Journal of Combinatorial Optimization最新文献

{"title":"Digital twin-augmented proximal policy optimization for real-time home energy management: addressing uncertainties in renewables, pricing, and loads","authors":"Ubaid ur Rehman","doi":"10.1007/s10878-025-01386-0","DOIUrl":"https://doi.org/10.1007/s10878-025-01386-0","url":null,"abstract":"The escalating global energy demand, combined with intermittent renewable energy sources and dynamic electricity pricing, highlights the critical requirement for intelligent home energy management systems to optimize consumption, minimize expenses, and promote sustainability. Conventional home energy management systems face difficulties in managing uncertainties in power generation, household loads, and tariff variations, resulting in operational inefficiencies and increased grid pressure. This paper introduces a novel framework that integrates a digital twin with Proximal Policy Optimization for real-time home energy management, building on previous efforts by adding a digital twin for predictive validation and adaptive control. The framework structures the home energy management system as a Markov decision process, featuring extensive state-action spaces that handle energy storage systems, electric vehicles, photovoltaic panels, wind turbines, heating ventilation air conditioning, water heater, and schedulable loads through clipped Proximal Policy Optimization and generalized advantage estimation for reliable learning. Trained on a 180-day dataset and validated on a 96-step real-time dataset, it delivers 42.8% cost reduction from 23.68 to 13.53 euros compared to the baseline, achieves complete comfort compliance at 100%, and operates with 125 ms inference time, exceeding benchmarks by 7–30% through digital twin-enhanced robustness. This advancement establishes a scalable, resilient home energy management system for prosumer environments.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"161 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148287963","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 unified bi-level framework for gene-knockout strategies","authors":"Ashwin Arulselvan, Mahdi Doostmohammadi, Jose Alexander Vindel Garduno","doi":"10.1007/s10878-026-01431-6","DOIUrl":"https://doi.org/10.1007/s10878-026-01431-6","url":null,"abstract":"In this work, we explore the concept of gene deletion (knockout) in metabolic engineering using the well-known bi-level optimisation approach. The objective is to identify a knockout strategy that maximises the production of a desired chemical without hindering the cell growth. The most popular method in the literature to solve this bi-level problem is to reformulate it into a single-level model using KKT conditions, which is then solved as a mixed integer programme after linearising the quadratic terms. In contrast, we propose a branch-and-cut algorithm starting from a high-point relaxation of the problem. Existing techniques that rely on KKT reformulations scale poorly as they introduce auxiliary binary variables for linearisation. We avoid the need of auxillary binary variables in our approach. We add cuts to prune off solutions that are neither optimal nor feasible within the branch-and-bound exploration. We propose this as a framework capable of handling both optimistic and pessimistic strategies by appropriately parametrising the separation problem and selecting the cuts. No unified framework currently exists that permits the decision maker to bracket the production ranges and select strategies that matches their biological assumptions. Our computational results indicate that both our strategies are competent when compared to the single-level reformulation with a slight deterioration in solution time for pessimistic strategies as the problem size increases. We show the convergence of the algorithm and share our computational experience.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148287964","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 multi-scenario efficient solutions for an uncertain bi-criteria parallel machine scheduling problem","authors":"Xiechen Zhang, Eric Angel, Feng Chu, Damien Regnault","doi":"10.1007/s10878-026-01436-1","DOIUrl":"https://doi.org/10.1007/s10878-026-01436-1","url":null,"abstract":"This paper studies a bi-criteria parallel machine scheduling problem with uncertain processing times to minimize makespan and total completion time. We adopt a scenario-based approach for uncertainty and assume that a scenario index can take any real value in an interval. This index value can represent, for example, the amount of congestion in a network. We assume that the job processing times are non-decreasing polynomial functions of fixed degrees with respect to the scenario index. To tackle multi-criteria optimization in the presence of uncertainty, researchers have proposed the concept of multi-scenario efficient set (Botte and Schöbel 2019, Engau and Sigler 2020). We adopt this framework and want to compute, in polynomial time, an approximation of this set with theoretical performance guarantees. Based on a dynamic programming framework, with carefully designed states, we obtain a Fully Polynomial-Time Approximation Scheme (FPTAS) for the problem.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"36 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355671","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":"Efficient exact algorithms for multi-Machine non-idling unit-job scheduling","authors":"Philippe Chretienne, Alain Quilliot, Hélène Toussaint","doi":"10.1007/s10878-026-01415-6","DOIUrl":"https://doi.org/10.1007/s10878-026-01415-6","url":null,"abstract":"We address here the problem of scheduling a set of jobs on a minimal number of identical machines while forbidding <italic>idleness</italic>: A machine is <italic>idle</italic> when it is waiting to process a job after processing another job. After analyzing the complexity of this <italic>Multi-Machine Non-Idling Scheduling</italic> problem, we focus on its preemptive relaxation obtained by restricting ourselves to unit-period jobs. It has been proved that the variant of this problem induced by a fixed number of machines can be solved in polynomial time by implementing an <italic>ad hoc</italic> recursive scheme. However, the high complexity of this recursive scheme makes that designing exact algorithms able to solve medium size instances remain an issue. We handle this by first reformulating our problem in a way that skips the machines and the jobs, and next designing an exact <inline-formula><alternatives><mml:math><mml:msup><mml:mi>A</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$A^*$$end{document}</tex-math></alternatives></inline-formula>-type path search algorithm enhanced by constraint propagation mechanisms. We test the efficiency of this approach via numerical tests.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"70 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288005","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":"Hamiltonian paths extending a set of matchings in hypercubes","authors":"Abid Ali, Lina Ba, Weihua Yang","doi":"10.1007/s10878-026-01418-3","DOIUrl":"https://doi.org/10.1007/s10878-026-01418-3","url":null,"abstract":"The hypercube &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Q_n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is a fundamental structure in interconnection networks, where Hamiltonian paths and cycles play a key role in supporting efficient communication and routing. It is known that &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Q_n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; contains a Hamiltonian path between two vertices &lt;italic&gt;x&lt;/italic&gt; and &lt;italic&gt;y&lt;/italic&gt; from opposite partite sets that includes a prescribed set of edges. Dvořák and Gregor resolved a problem posed by Caha and Koubek by proving that for every &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;≥&lt;/mml:mo&gt;&lt;mml:mn&gt;5&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n ge 5$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, there exist vertices &lt;italic&gt;x&lt;/italic&gt; and &lt;italic&gt;y&lt;/italic&gt; and a set of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mn&gt;4&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2n - 4$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; edges in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Q_n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; that can be extended to a Hamiltonian path between &lt;italic&gt;x&lt;/italic&gt; and &lt;italic&gt;y&lt;/italic&gt;. In this paper, we consider matchings &lt;italic&gt;M&lt;/italic&gt; in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} us","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288012","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 the Square Min-Sum Bin Packing Problem","authors":"Rachel Vanucchi Saraiva, Rafael C. S. Schouery","doi":"10.1007/s10878-026-01425-4","DOIUrl":"https://doi.org/10.1007/s10878-026-01425-4","url":null,"abstract":"In this work, we study the Square Min-Sum Bin Packing Problem (SMSBPP), where a list of &lt;italic&gt;n&lt;/italic&gt; square items has to be packed into square bins of dimensions &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;×&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$1 times 1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with no overlap between the areas of the items. The bins are indexed (starting at one) and the cost of packing each item is equal to the index of the bin in which it is placed. The objective is to minimize the total cost of packing all items, which is equivalent to minimizing the average cost of items. The problem has applications in minimizing the average time of logistic operations such as cutting stock and delivery of products. We prove that classic algorithms for two-dimensional bin packing that order items in non-increasing order of size, such as Next Fit Decreasing Height or Any Fit Decreasing Height heuristics, can have an arbitrarily bad performance for SMSBPP. We, then, present an algorithm with an approximation ratio of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:msqrt&gt;&lt;mml:mn&gt;205&lt;/mml:mn&gt;&lt;/mml:msqrt&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mn&gt;12&lt;/mml:mn&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mi&gt;δ&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$sqrt{205}-12+delta $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo&gt;≈&lt;/mml:mo&gt;&lt;mml:mn&gt;2.3178&lt;/mml:mn&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mi&gt;δ&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$approx 2.3178+delta $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;), for any &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;δ&lt;/mml:mi&gt;&lt;mml:mo&gt;&gt;&lt;/mml:mo&gt;&lt;mml:mn&gt;0&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$delta &gt; 0$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, and running time &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;O&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo&gt;log&lt;/mml:mo&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usep","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"68 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288008","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 the minimum weight cycle/path partition problem","authors":"Yaqi Li, Wei Yu, Zhaohui Liu","doi":"10.1007/s10878-026-01422-7","DOIUrl":"https://doi.org/10.1007/s10878-026-01422-7","url":null,"abstract":"Let &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;V&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;E&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G=(V,E)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; be an undirected complete graph on &lt;italic&gt;kn&lt;/italic&gt; vertices. Each edge is associated with a non-negative weight. The edge weights satisfy the triangle inequality. A &lt;italic&gt;k&lt;/italic&gt;-cycle partition is a set of &lt;italic&gt;n&lt;/italic&gt; vertex-disjoint &lt;italic&gt;k&lt;/italic&gt;-cycles, i.e. cycles containing exactly &lt;italic&gt;k&lt;/italic&gt; vertices. The minimum weight &lt;italic&gt;k&lt;/italic&gt;-cycle partition problem (MinWkCP) is to determine a &lt;italic&gt;k&lt;/italic&gt;-cycle partition with minimum total edge weight. In the MinWkCP, if we replace cycles with paths, we obtain the minimum weight &lt;italic&gt;k&lt;/italic&gt;-path partition problem (MinWkPP). In this paper, we first devise a tight &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mfrac&gt;&lt;mml:mn&gt;3&lt;/mml:mn&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:mfrac&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$frac{3}{2}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-approximation algorithm for the MinW4CP, improving on the best-known 3-approximation algorithm by Goemans and Williamson (&lt;xref ref-type=\"bibr\"&gt;1995&lt;/xref&gt;). Then we deal with a special case of the MinWkCP and MinWkPP, where the edge weights are either 1 or 2, and propose approximation algorithms with ratios &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mfrac&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;8&lt;/mml:mn&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;14&lt;/mml:mn&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mn&gt;8&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;7&lt;/mml:mn&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;/mml:mfrac&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$frac{8k^2+14k-8}{7k^2}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mfrac&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;8&lt;/mml:mn&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;7&lt;/mml:mn&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:mfrac&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} u","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288007","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":"The approximation algorithm and fast algorithm for constrained or-submodular maximization problem","authors":"Haifeng Huang, Qian Liu, Yang Zhou, Min Li","doi":"10.1007/s10878-026-01421-8","DOIUrl":"https://doi.org/10.1007/s10878-026-01421-8","url":null,"abstract":"The problem of maximizing <italic>k</italic>-submodular functions is a classical issue in the field of combinatorial optimization. In our work, we primarily consider an or-submodular function, which is a generalization of <italic>k</italic>-submodular. For maximizing the or-submodular function problem, we first propose a greedy algorithm to get a <inline-formula><alternatives><mml:math><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$frac{1}{r+1}$$end{document}</tex-math></alternatives></inline-formula>-approximation ratio under a matroid constraint and design a fast algorithm to reduce its complexity, where <inline-formula><alternatives><mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo>≤</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$1le r le k$$end{document}</tex-math></alternatives></inline-formula>. In addition, we also use a greedy algorithm to obtain an approximation solution of <inline-formula><alternatives><mml:math><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$frac{1}{r+1}(1-e^{-(r+1)})$$end{document}</tex-math></alternatives></inline-formula> for maximizing the or-submodular function with a knapsack constraint, where <inline-formula><alternatives><mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo>≤</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$1le r le k$$end{document}</tex-math></alternatives></inline-formula>.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"30 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288009","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":"Improved lower bounds on the general reduced second Zagreb index of trees and unicyclic graphs","authors":"Nasrin Dehgardi, Sandi Klavžar, Mahdieh Azari","doi":"10.1007/s10878-026-01428-1","DOIUrl":"https://doi.org/10.1007/s10878-026-01428-1","url":null,"abstract":"For a simple graph &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"normal\"&gt;Γ&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Gamma $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and a real number &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the general reduced second Zagreb index is defined by the formula &lt;disp-formula&gt;&lt;alternatives&gt;&lt;mml:math display=\"block\"&gt;&lt;mml:mrow&gt;&lt;mml:mtable&gt;&lt;mml:mtr&gt;&lt;mml:mtd columnalign=\"right\"&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mi&gt;R&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;M&lt;/mml:mi&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi mathvariant=\"normal\"&gt;Γ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:munder&gt;&lt;mml:mo&gt;∑&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;a&lt;/mml:mi&gt;&lt;mml:mi&gt;b&lt;/mml:mi&gt;&lt;mml:mo&gt;∈&lt;/mml:mo&gt;&lt;mml:mi&gt;E&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi mathvariant=\"normal\"&gt;Γ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:munder&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;[&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mo&gt;deg&lt;/mml:mo&gt;&lt;mml:mi mathvariant=\"normal\"&gt;Γ&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;a&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mo&gt;deg&lt;/mml:mo&gt;&lt;mml:mi mathvariant=\"normal\"&gt;Γ&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;b&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;]&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mspace width=\"0.166667em\"&gt;&lt;/mml:mspace&gt;&lt;mml:mo&gt;.&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;/mml:mtr&gt;&lt;/mml:mtable&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} GRM_lambda (Gamma )=sum _{abin E(Gamma )}[(deg _{Gamma }(a)+lambda )(deg _{Gamma }(b)+lambda )],. end{aligned}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/disp-formula&gt;A sharp lower bound for &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mi&gt;R&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;M&lt;/mml:mi&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackag","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288006","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":"Bond polytope under vertex- and edge-sums","authors":"Petr Kolman, Hans Raj Tiwary","doi":"10.1007/s10878-026-01426-3","DOIUrl":"https://doi.org/10.1007/s10878-026-01426-3","url":null,"abstract":"A cut in a graph &lt;italic&gt;G&lt;/italic&gt; is called a &lt;italic&gt;bond&lt;/italic&gt; if both parts of the cut induce connected subgraphs in &lt;italic&gt;G&lt;/italic&gt;, and the &lt;italic&gt;bond polytope&lt;/italic&gt; is the convex hull of all bonds. Computing the maximum weight bond is an NP-hard problem even for planar graphs. However, the problem is solvable in linear time on &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;K&lt;/mml:mi&gt;&lt;mml:mn&gt;5&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\"&gt;&lt;/mml:mo&gt;&lt;mml:mi&gt;e&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(K_5 setminus e)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-minor-free graphs, and in more general, on graphs of bounded treewidth, essentially due to clique-sum decomposition into simpler graphs. We show how to obtain the bond polytope of graphs that are 1- or 2-sum of graphs &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ G_2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; from the bond polytopes of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_1,G_2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. Using this we show that the extension complexity of the bond polytope of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;K&lt;/mml:mi&gt;&lt;mml:mn&gt;5&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\"&gt;&lt;/mml:mo&gt;&lt;mml:mi&gt;e&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} u","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288011","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}