Discrete Optimization最新文献

{"title":"Approximation algorithms for the cluster editing problem with small clusters","authors":"Alexander Kononov ,&nbsp;Victor Il’ev","doi":"10.1016/j.disopt.2025.100900","DOIUrl":"10.1016/j.disopt.2025.100900","url":null,"abstract":"<div><div>Clustering is the task of dividing objects into groups (called clusters) so that objects in the same group are similar to each other. The Cluster Editing problem is one of the most natural ways to model clustering on graphs. In this problem, the similarity relation between objects is given by an undirected graph whose vertices correspond to the objects, edges connect couples of similar objects, and it is required to partition the set of vertices into disjoint subsets minimizing the number of edges between clusters and the number of missing edges within clusters. We present new approximation algorithms with better worst-case performance guarantees when cluster sizes are upper bounded by three or four vertices.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100900"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279680","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":"Disjoint dominating and 2-dominating sets in graphs: Hardness and approximation results","authors":"Soumyashree Rana ,&nbsp;Sounaka Mishra ,&nbsp;Bhawani Sankar Panda","doi":"10.1016/j.disopt.2025.100902","DOIUrl":"10.1016/j.disopt.2025.100902","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A set &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of a graph &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a dominating set of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; if each vertex &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is adjacent to at least one vertex in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; whereas a set &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a 2-dominating (double dominating) set of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; if each vertex &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is adjacent to at least two vertices in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; A graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-graph if there exists a pair (&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) of dominating set and 2-dominating set of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; which are disjoint. In this paper, we give approximation algorithms for the problem of determining a minimal spanning &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-graph of minimum size (&lt;span&gt;Min-&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) with an approximation ratio of 3; a minimal spanning &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-graph of maximum size (&lt;span&gt;Max-&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) with an approximation ratio of 3; and for the problem of adding minimum number of edges to a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; to make it a &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;-graph (&lt;span&gt;Min-to-&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) with an &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; approximation ratio. The above three results answer the open problems mentioned in the paper, Miotk et al. (2020). Furthermore, we prove that &lt;span&gt;Min-&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;s","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100902"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696742","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":"Computation of lower tolerances of combinatorial bottleneck problems","authors":"Gerold Jäger ,&nbsp;Marcel Turkensteen","doi":"10.1016/j.disopt.2025.100887","DOIUrl":"10.1016/j.disopt.2025.100887","url":null,"abstract":"<div><div>This paper considers the computation of lower tolerances of combinatorial optimization problems with an objective of type bottleneck, in which the objective is to minimize the element with maximum cost of a feasible solution. A lower tolerance can be defined as the supremum decrease such that the objective value remains the same. We develop a computational approach for generic problems with objective of type bottleneck and two specific approaches for the Linear Bottleneck Assignment Problem and the Bottleneck Shortest Path Problem, which have a similar complexity as solution approaches for these two problems. Finally, we present some experimental results on random instances for these problems.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"56 ","pages":"Article 100887"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143816640","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 remark on the formulation given in “A note on the lifted Miller-Tucker-Zemlin subtour elimination constraints for routing problems with time windows”","authors":"İmdat Kara,&nbsp;Gözde Önder Uzun","doi":"10.1016/j.disopt.2025.100888","DOIUrl":"10.1016/j.disopt.2025.100888","url":null,"abstract":"<div><div>In this paper, we show that, the formulation given in a recent paper [1] for the travelling salesman problem with time windows (TSPTW), may not find the optimal solution and then we recommend to add a new constraint to the model.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"56 ","pages":"Article 100888"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143748273","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 k-way vertex cut problem on bipartite graphs: Complexity results and algorithms","authors":"Mohammed Lalou ,&nbsp;Hamamache Kheddouci","doi":"10.1016/j.disopt.2025.100889","DOIUrl":"10.1016/j.disopt.2025.100889","url":null,"abstract":"<div><div>We consider the <em>k-way vertex cut problem</em> that consists in finding a subset of vertices of a given cardinality, in a graph, whose removal partitions the graph into the maximum connected components. This problem has been proven to be NP-complete on general graphs, split and planar graphs. In this paper, we consider it on bipartite graphs and we show that it remains NP-complete even restricted on this class of graphs. However, for the subclass of bipartite-permutation graphs, we develop a polynomial-time algorithm using the dynamic programming approach for solving the problem. The algorithm runs in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> time and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mi>K</mi><mo>)</mo></mrow></mrow></math></span> space, where <span><math><mi>n</mi></math></span> is the graph order, and <span><math><mi>K</mi></math></span> is the number of deleted vertices. We also extend our attention by considering vertex deletion costs, and we adapt the proposed dynamic program to the case where non-negative costs are associated to vertex deletion. The obtained algorithm is of time and space complexity <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, respectively.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"56 ","pages":"Article 100889"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937847","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":"Trimming of finite subsets of the Manhattan plane","authors":"Gökçe Çakmak ,&nbsp;Ali Deniz ,&nbsp;Şahin Koçak","doi":"10.1016/j.disopt.2025.100880","DOIUrl":"10.1016/j.disopt.2025.100880","url":null,"abstract":"<div><div>V. Turaev defined recently an operation of “Trimming” for pseudo-metric spaces and analyzed the tight span of (pseudo-)metric spaces via this process. In this work we investigate the trimming of finite subspaces of the Manhattan plane. We show that this operation amounts for them to taking the metric center set and we give an algorithm to construct the tight spans via trimming.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"56 ","pages":"Article 100880"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143520774","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":"Linear time algorithm for the vertex-edge domination problem in convex bipartite graphs","authors":"Yasemin Büyükçolak","doi":"10.1016/j.disopt.2024.100877","DOIUrl":"10.1016/j.disopt.2024.100877","url":null,"abstract":"<div><div>Given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, a vertex <span><math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi></mrow></math></span> <em>ve-dominates</em> all edges incident to any vertex in the closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow></mrow></math></span>. A subset <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is <em>a vertex-edge dominating set</em> if, for each edge <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math></span>, there exists a vertex <span><math><mrow><mi>u</mi><mo>∈</mo><mi>D</mi></mrow></math></span> such that <span><math><mi>u</mi></math></span> ve-dominates <span><math><mi>e</mi></math></span>. The objective of the <em>ve-domination problem</em> is to find a minimum cardinality ve-dominating set in <span><math><mi>G</mi></math></span>. In this paper, we present a linear time algorithm to find a minimum cardinality ve-dominating set for convex bipartite graphs, which is a superclass of bipartite permutation graphs and a subclass of bipartite graphs, where the ve-domination problem is solvable in linear time and NP-complete, respectively. We also establish the relationship <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>v</mi><mi>e</mi></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>v</mi><mi>e</mi></mrow></msub></mrow></math></span> for convex bipartite graphs. Our approach leverages a chain decomposition of convex bipartite graphs, allowing for efficient identification of minimum ve-dominating sets and extending algorithmic insights into ve-domination for specific structured graph classes.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100877"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178624","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":"Generalized min-up/min-down polytopes","authors":"Cécile Rottner","doi":"10.1016/j.disopt.2024.100866","DOIUrl":"10.1016/j.disopt.2024.100866","url":null,"abstract":"<div><div>Consider a time horizon and a set of <span><math><mi>N</mi></math></span> possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an <span><math><mi>N</mi></math></span>-state system with <span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. The minimum-time constraints enforce that if the system switches to state <span><math><mi>i</mi></math></span> at time <span><math><mi>t</mi></math></span>, then it must remain in state <span><math><mi>i</mi></math></span> for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state <span><math><mi>i</mi></math></span> is switched on at time <span><math><mi>t</mi></math></span>, the system must remain on states <span><math><mrow><mi>j</mi><mo>≥</mo><mi>i</mi></mrow></math></span> during a minimum time. Symmetrically, they also enforce that if state <span><math><mi>i</mi></math></span> is switched off at time <span><math><mi>t</mi></math></span>, the system must remain on states <span><math><mrow><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow></math></span> during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities are facet-defining and lead to a complete description of polynomial size for each polytope.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100866"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745018","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":"Solving hard bi-objective knapsack problems using deep reinforcement learning","authors":"Hadi Charkhgard ,&nbsp;Hanieh Rastegar Moghaddam ,&nbsp;Ali Eshragh ,&nbsp;Sasan Mahmoudinazlou ,&nbsp;Kimia Keshanian","doi":"10.1016/j.disopt.2025.100879","DOIUrl":"10.1016/j.disopt.2025.100879","url":null,"abstract":"<div><div>We study a class of bi-objective integer programs known as bi-objective knapsack problems (BOKPs). Our research focuses on the development of innovative exact and approximate solution methods for BOKPs by synergizing algorithmic concepts from two distinct domains: multi-objective integer programming and (deep) reinforcement learning. While novel reinforcement learning techniques have been applied successfully to single-objective integer programming in recent years, a corresponding body of work is yet to be explored in the field of multi-objective integer programming. This study is an effort to bridge this existing gap in the literature. Through a computational study, we demonstrate that although it is feasible to develop exact reinforcement learning-based methods for solving BOKPs, they come with significant computational costs. Consequently, we recommend an alternative research direction: approximating the entire nondominated frontier using deep reinforcement learning-based methods. We introduce two such methods, which extend classical methods from the multi-objective integer programming literature, and illustrate their ability to rapidly produce high-quality approximations.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100879"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143372936","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":"Integer points in the degree-sequence polytope","authors":"Eleonore Bach ,&nbsp;Friedrich Eisenbrand ,&nbsp;Rom Pinchasi","doi":"10.1016/j.disopt.2024.100867","DOIUrl":"10.1016/j.disopt.2024.100867","url":null,"abstract":"<div><div>An integer vector <span><math><mrow><mi>b</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span> is a <em>degree sequence</em> if there exists a hypergraph with vertices <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>d</mi><mo>}</mo></mrow></math></span> such that each <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the number of hyperedges containing <span><math><mi>i</mi></math></span>. The <em>degree-sequence polytope</em> <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is the convex hull of all degree sequences. We show that all but a <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mi>Ω</mi><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></msup></math></span> fraction of integer vectors in the degree sequence polytope are degree sequences. Furthermore, the corresponding hypergraph of these points can be computed in time <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></msup></math></span> via linear programming techniques. This is substantially faster than the <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span> running time of the current-best algorithm for the degree-sequence problem. We also show that for <span><math><mrow><mi>d</mi><mo>⩾</mo><mn>98</mn></mrow></math></span>, <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> contains integer points that are not degree sequences. Furthermore, we prove that both the degree sequence problem itself and the linear optimization problem over <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> are <span><math><mi>NP</mi></math></span>-hard. The latter complements a recent result of Deza et al. (2018) who provide an algorithm that is polynomial in <span><math><mi>d</mi></math></span> and the number of hyperedges.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100867"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}