Discrete Optimization最新文献

{"title":"An efficient solution approach to capacitated bilevel time minimizing transportation problem","authors":"Nanlan Zhang ,&nbsp;Fanrong Xie","doi":"10.1016/j.disopt.2025.100915","DOIUrl":"10.1016/j.disopt.2025.100915","url":null,"abstract":"<div><div><em>Capacitated bilevel time minimizing transportation problem</em> (CBTMTP) is a generalization of <em>bilevel time minimizing transportation problem</em> (BTMTP). Because of route shipping capacity realistic finiteness, CBTMTP is an optimization problem of crucial importance in logistics and emergency and project management. In the literature, no research report on CBTMTP is available due to its intractability, with exception of one BTMTP solving approach with defects of difficult computer implementation and difficult extension and inability used directly for efficiently solving CBTMTP. In this paper, by creating CBTMTP’s mathematical model along with auxiliary models and constructing network to make sufficient exploitation of CBTMTP’s network flow structure, two algorithms with one as exact optimum algorithm called CBTMTP-OA while another as heuristic algorithm called CBTMTP-HA are developed to solve CBTMTP efficiently. It is proved that CBTMTP-OA finds CBTMTP’s optimum solution by calling a polynomial time algorithm for a finite number of times, but CBTMTP-HA finds CBTMTP’s near optimum even exact optimum solution in a polynomial computation time. Computation study comprising distinct tests is conducted to verify the practical performance of CBTMTP-OA and CBTMTP-HA. It is revealed that CBTMTP-OA is capable of solving small and medium size instances efficiently, and inapplicable to solving large size instances because of too time consuming and memory overflow. But CBTMTP-HA is always capable of finding CBTMTP’s near optimum even exact optimum solution in high efficiency, and especially applicable to solving large size instances, with significant superiority to extant BTMTP solving approach. Both algorithms can serve as powerful tool for solving other relevant complicated optimization problems.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100915"},"PeriodicalIF":1.6,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417170","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":"Computational aspects of lifted cover inequalities for knapsacks with few different weights","authors":"Christopher Hojny,&nbsp;Cédric Roy","doi":"10.1016/j.disopt.2025.100912","DOIUrl":"10.1016/j.disopt.2025.100912","url":null,"abstract":"<div><div>Cutting planes are frequently used for solving integer programs. A common strategy is to derive cutting planes from building blocks or a substructure of the integer program. In this paper, we focus on knapsack constraints that arise from single row relaxations. Among the most popular classes derived from knapsack constraints are lifted minimal cover inequalities. The separation problem for these inequalities is NP-hard though, and one usually separates them heuristically, therefore not fully exploiting their potential.</div><div>For many benchmarking instances however, it turns out that many knapsack constraints only have few different coefficients. This motivates the concept of sparse knapsacks where the number of different coefficients is a small constant, independent of the number of variables present. For such knapsacks, we observe that there are only polynomially many different classes of structurally equivalent minimal covers. This opens the door to specialized techniques for using lifted minimal cover inequalities.</div><div>In this article we will discuss two such techniques, which are based on specialized sorting methods. On the one hand, we present new separation routines that separate equivalence classes of inequalities rather than individual inequalities. On the other hand, we derive compact extended formulations that express all lifted minimal cover inequalities by means of a polynomial number of constraints. These extended formulations are based on tailored sorting networks that express our separation algorithm by linear inequalities. We conclude the article by a numerical investigation of the different techniques for popular benchmarking instances.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100912"},"PeriodicalIF":1.6,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved bound for the price of anarchy for related machine scheduling","authors":"André Berger,&nbsp;Arman Rouhani,&nbsp;Marc Schröder","doi":"10.1016/j.disopt.2025.100911","DOIUrl":"10.1016/j.disopt.2025.100911","url":null,"abstract":"<div><div>In this paper, we introduce an improved upper bound for the efficiency of Nash equilibria in utilitarian scheduling games on related machines. The machines have varying speeds and adhere to the shortest processing time first policy. The goal of each job is to minimize its completion time, while the social objective is to minimize the sum of completion times. Our main finding establishes an upper bound of <span><math><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mn>4</mn><mi>m</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> on the price of anarchy for the general case of <span><math><mi>m</mi></math></span> machines. We improve this bound to 3/2 for the case of two machines, and to <span><math><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mspace></mspace><mi>m</mi><mo>)</mo></mrow></mrow></math></span> for the general case of <span><math><mi>m</mi></math></span> machines when the machines have divisible speeds, i.e., if the speed of each machine is divisible by the speed of any slower machine.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100911"},"PeriodicalIF":1.6,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099139","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":"2025-08-01","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":"A criterion space search feasibility pump heuristic for solving maximum multiplicative programs","authors":"Ashim Khanal,&nbsp;Hadi Charkhgard","doi":"10.1016/j.disopt.2025.100903","DOIUrl":"10.1016/j.disopt.2025.100903","url":null,"abstract":"<div><div>We study a class of nonlinear optimization problems with diverse practical applications, particularly in cooperative game theory. These problems are referred to as Maximum Multiplicative Programs (MMPs), and can be conceived as instances of “Optimization Over the Frontier” in multi-objective optimization. To solve MMPs, we introduce a feasibility pump-based heuristic that is specifically designed to search the criterion space of their multi-objective optimization counterparts. Through a computational study, we show the efficacy of the proposed method.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100903"},"PeriodicalIF":1.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738317","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":"2025-08-01","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":"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":"2025-08-01","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":"2025-08-01","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}