Pub Date : 2025-04-10DOI: 10.1016/j.disopt.2025.100887
Gerold Jäger , Marcel Turkensteen
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.
{"title":"Computation of lower tolerances of combinatorial bottleneck problems","authors":"Gerold Jäger , 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}
Pub Date : 2025-04-02DOI: 10.1016/j.disopt.2025.100888
İmdat Kara, Gözde Önder Uzun
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.
{"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, 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}
Pub Date : 2025-03-01DOI: 10.1016/j.disopt.2025.100880
Gökçe Çakmak , Ali Deniz , Şahin Koçak
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.
{"title":"Trimming of finite subsets of the Manhattan plane","authors":"Gökçe Çakmak , Ali Deniz , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.disopt.2024.100877
Yasemin Büyükçolak
Given a graph , a vertex ve-dominates all edges incident to any vertex in the closed neighborhood . A subset is a vertex-edge dominating set if, for each edge , there exists a vertex such that ve-dominates . The objective of the ve-domination problem is to find a minimum cardinality ve-dominating set in . 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 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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.disopt.2025.100879
Hadi Charkhgard , Hanieh Rastegar Moghaddam , Ali Eshragh , Sasan Mahmoudinazlou , Kimia Keshanian
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.
{"title":"Solving hard bi-objective knapsack problems using deep reinforcement learning","authors":"Hadi Charkhgard , Hanieh Rastegar Moghaddam , Ali Eshragh , Sasan Mahmoudinazlou , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.disopt.2024.100876
Bjoern Andres , Silvia Di Gregorio , Jannik Irmai , Jan-Hendrik Lange
{"title":"Corrigendum to “A polyhedral study of lifted multicuts” [Discrete Optim. 47 (2023) 100757]","authors":"Bjoern Andres , Silvia Di Gregorio , Jannik Irmai , Jan-Hendrik Lange","doi":"10.1016/j.disopt.2024.100876","DOIUrl":"10.1016/j.disopt.2024.100876","url":null,"abstract":"","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100876"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143177281","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}
Pub Date : 2025-02-01DOI: 10.1016/j.disopt.2024.100867
Eleonore Bach , Friedrich Eisenbrand , Rom Pinchasi
An integer vector is a degree sequence if there exists a hypergraph with vertices such that each is the number of hyperedges containing . The degree-sequence polytope is the convex hull of all degree sequences. We show that all but a fraction of integer vectors in the degree sequence polytope are degree sequences. Furthermore, the corresponding hypergraph of these points can be computed in time via linear programming techniques. This is substantially faster than the running time of the current-best algorithm for the degree-sequence problem. We also show that for , contains integer points that are not degree sequences. Furthermore, we prove that both the degree sequence problem itself and the linear optimization problem over are -hard. The latter complements a recent result of Deza et al. (2018) who provide an algorithm that is polynomial in and the number of hyperedges.
{"title":"Integer points in the degree-sequence polytope","authors":"Eleonore Bach , Friedrich Eisenbrand , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.disopt.2025.100878
Rajni Dabas, Neelima Gupta
In this paper, we present a framework to design approximation algorithms for capacitated facility location problems with penalties/outliers. We apply our framework to obtain first approximations for capacitated -facility location problem with penalties (CFLwP) and capacitated facility location problem with outliers (CFLwO), for hard uniform capacities. Our solutions incur slight violations in capacities, () for the problems without cardinality() constraint and () for the problems with the cardinality constraint. For the outlier variant, we also incur a small loss () in outliers. To the best of our knowledge, no results are known for CFLwO and CFLwP in the literature. For uniform facility opening cost, we get rid of violation in capacities for CFLwO. Our approach is based on LP rounding technique.
{"title":"Uniform capacitated facility location with outliers/penalties","authors":"Rajni Dabas, Neelima Gupta","doi":"10.1016/j.disopt.2025.100878","DOIUrl":"10.1016/j.disopt.2025.100878","url":null,"abstract":"<div><div>In this paper, we present a framework to design approximation algorithms for capacitated facility location problems with penalties/outliers. We apply our framework to obtain first approximations for capacitated <span><math><mi>k</mi></math></span>-facility location problem with penalties (C<span><math><mi>k</mi></math></span>FLwP) and capacitated facility location problem with outliers (CFLwO), for hard uniform capacities. Our solutions incur slight violations in capacities, (<span><math><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow></math></span>) for the problems without cardinality(<span><math><mi>k</mi></math></span>) constraint and (<span><math><mrow><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></math></span>) for the problems with the cardinality constraint. For the outlier variant, we also incur a small loss (<span><math><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow></math></span>) in outliers. To the best of our knowledge, no results are known for CFLwO and C<span><math><mi>k</mi></math></span>FLwP in the literature. For uniform facility opening cost, we get rid of violation in capacities for CFLwO. Our approach is based on LP rounding technique.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100878"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143177068","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}
The pure fixed charge transportation problem is a well-known variant of the classic transportation problem where the cost of sending goods from a source to a destination only equals a fixed charge, regardless of the flow quantity. The objective is to minimize the total cost of shipping available goods to meet the required demands. Hence, we first demonstrate that this problem is NP-hard even when there are only two destinations, and it is Strong NP-hard when the number of destinations is input. These two new complexity results are an important supplement to the previous complexity results of this problem. Then, we propose two simple but novel approximation algorithms with a constant worst-case ratio, which is proved using an integer convex optimization model. Although our approximation algorithm applies to a few destinations, to our knowledge, it is the first approximation algorithm to handle the pure fixed-charge transportation problem.
{"title":"On the pure fixed charge transportation problem","authors":"Pengfei Zhu , Guangting Chen , Yong Chen , An Zhang","doi":"10.1016/j.disopt.2024.100875","DOIUrl":"10.1016/j.disopt.2024.100875","url":null,"abstract":"<div><div>The <em>pure fixed charge transportation problem</em> is a well-known variant of the classic transportation problem where the cost of sending goods from a source to a destination only equals a fixed charge, regardless of the flow quantity. The objective is to minimize the total cost of shipping available goods to meet the required demands. Hence, we first demonstrate that this problem is NP-hard even when there are only two destinations, and it is Strong NP-hard when the number of destinations is input. These two new complexity results are an important supplement to the previous complexity results of this problem. Then, we propose two simple but novel approximation algorithms with a constant worst-case ratio, which is proved using an integer convex optimization model. Although our approximation algorithm applies to a few destinations, to our knowledge, it is the first approximation algorithm to handle the pure fixed-charge transportation problem.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100875"},"PeriodicalIF":0.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178626","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}
Pub Date : 2024-11-29DOI: 10.1016/j.disopt.2024.100865
Luerbio Faria, Mauro Nigro, Diana Sasaki
In 1988, Chetwynd and Hilton observed that a -total coloring induces a vertex coloring in the graph, they called it conformable. A -vertex coloring of a graph is called conformable if the number of color classes of parity different from that of is at most the deficiency of , where is the degree of a vertex of . In 1994, McDiarmid and Sánchez-Arroyo proved that deciding whether a graph has -total coloring is NP-complete even when is -regular bipartite with . However, the time-complexity of the problem of determining whether a graph admits a conformable coloring (Conformability problem) remains unknown. In this paper, we prove that Conformability problem is polynomial solvable for the class of -regular bipartite and for the class of subcubic graphs.
{"title":"A polynomial-time algorithm for conformable coloring on regular bipartite and subcubic graphs","authors":"Luerbio Faria, Mauro Nigro, Diana Sasaki","doi":"10.1016/j.disopt.2024.100865","DOIUrl":"10.1016/j.disopt.2024.100865","url":null,"abstract":"<div><div>In 1988, Chetwynd and Hilton observed that a <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-total coloring induces a vertex coloring in the graph, they called it conformable. A <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-vertex coloring 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 called <em>conformable</em> if the number of color classes of parity different from that of <span><math><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></math></span> is at most the deficiency <span><math><mrow><mo>def</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><mrow><mo>(</mo><mi>Δ</mi><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>, where <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> is the degree of a vertex <span><math><mi>v</mi></math></span> of <span><math><mi>V</mi></math></span>. In 1994, McDiarmid and Sánchez-Arroyo proved that deciding whether a graph <span><math><mi>G</mi></math></span> has <span><math><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-total coloring is NP-complete even when <span><math><mi>G</mi></math></span> is <span><math><mi>k</mi></math></span>-regular bipartite with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. However, the time-complexity of the problem of determining whether a graph admits a conformable coloring (<span>Conformability</span> problem) remains unknown. In this paper, we prove that <span>Conformability</span> problem is polynomial solvable for the class of <span><math><mi>k</mi></math></span>-regular bipartite and for the class of subcubic graphs.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"55 ","pages":"Article 100865"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745017","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}
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