Discrete Optimization最新文献

{"title":"Disjoint dominating and 2-dominating sets in graphs: Hardness and approximation results","authors":"Soumyashree Rana , Sounaka Mishra , Bhawani Sankar Panda","doi":"10.1016/j.disopt.2025.100902","DOIUrl":"10.1016/j.disopt.2025.100902","url":null,"abstract":"<div><div>A set <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> of 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> is a dominating set of <span><math><mi>G</mi></math></span> if each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>∖</mo><mi>D</mi></mrow></math></span> is adjacent to at least one vertex in <span><math><mrow><mi>D</mi><mo>,</mo></mrow></math></span> whereas a set <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mi>V</mi></mrow></math></span> is a 2-dominating (double dominating) set of <span><math><mi>G</mi></math></span> if each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>∖</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> is adjacent to at least two vertices in <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>.</mo></mrow></math></span> A graph <span><math><mi>G</mi></math></span> is a <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph if there exists a pair (<span><math><mrow><mi>D</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) of dominating set and 2-dominating set of <span><math><mi>G</mi></math></span> which are disjoint. In this paper, we give approximation algorithms for the problem of determining a minimal spanning <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph of minimum size (<span>Min-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) with an approximation ratio of 3; a minimal spanning <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph of maximum size (<span>Max-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) with an approximation ratio of 3; and for the problem of adding minimum number of edges to a graph <span><math><mi>G</mi></math></span> to make it a <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph (<span>Min-to-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) with an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> approximation ratio. The above three results answer the open problems mentioned in the paper, Miotk et al. (2020). Furthermore, we prove that <span>Min-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> and <s","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100902"},"PeriodicalIF":0.9,"publicationDate":"2025-07-24","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":"On the integrality gap of small Asymmetric Traveling Salesman Problems: A polyhedral and computational approach","authors":"Eleonora Vercesi ,&nbsp;Janos Barta ,&nbsp;Luca Maria Gambardella ,&nbsp;Stefano Gualandi ,&nbsp;Monaldo Mastrolilli","doi":"10.1016/j.disopt.2025.100901","DOIUrl":"10.1016/j.disopt.2025.100901","url":null,"abstract":"<div><div>In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for instances with <span><math><mi>n</mi></math></span> nodes, where <span><math><mi>n</mi></math></span> is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope (<span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ASEP</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>) and its vertices. The polytope’s symmetries are exploited to design a heuristic pivoting algorithm to search vertices where the integrality gap is maximized. Furthermore, a general procedure for the extension of vertices from <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ASEP</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ASEP</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span> is defined. The generated vertices improve the known lower bounds of the integrality gap for <span><math><mrow><mn>16</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>22</mn></mrow></math></span> and, provide small hard-to-solve ATSP instances.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100901"},"PeriodicalIF":0.9,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549222","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 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":"2025-06-13","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":"On degeneracy in the P-matroid oriented matroid complementarity problem","authors":"Michaela Borzechowski ,&nbsp;Simon Weber","doi":"10.1016/j.disopt.2025.100891","DOIUrl":"10.1016/j.disopt.2025.100891","url":null,"abstract":"<div><div>Klaus showed that the <span>Oriented Matroid Complementarity Problem</span> (<span>OMCP</span>) can be solved by a reduction to the problem of sink-finding in a <em>unique sink orientation (USO)</em> if the input is promised to be given by a <em>non-degenerate</em> extension of a <em>P-matroid</em>. In this paper, we investigate the effect of degeneracy on this reduction. On the one hand, this understanding of degeneracies allows us to prove a linear lower bound on the number of vertex evaluations required for sink-finding in <em>P-matroid USOs</em>, the set of USOs obtainable through Klaus’ reduction. On the other hand, it allows us to adjust Klaus’ reduction to also work with degenerate instances. Furthermore, we introduce a total search version of the <span>P-Matroid Oriented Matroid Complementarity Problem</span> (<span>P-OMCP</span>). Given <em>any</em> extension of <em>any</em> oriented matroid <span><math><mi>M</mi></math></span>, by reduction to a total search version of USO sink-finding we can either solve the <span>OMCP</span>, or provide a polynomial-time verifiable certificate that <span><math><mi>M</mi></math></span> is <em>not</em> a P-matroid. This places the total search version of the <span>P-OMCP</span> in the complexity class <span>Unique End of Potential Line</span> (<span>UEOPL</span>).</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100891"},"PeriodicalIF":0.9,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144253723","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":"Optimal partitions of the flat torus into parts of smaller diameter","authors":"D.S. Protasov ,&nbsp;A.D. Tolmachev ,&nbsp;V.A. Voronov","doi":"10.1016/j.disopt.2025.100890","DOIUrl":"10.1016/j.disopt.2025.100890","url":null,"abstract":"<div><div>We consider the problem of partitioning a two-dimensional flat torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> into <span><math><mi>m</mi></math></span> sets in order to minimize the maximum diameter of a part. For <span><math><mrow><mi>m</mi><mo>⩽</mo><mn>25</mn></mrow></math></span> we give numerical estimates for the maximum diameter <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> at which the partition exists. Several approaches are proposed to obtain such estimates. In particular, we use the search for mesh partitions via the SAT solver, the global optimization approach for polygonal partitions, and the optimization of periodic hexagonal tilings. For <span><math><mrow><mi>m</mi><mo>=</mo><mn>3</mn></mrow></math></span>, the exact estimate is proved using elementary topological reasoning.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100890"},"PeriodicalIF":0.9,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116524","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":"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":"2025-04-10","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":"2025-04-02","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":"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":"2025-03-01","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}