Journal of Combinatorial Optimization最新文献

{"title":"A Branch–Reduction–Bound algorithm for linear fractional multi-product planning problems","authors":"Xianfeng Ding, Meiling Hu","doi":"10.1007/s10878-025-01333-z","DOIUrl":"https://doi.org/10.1007/s10878-025-01333-z","url":null,"abstract":"<p>In this paper, we propose a Branch–Reduction–Bound (BRB) algorithm to solve fractional multiplicative product programming problems, with the aim of finding globally optimal solutions. The method introduces two innovative linear transformation techniques that simplify the solution process by converting the original problem into two equivalent linear relaxation problems. Building on this, a novel branch-and-delete rule is developed to efficiently manage sub-problem selection using a dynamic priority queue approach, and the computational process is further optimized through a region deletion rule. The synergy of these techniques significantly accelerates the algorithm's convergence rate, providing an efficient global optimization strategy. We compare the BRB algorithm with four other algorithms through numerical experiments, and the results confirm its feasibility, effectiveness, and superior computational efficiency, highlighting its advantages in solving complex optimization problems.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"33 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003502","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":"Hardness and algorithms for several new optimization problems on the weighted massively parallel computation model","authors":"Hengzhao Ma, Jianzhong Li","doi":"10.1007/s10878-025-01297-0","DOIUrl":"https://doi.org/10.1007/s10878-025-01297-0","url":null,"abstract":"<p>The topology-aware Massively Parallel Computation (MPC) model is proposed and studied recently, which enhances the classical MPC model by the awareness of network topology. The work of Hu et. al. on topology-aware MPC model considers only the tree topology. In this paper a more general case is considered, where the underlying network is a weighted complete graph. We then call this model as Weighted Massively Parallel Computation (WMPC) model, and study the problem of minimizing communication cost under it. Three communication cost minimization problems are defined based on different patterns of communication, which are the Data Redistribution Problem, Data Allocation Problem on Continuous data, and Data Allocation Problem on Categorized data. We also define four kinds of objective functions for communication cost, which consider the total cost, bottleneck cost, maximum of send and receive cost, and summation of send and receive cost, respectively. Combining the three problems in different communication patterns with the four kinds of objective cost functions, 12 problems are obtained. The hardness results and algorithms of the 12 problems make up the content of this paper. With rigorous proof, we prove that some of the 12 problems are in P, some FPT, some NP-complete, and some W[1]-complete. Approximate algorithms are proposed for several selected problems.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"51 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003451","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":"New Challenges in Combinatorial Optimization","authors":"Bo Chen, Alexander Kulikov, Silvano Martello","doi":"10.1007/s10878-025-01330-2","DOIUrl":"https://doi.org/10.1007/s10878-025-01330-2","url":null,"abstract":"","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"277 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003457","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":"Synchronizing production planning and job scheduling: MILP models and exact algorithms","authors":"Aurélien Mombelli, Alain Quilliot","doi":"10.1007/s10878-025-01326-y","DOIUrl":"https://doi.org/10.1007/s10878-025-01326-y","url":null,"abstract":"<p>We address the synchronization of a resource production process with the consumption of related resources by jobs. Both processes interact through <i>transfer transactions</i>, which become the key components of the resulting scheduling problem. This <i>Synchronized Resource Production/Job Processing problem</i> (<b>SRPJP</b>) problem typically arises when the resource is a form of renewable energy (e.g., hydrogen, photovoltaic) stored in tanks or batteries. We first cast <b>SRPJP</b> into the Mixed-Integer Linear Programming (MILP) format and handle it through a branch-and-cut process involving specific <i>No</i>_<i>Antichain</i> constraints derived from the structure of the feasible <i>transfer transactions</i>. Subsequently, we explore another approach, which involves eliminating non-binary decision variables and applying a Benders decomposition scheme. Finally, we reformulate the <b>SRPJP</b> problem as a path search problem, which we efficiently handle by designing a tailored adaptation of the A* algorithm.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003454","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":"Approximate maximin share allocation for indivisible goods under a knapsack constraint","authors":"Bin Deng, Weidong Li","doi":"10.1007/s10878-025-01331-1","DOIUrl":"https://doi.org/10.1007/s10878-025-01331-1","url":null,"abstract":"&lt;p&gt;The maximin share (MMS) allocation problem under a knapsack constraint is to allocate a set of indivisible goods to a set of &lt;i&gt;n&lt;/i&gt; heterogeneous agents, such that the total cost of the allocated goods does not exceed the given budget, and the approximation ratio of the MMS allocation is as large as possible. For any &lt;span&gt;&lt;span style=\"\"&gt;&lt;/span&gt;&lt;span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;&amp;#x03F5;&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"&gt;&lt;svg aria-hidden=\"true\" focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 3854.7 1125.3\" width=\"8.953ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"&gt;&lt;g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"&gt;&lt;use x=\"0\" xlink:href=\"#MJMATHI-3F5\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"684\" xlink:href=\"#MJMAIN-2208\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"1629\" xlink:href=\"#MJMAIN-28\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"2019\" xlink:href=\"#MJMAIN-30\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"2519\" xlink:href=\"#MJMAIN-2C\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"2964\" xlink:href=\"#MJMAIN-31\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"3465\" xlink:href=\"#MJMAIN-29\" y=\"0\"&gt;&lt;/use&gt;&lt;/g&gt;&lt;/svg&gt;&lt;span role=\"presentation\"&gt;&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;script type=\"math/tex\"&gt;epsilon in (0, 1)&lt;/script&gt;&lt;/span&gt;, we prove that &lt;span&gt;&lt;span style=\"\"&gt;&lt;/span&gt;&lt;span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;93&lt;/mn&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03F5;&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"&gt;&lt;svg aria-hidden=\"true\" focusable=\"false\" height=\"3.215ex\" role=\"img\" style=\"vertical-align: -1.006ex;\" viewbox=\"0 -950.8 3476.3 1384.1\" width=\"8.074ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"&gt;&lt;g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"&gt;&lt;use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"&gt;&lt;/use&gt;&lt;g transform=\"translate(389,0)\"&gt;&lt;g transform=\"translate(120,0)\"&gt;&lt;rect height=\"60\" stroke=\"none\" width=\"827\" x=\"0\" y=\"220\"&gt;&lt;/rect&gt;&lt;g transform=\"translate(60,407)\"&gt;&lt;use transform=\"scale(0.707)\" xlink:href=\"#MJMAIN-39\"&gt;&lt;/use&gt;&lt;use transform=\"scale(0.707)\" x=\"500\" xlink:href=\"#MJMAIN-33\" y=\"0\"&gt;&lt;/use&gt;&lt;/g&gt;&lt;g transform=\"translate(60,-363)\"&gt;&lt;use transform=\"scale(0.707)\" xlink:href=\"#MJMAIN-39\"&gt;&lt;/use&gt;&lt;use transform=\"scale(0.707)\" x=\"500\" xlink:href=\"#MJMAIN-35\" y=\"0\"&gt;&lt;/use&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;use x=\"1679\" xlink:href=\"#MJMAIN-2B\" y=\"0\"&gt;&lt;/use&gt;&lt;use x=\"2680\" xlink:href=\"#MJMAT","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"50 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003456","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":"On the initial transition of graphs of Kirkman schedules by the partial team swap","authors":"Yusuke Kashiwagi, Masaki Yamamoto, Takamasa Yashima","doi":"10.1007/s10878-025-01329-9","DOIUrl":"https://doi.org/10.1007/s10878-025-01329-9","url":null,"abstract":"<p>Kirkman schedule is one of the typical single round-robin (abbrev. SRR) tournaments. The partial team swap (abbrev. PTS) is one of the typical procedures of changing from an SRR tournament to another SRR tournament, which is used in local search for solving the traveling tournament problem. An SRR of <i>n</i> teams (of even number) can be represented by a 1-factorization of the complete graph <span><span style=\"\">K_n</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.213ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -735.2 1374.1 952.8\" width=\"3.192ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-4B\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1201\" xlink:href=\"#MJMATHI-6E\" y=\"-213\"></use></g></svg></span><script type=\"math/tex\">K_n</script></span>. It is known that the 1-factorization of any Kirkman schedule is “perfect” when <span><span style=\"\">n=p+1</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.309ex\" role=\"img\" style=\"vertical-align: -0.605ex;\" viewbox=\"0 -733.9 4161.5 994.3\" width=\"9.665ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"878\" xlink:href=\"#MJMAIN-3D\" y=\"0\"></use><use x=\"1934\" xlink:href=\"#MJMATHI-70\" y=\"0\"></use><use x=\"2660\" xlink:href=\"#MJMAIN-2B\" y=\"0\"></use><use x=\"3661\" xlink:href=\"#MJMAIN-31\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">n=p+1</script></span> for prime numbers <i>p</i>, meaning that any pair of 1-factors in the 1-factorization forms a Hamilton cycle <span><span style=\"\">C_n</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.313ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -778.3 1240.1 995.9\" width=\"2.88ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-43\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1011\" xlink:href=\"#MJMATHI-6E\" y=\"-213\"></use></g></svg></span><script type=\"math/tex\">C_n</script></span> in <span><span style=\"\">K_n</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.213ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -735.2 1374.1 952.8\" width=\"3.192ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-4B\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1201\" xlink:href=\"#MJMATHI-6E\" y=\"-213\"></use></g></svg></span><script type=\"math/tex\">K_n</script></span>, called a 2-edge-colored Hamilton cycle. We are concerned","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"43 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003471","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":"Single machine lot scheduling to minimize maximum weighted completion time","authors":"Feifeng Zheng, Na Li, Ming Liu, Yinfeng Xu","doi":"10.1007/s10878-025-01327-x","DOIUrl":"https://doi.org/10.1007/s10878-025-01327-x","url":null,"abstract":"<p>The development of artificial intelligence is a significant factor in the surge in demand for micro-products. Consequently, optimizing production scheduling for micro-products has become crucial in improving efficiency, quality, and competitiveness, which is essential for the sustainable development of the industry. In micro-product manufacturing, it is common for manufacturers to receive customized orders with varying quantities and priority levels. This work focuses on situations where orders are processed in lots with unified capacity on a single machine. Each lot has the potential to accommodate multiple orders, and if necessary, any order can be split and processed in consecutive lots. Each order is characterized by its size and weight. The objective of the problem is to minimize the maximum weighted completion time. In order to investigate the differences in the calculation of completion times for split orders, two mixed-integer linear programming models are established, and the optimal characteristics of these problems are subsequently analyzed. Furthermore, in consideration of the inherent unpredictability of order arrival over time in practice, we also explore the potential of online versions of these problems and propose an online algorithm for online problems. Finally, the experimental results assess the efficacy of the proposed optimality rules and the online algorithm and derive several managerial insights.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003453","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":"Algorithms for 2-balanced connected k-partition problem in graphs","authors":"Junran Yu, Jing Hu, Jiaquan Gao, Donglei Du, Xiaoyan Zhang","doi":"10.1007/s10878-025-01332-0","DOIUrl":"https://doi.org/10.1007/s10878-025-01332-0","url":null,"abstract":"<p>Motivated by the result of balanced connected graph edge partition problem for trees, we investigate the 2-balanced connected graph vertex <i>k</i>-partition problem. This paper leverages the charity vertex method and proposes several algorithms for 2-balanced vertex-connected partitioning. Furthermore, we prove that these algorithms are polynomial-time solvable on degree-bounded graphs, thereby refining and extending the results of Caragiannis et al.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"103 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003455","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":"Robust static and dynamic maximum flows","authors":"Christian Biefel, Martina Kuchlbauer, Frauke Liers, Lisa Waldmüller","doi":"10.1007/s10878-025-01298-z","DOIUrl":"https://doi.org/10.1007/s10878-025-01298-z","url":null,"abstract":"<p>We study the robust maximum flow problem and the robust maximum flow over time problem where a given number of arcs <span>(Gamma )</span> may fail or may be delayed. Two prominent models have been introduced for these problems: either one assigns flow to arcs fulfilling weak flow conservation in any scenario, or one assigns flow to paths where an arc failure or delay affects a whole path. We provide a unifying framework by presenting novel general models, in which we assign flow to subpaths. These models contain the known models as special cases and unify their advantages in order to obtain less conservative robust solutions.</p><p>We give a thorough analysis with respect to complexity of the general models. In particular, we show that the general models are essentially NP-hard, whereas, e.g., in the static case with <span>(Gamma =1)</span> an optimal solution can be computed in polynomial time. Further, we answer the open question about the complexity of the dynamic path model for <span>(Gamma =1)</span>. We also compare the solution quality of the different models. In detail, we show that the general models have better robust optimal values than the known models and we prove bounds on these gaps.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144133661","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":"Approximating the maximum weight cycle/path partition in graphs with weights one and two","authors":"Xinmeng Guo, Wei Yu, Zhaohui Liu","doi":"10.1007/s10878-025-01322-2","DOIUrl":"https://doi.org/10.1007/s10878-025-01322-2","url":null,"abstract":"<p>In this paper, we investigate the maximum weight <i>k</i>-cycle (<i>k</i>-path) partition problem (MaxWkCP/MaxWkPP for short). The input consists of an undirected complete graph <span>(G=(V,E))</span> with <span>(|V|=kn)</span>, where <i>k</i>, <i>n</i> are positive integers, and a non-negative weight function on <i>E</i>, the objective is to determine <i>n</i> vertex disjoint <i>k</i>-cycles (<i>k</i>-paths), which are cycles (paths) containing exactly <i>k</i> vertices, covering all the vertices such that the total edge weight of these cycles (paths) is as large as possible. We propose improved approximation algorithms for the MaxWkCP/MaxWkPP in graphs with weights one and two. For the MaxWkCP in graphs with weights one and two, we obtain an approximation algorithm having an approximation ratio of <span>(frac{37}{48})</span> for <span>(k=6)</span>, which improves upon the best available <span>(frac{91}{120})</span>-approximation algorithm by Zhao and Xiao 2024a. When <span>(k=4)</span>, we show that the same algorithm is a <span>(frac{7}{8})</span>-approximation algorithm and give a tight example. This ratio ties with the state-of-the-art result, also given by Zhao and Xiao 2024a. However, we demonstrate that our algorithm can be applied to the minimization variant of MaxWkCP in graphs with weights one and two and achieve a tight approximation ratio of <span>(frac{5}{4})</span>. For the MaxW5PP in graphs with weights one and two, we devise a novel <span>(frac{19}{24})</span>-approximation algorithm by combining two separate algorithms, each of which handles one of the two complementary scenarios of the optimal solution well. This ratio is better than the previous best ratio of <span>(frac{3}{4})</span> due to Li and Yu 2023.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144137116","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}