both runtime and space by a factor of Θ(log <em>n</em>).</div><div>As an additional application of our cloud decomposition we show that, for <em>H</em>-minor-free graphs, a tree decomposition of width <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></mrow></math></span> for any ϵ > 0 can be constructed in <em>O</em>(<em>n</em>) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width <span><math><mrow><mi>O</mi><mo>(</mo><mi>k</mi><msqrt><mi>n</mi></msqrt><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> for graphs of treewidth <span><math><mrow><mi>k</mi><mo>≤</mo><msqrt><mi>n</mi></msqrt></mrow></math></span> in sublinear space and polynomial time.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115682"},"PeriodicalIF":1.0,"publicationDate":"2025-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791948","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 : 2025-12-12DOI: 10.1016/j.tcs.2025.115702
Julien David , Mostafa Gholami , Loïck Lhote
Given a hypergraph, computing its transversal hypergraph is a classical problem whose exact worst-case complexity is still unknown today. In this article, for Erdös and Rényi random hypergraph models, we compute the average number of minimal transversals, that is the average size of the transversal hypergraph. We also study the average-case complexity of Berge algorithm, which sequentially enumerates the minimal transversals. We prove that the average-case complexity of Berge algorithm is quasi-linear in the average number of minimal traverses. Our contribution is also methodological since a wide range of analytic combinatorics techniques is for the first time used in this context.
{"title":"On the average-case complexity of Berge algorithm","authors":"Julien David , Mostafa Gholami , Loïck Lhote","doi":"10.1016/j.tcs.2025.115702","DOIUrl":"10.1016/j.tcs.2025.115702","url":null,"abstract":"<div><div>Given a hypergraph, computing its transversal hypergraph is a classical problem whose exact worst-case complexity is still unknown today. In this article, for Erdös and Rényi random hypergraph models, we compute the average number of minimal transversals, that is the average size of the transversal hypergraph. We also study the average-case complexity of Berge algorithm, which sequentially enumerates the minimal transversals. We prove that the average-case complexity of Berge algorithm is quasi-linear in the average number of minimal traverses. Our contribution is also methodological since a wide range of analytic combinatorics techniques is for the first time used in this context.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115702"},"PeriodicalIF":1.0,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791947","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 : 2025-12-11DOI: 10.1016/j.tcs.2025.115701
Vittorio Bilò , Lucaleonardo Bove , Cosimo Vinci
In this work, we consider the problem of improving the efficiency of utility-sharing games, by resorting to a limited amount of subsidies. Utility-sharing games model scenarios in which strategic and self-interested players interact with each other by selecting resources. Each resource produces a utility that depends on the number of players selecting it, as a non-negative, non-decreasing and concave function, and each of these players receives an equal share of this utility. As the players’ selfish behavior may lead to pure Nash equilibria whose total utility is sub-optimal, previous work has resorted to subsidies, incentivizing the use of some resources, to contrast this phenomenon. We focus on the case in which the budget used to provide subsidies is bounded. We consider a class of mechanisms, called α-subsidy mechanisms, that allocate the budget in such a way that each player’s payoff is re-scaled up to a factor α ≥ 1. We design a specific sub-class of α-subsidy mechanisms, that can be implemented efficiently and distributedly by each resource, and evaluate their efficiency by providing upper bounds on their price of anarchy. These bounds are parametrized by both α and the underlying utility functions and are shown to be best-possible for α-subsidy mechanisms. Finally, we apply our results to the particular case of monomial utility functions of degree p ∈ (0, 1), and derive bounds on the price of anarchy that are parametrized by p and α.
{"title":"Utility-sharing games: How to improve the efficiency with limited subsidies","authors":"Vittorio Bilò , Lucaleonardo Bove , Cosimo Vinci","doi":"10.1016/j.tcs.2025.115701","DOIUrl":"10.1016/j.tcs.2025.115701","url":null,"abstract":"<div><div>In this work, we consider the problem of improving the efficiency of utility-sharing games, by resorting to a limited amount of subsidies. Utility-sharing games model scenarios in which strategic and self-interested players interact with each other by selecting resources. Each resource produces a utility that depends on the number of players selecting it, as a non-negative, non-decreasing and concave function, and each of these players receives an equal share of this utility. As the players’ selfish behavior may lead to pure Nash equilibria whose total utility is sub-optimal, previous work has resorted to subsidies, incentivizing the use of some resources, to contrast this phenomenon. We focus on the case in which the budget used to provide subsidies is bounded. We consider a class of mechanisms, called <em>α</em>-subsidy mechanisms, that allocate the budget in such a way that each player’s payoff is re-scaled up to a factor <em>α</em> ≥ 1. We design a specific sub-class of <em>α</em>-subsidy mechanisms, that can be implemented efficiently and distributedly by each resource, and evaluate their efficiency by providing upper bounds on their price of anarchy. These bounds are parametrized by both <em>α</em> and the underlying utility functions and are shown to be best-possible for <em>α</em>-subsidy mechanisms. Finally, we apply our results to the particular case of monomial utility functions of degree <em>p</em> ∈ (0, 1), and derive bounds on the price of anarchy that are parametrized by <em>p</em> and <em>α</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115701"},"PeriodicalIF":1.0,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841933","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 : 2025-12-11DOI: 10.1016/j.tcs.2025.115700
Fengqin Zhang , Mingzu Zhang , Eddie Cheng
It is indispensable for a data center network to evaluate the number of processors in the remaining undamaged networks when it is attacked and links are faulted. The h-extra edge-connectivity of a connected graph G (λh(G)), the minimum cardinality of an edge set (F) of G whose removal will disconnect G and each remaining component of has at least h vertices, is a significant and accurate assessment to measure the fault tolerance and reliability of the interconnection network G in a data center network. It is usually closely related to the optimal solution of the edge isoperimetric problem for corresponding graphs. For the majority of graphs investigated to date, the lexicographic order yields the optimal solution to the edge isoperimetric problem; yet, this result does not extend to the quinary n-cube graph . Actually, in 2000, Carlson found a non-lexicographic order that provides an optimal solution to the edge isoperimetric problem for . Based on this fact, in this paper, by analyzing some properties of the non-lexicographic order optimal solution of the edge isoperimetric problem for , the following two conclusions are given: the exact values of h-extra edge-connectivity for are λh-optimal for , except for , and , and concentrate on a constant value for .
{"title":"Reliability analysis of the quinary n-cube networks with non-lexicographic order optimal solution of the edge isoperimetric problem","authors":"Fengqin Zhang , Mingzu Zhang , Eddie Cheng","doi":"10.1016/j.tcs.2025.115700","DOIUrl":"10.1016/j.tcs.2025.115700","url":null,"abstract":"<div><div>It is indispensable for a data center network to evaluate the number of processors in the remaining undamaged networks when it is attacked and links are faulted. The <em>h</em>-extra edge-connectivity of a connected graph <em>G</em> (<em>λ<sub>h</sub></em>(<em>G</em>)), the minimum cardinality of an edge set (<em>F</em>) of <em>G</em> whose removal will disconnect <em>G</em> and each remaining component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> has at least <em>h</em> vertices, is a significant and accurate assessment to measure the fault tolerance and reliability of the interconnection network <em>G</em> in a data center network. It is usually closely related to the optimal solution of the edge isoperimetric problem for corresponding graphs. For the majority of graphs investigated to date, the lexicographic order yields the optimal solution to the edge isoperimetric problem; yet, this result does not extend to the quinary <em>n</em>-cube graph <span><math><msubsup><mi>C</mi><mn>5</mn><mi>n</mi></msubsup></math></span>. Actually, in 2000, Carlson found a non-lexicographic order that provides an optimal solution to the edge isoperimetric problem for <span><math><msubsup><mi>C</mi><mn>5</mn><mi>n</mi></msubsup></math></span>. Based on this fact, in this paper, by analyzing some properties of the non-lexicographic order optimal solution of the edge isoperimetric problem for <span><math><msubsup><mi>C</mi><mn>5</mn><mi>n</mi></msubsup></math></span>, the following two conclusions are given: the exact values of <em>h</em>-extra edge-connectivity for <span><math><msubsup><mi>C</mi><mn>5</mn><mi>n</mi></msubsup></math></span> are <em>λ<sub>h</sub></em>-optimal for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mn>5</mn><mo>×</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>, except for <span><math><mrow><mi>h</mi><mo>=</mo><mn>5</mn><mo>·</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>h</mi><mo>=</mo><mn>5</mn><mo>·</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>h</mi><mo>=</mo><mn>5</mn><mo>·</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>3</mn></mrow></math></span>, and concentrate on a constant value <span><math><mrow><mn>2</mn><mo>×</mo><msup><mn>5</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> for <span><math><mrow><msup><mn>5</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>≤</mo><mi>h</mi><mo>≤</mo><mn>2</mn><mo>×</mo><msup><mn>5</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1065 ","pages":"Article 115700"},"PeriodicalIF":1.0,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842806","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 : 2025-12-10DOI: 10.1016/j.tcs.2025.115683
Ugo Dal Lago , Gabriele Vanoni
The interaction abstract machine is an automata-theoretic implementation of Girard’s geometry of interaction. We study one of its two formulations for the λ-calculus, namely the one obtained from the so-called call-by-value (or “boring”) translation of intuitionistic logic into linear logic. We prove the correctness of the resulting call-by-name machine, at the same time establishing an improvement bisimulation with Krivine’s abstract machine. The proof makes essential use of the definition of a novel relational property linking configurations of the two machines. Finally, exploiting the correspondence with non-idempotent intersection types, we prove that the interaction abstract machines coming from Girard’s two translations are strongly bisimilar.
{"title":"(Definitely not) Boring interaction abstract machines","authors":"Ugo Dal Lago , Gabriele Vanoni","doi":"10.1016/j.tcs.2025.115683","DOIUrl":"10.1016/j.tcs.2025.115683","url":null,"abstract":"<div><div>The interaction abstract machine is an automata-theoretic implementation of Girard’s geometry of interaction. We study one of its two formulations for the <em>λ</em>-calculus, namely the one obtained from the so-called call-by-value (or “boring”) translation of intuitionistic logic into linear logic. We prove the correctness of the resulting <em>call-by-name</em> machine, at the same time establishing an improvement bisimulation with Krivine’s abstract machine. The proof makes essential use of the definition of a novel relational property linking configurations of the two machines. Finally, exploiting the correspondence with non-idempotent intersection types, we prove that the interaction abstract machines coming from Girard’s two translations are strongly bisimilar.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115683"},"PeriodicalIF":1.0,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841935","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}
<div><div>In this paper, we consider the partial gathering problem of mobile agents in synchronous dynamic bidirectional ring networks. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all <em>k</em> agents distributed in the network terminate at a non-predetermined single node. The partial gathering problem requires, for a given positive integer <em>g</em> ( < <em>k</em>), that agents terminate in a configuration such that either at least <em>g</em> agents or no agent exists at each node. When <em>k</em> ≥ 2<em>g</em>, the requirement for the partial gathering problem is strictly weaker than that for the total gathering problem, and thus it is interesting to clarify the difference in the move complexity between them. So far, the partial gathering problem has been considered in static graphs. In this paper, we start considering partial gathering in dynamic graphs. As a first step, we consider this problem in 1-interval connected rings, that is, one of the links in a ring may be missing at each time step. In such networks, focusing on the relationship between the values of <em>k</em> and <em>g</em>, we fully characterize the solvability of the partial gathering problem and analyze the move complexity of the proposed algorithms when the problem can be solved. First, we show that the <em>g</em>-partial gathering problem is unsolvable when <em>k</em> ≤ 2<em>g</em>. Second, we show that the problem can be solved with <em>O</em>(<em>n</em>log <em>g</em>) time and the total number of <em>O</em>(<em>gn</em>log <em>g</em>) moves when <span><math><mrow><mn>2</mn><mi>g</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>3</mn><mi>g</mi><mo>−</mo><mn>2</mn></mrow></math></span>. Third, we show that the problem can be solved with <em>O</em>(<em>n</em>) time and the total number of <em>O</em>(<em>kn</em>) moves when <span><math><mrow><mn>3</mn><mi>g</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>8</mn><mi>g</mi><mo>−</mo><mn>4</mn></mrow></math></span>. Notice that since <span><math><mrow><mi>k</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>g</mi><mo>)</mo></mrow></math></span> holds when <span><math><mrow><mn>3</mn><mi>g</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>8</mn><mi>g</mi><mo>−</mo><mn>4</mn></mrow></math></span>, the move complexity <em>O</em>(<em>kn</em>) in this case can be represented also as <em>O</em>(<em>gn</em>). Finally, we show that the problem can be solved with <em>O</em>(<em>n</em>) time and the total number of <em>O</em>(<em>gn</em>) moves when <span><math><mrow><mi>k</mi><mo>≥</mo><mn>8</mn><mi>g</mi><mo>−</mo><mn>3</mn></mrow></math></span>. These results mean that the partial gathering problem can be solved also in dynamic rings when <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mi>g</mi><mo>+</mo><mn>1</mn></mrow></math></span>. In addition, agents require a total number of Ω(<em>gn</em>) (resp., Ω(<em>kn</em>)) moves to solve the par
{"title":"Partial gathering of mobile agents in dynamic rings","authors":"Masahiro Shibata , Yuichi Sudo , Junya Nakamura , Yonghwan Kim","doi":"10.1016/j.tcs.2025.115684","DOIUrl":"10.1016/j.tcs.2025.115684","url":null,"abstract":"<div><div>In this paper, we consider the partial gathering problem of mobile agents in synchronous dynamic bidirectional ring networks. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all <em>k</em> agents distributed in the network terminate at a non-predetermined single node. The partial gathering problem requires, for a given positive integer <em>g</em> ( < <em>k</em>), that agents terminate in a configuration such that either at least <em>g</em> agents or no agent exists at each node. When <em>k</em> ≥ 2<em>g</em>, the requirement for the partial gathering problem is strictly weaker than that for the total gathering problem, and thus it is interesting to clarify the difference in the move complexity between them. So far, the partial gathering problem has been considered in static graphs. In this paper, we start considering partial gathering in dynamic graphs. As a first step, we consider this problem in 1-interval connected rings, that is, one of the links in a ring may be missing at each time step. In such networks, focusing on the relationship between the values of <em>k</em> and <em>g</em>, we fully characterize the solvability of the partial gathering problem and analyze the move complexity of the proposed algorithms when the problem can be solved. First, we show that the <em>g</em>-partial gathering problem is unsolvable when <em>k</em> ≤ 2<em>g</em>. Second, we show that the problem can be solved with <em>O</em>(<em>n</em>log <em>g</em>) time and the total number of <em>O</em>(<em>gn</em>log <em>g</em>) moves when <span><math><mrow><mn>2</mn><mi>g</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>3</mn><mi>g</mi><mo>−</mo><mn>2</mn></mrow></math></span>. Third, we show that the problem can be solved with <em>O</em>(<em>n</em>) time and the total number of <em>O</em>(<em>kn</em>) moves when <span><math><mrow><mn>3</mn><mi>g</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>8</mn><mi>g</mi><mo>−</mo><mn>4</mn></mrow></math></span>. Notice that since <span><math><mrow><mi>k</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>g</mi><mo>)</mo></mrow></math></span> holds when <span><math><mrow><mn>3</mn><mi>g</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>8</mn><mi>g</mi><mo>−</mo><mn>4</mn></mrow></math></span>, the move complexity <em>O</em>(<em>kn</em>) in this case can be represented also as <em>O</em>(<em>gn</em>). Finally, we show that the problem can be solved with <em>O</em>(<em>n</em>) time and the total number of <em>O</em>(<em>gn</em>) moves when <span><math><mrow><mi>k</mi><mo>≥</mo><mn>8</mn><mi>g</mi><mo>−</mo><mn>3</mn></mrow></math></span>. These results mean that the partial gathering problem can be solved also in dynamic rings when <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mi>g</mi><mo>+</mo><mn>1</mn></mrow></math></span>. In addition, agents require a total number of Ω(<em>gn</em>) (resp., Ω(<em>kn</em>)) moves to solve the par","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1063 ","pages":"Article 115684"},"PeriodicalIF":1.0,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798611","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 : 2025-12-09DOI: 10.1016/j.tcs.2025.115677
A. Hosseinabadi, M. Haddadi
This paper investigates the support-preorder, a reflexive and transitive relation that can be defined on any finitely supported Cb-set using the notion of support. Finitely supported Cb-sets extend the concept of nominal sets. We introduce the category consisting of finitely supported Cb-sets and sp-preserving morphisms. We analyze its characteristics, including the existence of products and coproducts, and regularity, highlighting differences from the category of support-preordered nominal sets. A key distinction is the non-equivariance of the support-preorder on finitely supported Cb-sets. We also characterize strongly finitely supported Cb-sets, where the support-preorder is equivariant, and investigate the behavior of support under actions within these sets. Our research contributes to a deeper understanding of the structure and interrelations between nominal sets and finitely supported Cb-sets, with significant implications for fields such as formal systems and homotopy type theory.
{"title":"Support preorder on finitely supported Cb-Sets","authors":"A. Hosseinabadi, M. Haddadi","doi":"10.1016/j.tcs.2025.115677","DOIUrl":"10.1016/j.tcs.2025.115677","url":null,"abstract":"<div><div>This paper investigates the support-preorder, a reflexive and transitive relation that can be defined on any finitely supported <strong>Cb</strong>-set using the notion of support. Finitely supported <strong>Cb</strong>-sets extend the concept of nominal sets. We introduce the category <span><math><mrow><mtext>sp-</mtext><msub><mrow><mo>(</mo><mi>Cb</mi><mtext>-</mtext><mtext>Set</mtext><mo>)</mo></mrow><mrow><mrow><mi>f</mi></mrow><mi>s</mi></mrow></msub></mrow></math></span> consisting of finitely supported <strong>Cb</strong>-sets and sp-preserving morphisms. We analyze its characteristics, including the existence of products and coproducts, and regularity, highlighting differences from the category of support-preordered nominal sets. A key distinction is the non-equivariance of the support-preorder on finitely supported <strong>Cb</strong>-sets. We also characterize strongly finitely supported <strong>Cb</strong>-sets, where the support-preorder is equivariant, and investigate the behavior of support under actions within these sets. Our research contributes to a deeper understanding of the structure and interrelations between nominal sets and finitely supported <strong>Cb</strong>-sets, with significant implications for fields such as formal systems and homotopy type theory.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1064 ","pages":"Article 115677"},"PeriodicalIF":1.0,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792173","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 : 2025-12-08DOI: 10.1016/j.tcs.2025.115654
Shai Michael Dimant, Sven O. Krumke
In the Partial Scenario Set Cover problem (PSSC), given a finite ground set Q, a collection of subsets of Q with associated nonnegative costs, and a second collection of subsets of Q, the goal is to select a minimum-cost sub-collection of that covers at least l sets from . In this paper, we focus on two natural generalizations of PSSC. In the first generalization, called Partial PSSC (PPSSC), each scenario has a demand, and the task is to find a minimum-cost sub-collection of that satisfies the demand of at least l scenarios. We present a primal-dual algorithm for this problem whose approximation ratio improves over the previously best known for a special case of PPSSC. In the second generalization, given a finite set N, a non-negative cost function c over N, and integer-valued submodular non-decreasing functions over 2N together with integer demands and an additional integer l, we ask for a minimum-cost subset S⊆N such that at least l covering constraints of the form fi(S) ≥ di are satisfied. This is the partial cover version of the Multi-Submod-Cover problem studied in the literature, which we call Partial-Multi-Submod-Cover (PMSC). From the presented primal-dual algorithm we derive an algorithm for PMSC, which can be viewed as a generalization of Wolsey’s greedy algorithm for Submodular Set Cover. Finally, we highlight the broad applicability of the presented algorithms, which unlike previous algorithms for PSSC are purely combinatorial.
{"title":"On generalizations of partial scenario set cover","authors":"Shai Michael Dimant, Sven O. Krumke","doi":"10.1016/j.tcs.2025.115654","DOIUrl":"10.1016/j.tcs.2025.115654","url":null,"abstract":"<div><div>In the <em>Partial Scenario Set Cover problem</em> (PSSC), given a finite ground set <em>Q</em>, a collection <span><math><mi>S</mi></math></span> of subsets of <em>Q</em> with associated nonnegative costs, and a second collection <span><math><mi>U</mi></math></span> of subsets of <em>Q</em>, the goal is to select a minimum-cost sub-collection of <span><math><mi>S</mi></math></span> that covers at least <em>l</em> sets from <span><math><mi>U</mi></math></span>. In this paper, we focus on two natural generalizations of PSSC. In the first generalization, called <em>Partial PSSC</em> (PPSSC), each scenario has a demand, and the task is to find a minimum-cost sub-collection of <span><math><mi>S</mi></math></span> that satisfies the demand of at least <em>l</em> scenarios. We present a primal-dual algorithm for this problem whose approximation ratio improves over the previously best known for a special case of PPSSC. In the second generalization, given a finite set <em>N</em>, a non-negative cost function <em>c</em> over <em>N</em>, and integer-valued submodular non-decreasing functions <span><math><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>f</mi><mi>r</mi></msub></mrow></math></span> over 2<sup><em>N</em></sup> together with integer demands <span><math><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>d</mi><mi>r</mi></msub></mrow></math></span> and an additional integer <em>l</em>, we ask for a minimum-cost subset <em>S</em>⊆<em>N</em> such that at least <em>l</em> covering constraints of the form <em>f<sub>i</sub></em>(<em>S</em>) ≥ <em>d<sub>i</sub></em> are satisfied. This is the partial cover version of the Multi-Submod-Cover problem studied in the literature, which we call <em>Partial-Multi-Submod-Cover</em> (PMSC). From the presented primal-dual algorithm we derive an algorithm for PMSC, which can be viewed as a generalization of Wolsey’s greedy algorithm for Submodular Set Cover. Finally, we highlight the broad applicability of the presented algorithms, which unlike previous algorithms for PSSC are purely combinatorial.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1063 ","pages":"Article 115654"},"PeriodicalIF":1.0,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884648","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 : 2025-12-08DOI: 10.1016/j.tcs.2025.115679
Chunjian Liang , Jiafei Liu , Chia-Wei Lee , Jingli Wu , Gaoshi Li
Assessing the diagnosability of multiprocessor systems is vital for maintaining reliability and fault-tolerance, especially in extensive interconnection networks where precise reliability assessments are crucial for system stability and resilience against processor failures. In this paper, we introduce a novel diagnosability metric called h-extra r-component diagnosability, which extends traditional models by considering both component-level robustness and structural constraints. Specifically, for a graph G, a vertex subset F ⊂ V(G) is termed an h-extra r-component vertex-cut if is disconnected with at least r connected components, each containing at least vertices. The h-extra r-component diagnosability of G, denoted by , is defined as the maximum integer t such that G is conditionally t-diagnosable under this constraint. We establish theoretical characterization for hypercube networks Qn under the condition that there does not exist exactly one isolated node in for two distinct sets F1, F2. Specifically, we show that for n ≥ 7 and for n ≥ 13 under the MM* diagnostic model. To enhance fault identification efficiency, we propose a two-stage diagnosis algorithm (TSDA-MM*), leveraging network structural properties to improve diagnostic accuracy and efficiency. Extensive simulation experiments on hypercube networks and the data center networks Bcube(n, k) demonstrate that TSDA-MM* achieves high performance in terms of Accuracy, TrueNegativeRate, TruePositiveRate, and Precision, thereby providing a promising solution for practical fault diagnosis in large-scale systems.
{"title":"An efficient two-stage diagnostic algorithm for assessing system reliability","authors":"Chunjian Liang , Jiafei Liu , Chia-Wei Lee , Jingli Wu , Gaoshi Li","doi":"10.1016/j.tcs.2025.115679","DOIUrl":"10.1016/j.tcs.2025.115679","url":null,"abstract":"<div><div>Assessing the diagnosability of multiprocessor systems is vital for maintaining reliability and fault-tolerance, especially in extensive interconnection networks where precise reliability assessments are crucial for system stability and resilience against processor failures. In this paper, we introduce a novel diagnosability metric called <em>h</em>-extra <em>r</em>-component diagnosability, which extends traditional models by considering both component-level robustness and structural constraints. Specifically, for a graph <em>G</em>, a vertex subset <em>F</em> ⊂ <em>V</em>(<em>G</em>) is termed an <em>h</em>-extra <em>r</em>-component vertex-cut if <span><math><mrow><mi>G</mi><mo>−</mo><mi>F</mi></mrow></math></span> is disconnected with at least <em>r</em> connected components, each containing at least <span><math><mrow><mi>h</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. The <em>h</em>-extra <em>r</em>-component diagnosability of <em>G</em>, denoted by <span><math><mrow><msubsup><mi>t</mi><mrow><mi>r</mi></mrow><mi>h</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is defined as the maximum integer <em>t</em> such that <em>G</em> is conditionally <em>t</em>-diagnosable under this constraint. We establish theoretical characterization for hypercube networks <em>Q<sub>n</sub></em> under the condition that there does not exist exactly one isolated node in <span><math><mrow><msub><mi>Q</mi><mi>n</mi></msub><mo>−</mo><mrow><mo>(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></math></span> for two distinct sets <em>F</em><sub>1</sub>, <em>F</em><sub>2</sub>. Specifically, we show that <span><math><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>8</mn><mo>≤</mo><msubsup><mi>t</mi><mrow><mn>2</mn></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><msub><mi>Q</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>≤</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>7</mn></mrow></math></span> for <em>n</em> ≥ 7 and <span><math><mrow><msubsup><mi>t</mi><mrow><mn>3</mn></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><msub><mi>Q</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>6</mn><mi>n</mi><mo>−</mo><mn>15</mn></mrow></math></span> for <em>n</em> ≥ 13 under the MM* diagnostic model. To enhance fault identification efficiency, we propose a two-stage diagnosis algorithm (TSDA-MM*), leveraging network structural properties to improve diagnostic accuracy and efficiency. Extensive simulation experiments on hypercube networks and the data center networks <em>Bcube</em>(<em>n, k</em>) demonstrate that TSDA-MM* achieves high performance in terms of <em>Accuracy, True</em> <em>Negative</em> <em>Rate, True</em> <em>Positive</em> <em>Rate</em>, and <em>Precision</em>, thereby providing a promising solution for practical fault diagnosis in large-scale systems.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1063 ","pages":"Article 115679"},"PeriodicalIF":1.0,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145746976","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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