Journal of Combinatorial Optimization最新文献

{"title":"On the parenthesisations of matrix chains: All are useful, few are essential","authors":"Francisco López, Lars Karlsson, Paolo Bientinesi","doi":"10.1007/s10878-025-01290-7","DOIUrl":"https://doi.org/10.1007/s10878-025-01290-7","url":null,"abstract":"<p>The product of a matrix chain consisting of <i>n</i> matrices can be computed in <span>(C_{n-1})</span> (Catalan’s number) different ways, each identified by a distinct parenthesisation of the chain. The best algorithm to select a parenthesisation that minimises the cost runs in <span>(O(n log n))</span> time. Approximate algorithms run in <i>O</i>(<i>n</i>) time and find solutions that are guaranteed to be within a certain factor from optimal; the best factor is currently 1.155. In this article, we first prove two results that characterise different parenthesisations, and then use those results to improve on the best known approximation algorithms. Specifically, we show that (a) each parenthesisation is optimal somewhere in the problem domain, and (b) exactly <span>(n + 1)</span> parenthesisations are essential in the sense that the removal of any one of them causes an unbounded penalty for an infinite number of problem instances. By focusing on essential parenthesisations, we improve on the best known approximation algorithm and show that the approximation factor is at most 1.143.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"60 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143832389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maximizing utilitarian and Egalitarian welfare of fractional hedonic games on tree-like graphs","authors":"Tesshu Hanaka, Airi Ikeyama, Hirotaka Ono","doi":"10.1007/s10878-025-01283-6","DOIUrl":"https://doi.org/10.1007/s10878-025-01283-6","url":null,"abstract":"<p>Fractional hedonic games are coalition formation games where a player’s utility is determined by the average value they assign to the members of their coalition. These games are a variation of graph hedonic games, which are a class of coalition formation games that can be succinctly represented. Due to their applicability in network clustering and their relationship to graph hedonic games, fractional hedonic games have been extensively studied from various perspectives. However, finding welfare-maximizing partitions in fractional hedonic games is a challenging task due to the nonlinearity of utilities. In fact, it has been proven to be NP-hard and can be solved in polynomial time only for a limited number of graph classes, such as trees. This paper presents (pseudo)polynomial-time algorithms to compute welfare-maximizing partitions in fractional hedonic games on tree-like graphs. We consider two types of social welfare measures: utilitarian and egalitarian. Tree-like graphs refer to graphs with bounded treewidth and block graphs. A hardness result is provided, demonstrating that the pseudopolynomial-time solvability is the best possible under the assumption P <span>(ne )</span> NP.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143832390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Testing Higher-order Clusterability on Graphs","authors":"Yifei Li, Donghua Yang, Jianzhong Li","doi":"10.1007/s10878-025-01262-x","DOIUrl":"https://doi.org/10.1007/s10878-025-01262-x","url":null,"abstract":"<p>Analysis of higher-order organizations, represented as small connected subgraphs, is a fundamental task on complex networks. This paper studies a new problem of testing higher-order clusterability: given neighbor query access to an undirected graph, can we judge whether this graph can be partitioned into a few clusters of highly-connected cliques? This problem is an extension of the former work proposed by Czumaj et al. (STOC’ 15), who recognized cluster structure on graphs using the framework of property testing. In this paper, the problem of testing whether a well-defined higher-order cluster exists is first defined. Then, an <span>(varOmega (sqrt{n}))</span> query lower bound of this problem is given. After that, a baseline algorithm is provided by an edge-cluster tester on <i>k</i>-clique dual graph. Finally, an optimized <span>(tilde{O}(sqrt{n}))</span>-time algorithm is developed for testing clusterability based on triangles.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation algorithm for dynamic facility location problem","authors":"Li Zhang, Qiaoliang Li","doi":"10.1007/s10878-025-01282-7","DOIUrl":"https://doi.org/10.1007/s10878-025-01282-7","url":null,"abstract":"<p>In this paper, we consider dynamic facility location problem with unit demand (DFLPUD). We propose a 1.52-approximation algorithm that skillfully integrates dual-fitting and greedy augmentation schemes. Our algorithmic framework begins by formulating DFLPUD as a set covering linear integer programming problem. Then we scale the opening cost of all facilities and use the solution of dual-fitting algorithm to seed a local search to yield an improved performance guarantee 1.52. To the best of our knowledge, this is the best known approximation ratio for DFLPUD.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"117 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143822885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A sharp upper bound for the edge dominating number of hypergraphs with minimum degree","authors":"Zhongzheng Tang, Zhuo Diao","doi":"10.1007/s10878-025-01284-5","DOIUrl":"https://doi.org/10.1007/s10878-025-01284-5","url":null,"abstract":"<p>In a hypergraph <i>H</i>(<i>V</i>, <i>E</i>), a subset of edges <span>(Asubseteq E)</span> forms an edge dominating set if each edge <span>(ein Esetminus A)</span> is adjacent to at least one edge in <i>A</i>. The edge dominating number <span>(gamma '(H))</span> represents the smallest size of an edge dominating set in <i>H</i>. In this paper, we establish upper bounds on the edge dominating number for hypergraphs with minimum degree <span>(delta )</span>: (1) For <span>(delta le 4)</span>, <span>(gamma '(H)le frac{m}{delta })</span>; (2) For <span>(delta ge 5)</span>, <span>(gamma '(H)le frac{m}{delta })</span> holds for hypertrees and uniform hypergraphs; (3) For a random hypergraph model <span>(mathcal H(n,m))</span>, for any positive number <span>(varepsilon &gt;0)</span>, <span>(gamma ' (H)le (1+varepsilon )frac{m}{delta })</span> holds with high probability when <i>m</i> is bounded by some polynomial function of <i>n</i>. Based on the proofs, some combinatorial algorithms on the edge dominating number of hypergraphs with minimum degree are designed.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"34 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143822881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Link fault tolerability of the Cartesian product power graph $$(K_{9}-C_{9})^{n}$$ : conditional edge-connectivities under six link fault patterns","authors":"Zhaoman Huang, Yayu Yang, Mingzu Zhang, Weihua Yang","doi":"10.1007/s10878-025-01273-8","DOIUrl":"https://doi.org/10.1007/s10878-025-01273-8","url":null,"abstract":"<p>High-performance computing extensively depends on parallel and distributed systems, necessitating the establishment of quantitative parameters to evaluate the fault tolerability of interconnection networks. The topological structures of interconnection networks in some parallel and distributed systems are designed as <i>n</i>-dimensional <span>((K_{9}-C_{9})^{n})</span>, obtained through the repeatedly application of the <i>n</i>-th Cartesian product operation. Since the <span>(mathcal {P})</span>-conditional edge-connectivity is proposed by Harary, as a parameter for evaluating the link fault tolerability of the underlying topology graph of the interconnection network system, it has been widely studied in many interconnection networks. The <span>(mathcal {P})</span>-conditional edge-connectivity of a connected graph <i>G</i>, denoted by <span>(lambda (mathcal {P};G))</span>, if any, describes the minimum cardinality of the fault edge-cut of the graph <i>G</i>, whose malfunction divides <i>G</i> into multiple components, with each component satisfying a given property <span>(mathcal {P})</span> of the graph. In this paper, we primarily define <span>(mathcal {P}_{i}^{t})</span> to be properties of containing at least <span>(9^t)</span> processors, every remaining processor lying in a lower dimensional subnetwork of the <span>((K_{9}-C_{9})^{n})</span>, <span>((K_{9}-C_{9})^{t})</span>, having a minimum degree or average degree of at least 6<i>t</i>, existing two components with each component having at least <span>(9^t)</span> processors, and containing at least one cycle, respectively. We use the properties of the optimal solution to the edge isoperimetric problem of <span>((K_{9}-C_{9})^{n})</span> and find that the exact values of the <span>(mathcal {P}_{i})</span>-conditional edge-connectivities of the graph <span>((K_{9}-C_{9})^{n})</span> share a common value of <span>(6(n-t)9^t)</span> for <span>(1le ile 5)</span> and <span>(0le tle n-1)</span>, except for <span>(i=6)</span>, the value is <span>(18n - 6)</span>.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"66 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143822884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uav trajectory optimization for maximizing the ToI-based data utility in wireless sensor networks","authors":"Qing Zhao, Zhen Li, Jianqiang Li, Jianxiong Guo, Xingjian Ding, Deying Li","doi":"10.1007/s10878-025-01286-3","DOIUrl":"https://doi.org/10.1007/s10878-025-01286-3","url":null,"abstract":"<p>It’s a promising way to use Unmanned Aerial Vehicles (UAVs) as mobile base stations to collect data from sensor nodes, especially for large-scale wireless sensor networks. There are a lot of works that focus on improving the freshness of the collected data or the data collection efficiency by scheduling UAVs. Given that sensing data in certain applications is time-sensitive, with its value diminishing as time progresses based on Timeliness of Information (ToI), this paper delves into the UAV Trajectory optimization problem for Maximizing the ToI-based data utility (TMT). We give the formal definition of the problem and prove its NP-Hardness. To solve the TMT problem, we propose a deep reinforcement learning-based algorithm that combines the Action Rejection Mechanism and the Deep Q-Network with Priority Experience Replay (ARM-PER-DQN). Where the action rejection mechanism could reduce the action space and PER helps improve the utilization of experiences with high value, thus increasing the training efficiency. To avoid the unbalanced data collection problem, we also investigate a variant problem of TMT (named V-TMT), i.e., each sensor node can be visited by the UAV at most once. We prove that the V-TMT problem is also NP-Hard, and propose a 2-approximation algorithm as the baseline of the ARM-PER-DQN algorithm. We conduct extensive simulations for the two problems to validate the performance of our designs, and the results show that our ARM-PER-DQN algorithm outperforms other baselines, especially in the V-TMT problem, the ARM-PER-DQN algorithm always outperforms the proposed 2-approximation algorithm, which suggests the effectiveness of our algorithm.\u0000</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"119 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143822880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved approximation algorithm for covering vertices by $$4^+$$ -paths","authors":"Mingyang Gong, Zhi-Zhong Chen, Guohui Lin, Lusheng Wang","doi":"10.1007/s10878-025-01279-2","DOIUrl":"https://doi.org/10.1007/s10878-025-01279-2","url":null,"abstract":"<p><i>Path cover</i> is one of the well-known NP-hard problems that has received much attention. In this paper, we study a variant of path cover, denoted by <span>(hbox {MPC}^{{4}+}_v)</span>, to cover as many vertices in a given graph <span>(G = (V, E))</span> as possible by a collection of vertex-disjoint paths each of order four or above. The problem admits an existing <span>(O(|V|^8))</span>-time 2-approximation algorithm by applying several time-consuming local improvement operations (Gong et al.: Proceedings of MFCS 2022, LIPIcs 241, pp 53:1–53:14, 2022). In contrast, our new algorithm uses a completely different method and it is an improved <span>(O(min {|E|^2|V|^2, |V|^5}))</span>-time 1.874-approximation algorithm, which answers the open question in Gong et al. (2022) in the affirmative. An important observation leading to the improvement is that the number of vertices in a maximum matching <i>M</i> of <i>G</i> is relatively large compared to that in an optimal solution of <span>(hbox {MPC}^{{4}+}_v)</span>. Our new algorithm forms a feasible solution of <span>(hbox {MPC}^{{4}+}_v)</span> from a maximum matching <i>M</i> by computing a maximum-weight path-cycle cover in an auxiliary graph to connect as many edges in <i>M</i> as possible.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143822883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation algorithms for solving the heterogeneous rooted tree/path cover problems","authors":"Pengxiang Pan, Junran Lichen, Ping Yang, Jianping Li","doi":"10.1007/s10878-025-01278-3","DOIUrl":"https://doi.org/10.1007/s10878-025-01278-3","url":null,"abstract":"<p>In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph <span>(G=(V,E; w,f; r))</span> and <i>k</i> construction teams, having nonuniform construction speeds <span>(lambda _{1})</span>, <span>(lambda _{2})</span>, <span>(ldots )</span>, <span>(lambda _{k})</span>, where <span>(rin V)</span> is a fixed common root, <span>(w:Erightarrow {mathbb {R}}^{+})</span> is an edge-weight function, satisfying the triangle inequality, and <span>(f:Vrightarrow {mathbb {R}}^{+}_{0})</span> (<i>i.e., </i> <span>({mathbb {R}}^{+}cup {0}))</span> is a vertex-weight function with <span>(f(r)=0)</span>, we are asked to find <i>k</i> trees for these <i>k</i> construction teams, each tree having the same root <i>r</i>, and collectively covering all vertices in <i>V</i>, the objective is to minimize the maximum completion time of <i>k</i> construction teams, where the completion time of each team is the total construction weight of its related tree divided by its construction speed. In addition, substituting <i>k</i> paths for <i>k</i> trees in the HRTC problem, we also consider the heterogeneous rooted path cover (HRPC) problem. Our main contributions are as follows. (1) Given any small constant <span>(delta &gt;0)</span>, we first design a <span>(58.3286(1+delta ))</span>-approximation algorithm to solve the HRTC problem, and this algorithm runs in time <span>(O(n^{2}(n+frac{log n}{delta })+log (w(E)+f(V))))</span>. Meanwhile, we present a simple <span>(116.6572(1+delta ))</span>-approximation algorithm to solve the HRPC problem, whose time complexity is the same as the preceding algorithm. (2) We provide a <span>(max {2rho , 2+rho -frac{2}{k}})</span>-approximation algorithm to resolve the HRTC problem, and that algorithm runs in time <span>(O(n^{2}))</span>, where <span>(rho )</span> is the ratio of the largest team speed to the smallest one. At the same time, we can prove that the preceding <span>(max {2rho , 2+rho -frac{2}{k}})</span>-approximation algorithm also resolves the HRPC problem.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"39 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143822882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical models for the one-dimensional cutting stock problem with setups and open stacks","authors":"Gabriel Gazzinelli Guimarães, Kelly Cristina Poldi, Mateus Martin","doi":"10.1007/s10878-025-01276-5","DOIUrl":"https://doi.org/10.1007/s10878-025-01276-5","url":null,"abstract":"<p>In real-life production, the cutting stock problem is often associated with additional constraints and objectives. Among the auxiliary objectives, two of the most relevant are the minimization of the number of different cutting patterns used and the minimization of the maximum number of simultaneously open stacks. The first auxiliary objective arises in manufacturing environments where the adjustment of the cutting tools when changing the cutting patterns incurs increased costs and time spent in production. The second is crucial to face scenarios where the space near the cutting machine or the number of automatic unloading stations is limited. In this paper, we address the one-dimensional cutting stock problem, considering the additional goals of minimizing the number of different cutting patterns used and the maximum number of simultaneously open stacks. We propose two Integer Linear Programming (ILP) formulations and a Constraint Programming (CP) model for the problem. Moreover, we develop new upper bounds on the frequency of the cutting patterns in a solution and address some special cases in which the problem may be simplified. All three approaches are embedded into an iterative exact framework to find efficient solutions. We perform computational experiments using two sets of instances from the literature. The proposed approaches proved effective in determining the entire Pareto front for small problem instances, and several solutions for medium-sized instances with minimum trim loss, a reduced maximum number of simultaneously open stacks, and a small number of different used cutting patterns.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"217 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143782634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}