Pub Date : 2026-02-03DOI: 10.1016/j.disopt.2026.100937
Nils Hausbrandt, Stefan Ruzika
In this article, we introduce the parametric matroid -interdiction problem, where is a fixed number of elements allowed to be interdicted. Each element of the matroid’s ground set is assigned a weight that depends linearly on a real parameter from a given interval. The goal is to compute, for each possible parameter value, a set of -most vital elements with corresponding objective value the deletion of which causes a maximum increase of the weight of a minimal basis. We show that such a set, which of course depends on the parameter, can only change polynomially often if the parameter varies. We develop several exact algorithms to solve the problem that have polynomial running times if an independence test can be performed in polynomial time.
{"title":"The parametric matroid ℓ-interdiction problem","authors":"Nils Hausbrandt, Stefan Ruzika","doi":"10.1016/j.disopt.2026.100937","DOIUrl":"10.1016/j.disopt.2026.100937","url":null,"abstract":"<div><div>In this article, we introduce the parametric matroid <span><math><mi>ℓ</mi></math></span>-interdiction problem, where <span><math><mrow><mi>ℓ</mi><mo>∈</mo><mi>N</mi></mrow></math></span> is a fixed number of elements allowed to be interdicted. Each element of the matroid’s ground set is assigned a weight that depends linearly on a real parameter from a given interval. The goal is to compute, for each possible parameter value, a set of <span><math><mi>ℓ</mi></math></span>-most vital elements with corresponding objective value the deletion of which causes a maximum increase of the weight of a minimal basis. We show that such a set, which of course depends on the parameter, can only change polynomially often if the parameter varies. We develop several exact algorithms to solve the problem that have polynomial running times if an independence test can be performed in polynomial time.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"60 ","pages":"Article 100937"},"PeriodicalIF":1.6,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146098579","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 : 2026-01-16DOI: 10.1016/j.disopt.2026.100930
Xichan Liu, Ligong Wang
<div><div>Let <span><math><mi>G</mi></math></span> be a graph with adjacency matrix <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and degree diagonal matrix <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In 2017, Nikiforov (2017) defined the matrix <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for any real <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>. The largest eigenvalue of <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called the spectral radius of <span><math><mi>G</mi></math></span>, while the largest eigenvalue of <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectral radius of <span><math><mi>G</mi></math></span>. Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span> be the set of graphs of order <span><math><mi>n</mi></math></span> with independence number <span><math><mi>i</mi></math></span>. Recently, for all graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span> having the minimum or the maximum of <span><math><mi>A</mi></math></span>, <span><math><mi>Q</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectral radius where <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mspace></mspace><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> there are some results given by Xu, Li and Sun et al., respectively. In 2022, Luo and Guo (2022) determined all graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msub></math></span> having the minimum spectral radius. In this paper, we characterize the graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msub></math></span> having the minimum and the maximum <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectr
设G为具有邻接矩阵a (G)和度对角矩阵D(G)的图。2017年，Nikiforov（2017）定义了对于任意实数α∈[0,1]，矩阵Aα(G)=α d (G)+(1−α)A(G)。A(G)的最大特征值称为G的谱半径，而Aα(G)的最大特征值称为G的Aα谱半径。设Gn i为独立数为i的n阶图的集合。最近，对于Gn i中所有具有A， Q和Aα谱半径最小或最大的图，其中i∈{1,2，⌊n2⌋≤n2≤+1,n−3,n−2,n−1},Xu， Li和Sun等人分别给出了一些结果。2022年，Luo和Guo（2022）确定了Gn，n−4中光谱半径最小的所有图。在本文中，我们分别刻画了在Gn，n−4中对于α∈[12,1]具有最小和最大Aα谱半径的图。
{"title":"The Aα spectral radius of graphs with given independence number n−4","authors":"Xichan Liu, Ligong Wang","doi":"10.1016/j.disopt.2026.100930","DOIUrl":"10.1016/j.disopt.2026.100930","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with adjacency matrix <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and degree diagonal matrix <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In 2017, Nikiforov (2017) defined the matrix <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for any real <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>. The largest eigenvalue of <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called the spectral radius of <span><math><mi>G</mi></math></span>, while the largest eigenvalue of <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is called the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectral radius of <span><math><mi>G</mi></math></span>. Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span> be the set of graphs of order <span><math><mi>n</mi></math></span> with independence number <span><math><mi>i</mi></math></span>. Recently, for all graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span> having the minimum or the maximum of <span><math><mi>A</mi></math></span>, <span><math><mi>Q</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectral radius where <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mspace></mspace><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> there are some results given by Xu, Li and Sun et al., respectively. In 2022, Luo and Guo (2022) determined all graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msub></math></span> having the minimum spectral radius. In this paper, we characterize the graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msub></math></span> having the minimum and the maximum <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectr","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100930"},"PeriodicalIF":1.6,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978198","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 : 2026-01-12DOI: 10.1016/j.disopt.2026.100929
Brahim Chaourar
Let be an undirected graph. An independent result of Maurras (Maurras, 1975), and Grötschel and Padberg (Grötschel and Padberg, 1979b) implies a characterization of the facets of the subtour polytope when is complete. In this paper, we generalize this result to arbitrary graphs.
{"title":"The facets of the subtour polytope","authors":"Brahim Chaourar","doi":"10.1016/j.disopt.2026.100929","DOIUrl":"10.1016/j.disopt.2026.100929","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be an undirected graph. An independent result of Maurras (Maurras, 1975), and Grötschel and Padberg (Grötschel and Padberg, 1979b) implies a characterization of the facets of the subtour polytope when <span><math><mi>G</mi></math></span> is complete. In this paper, we generalize this result to arbitrary graphs.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100929"},"PeriodicalIF":1.6,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978273","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 : 2026-01-05DOI: 10.1016/j.disopt.2025.100926
Ankit Bansal
This paper introduces a novel class of valid inequalities for the Time-Indexed Non-Preemptive Single Machine Scheduling Problem (T-SMSP). Under some assumptions on the length of the planning horizon, these inequalities are proven to define facets for the convex hull of a relaxation of T-SMSP that is tighter than the one considered in the literature. Furthermore, it is shown that the proposed set of valid inequalities are not dominated by certain existing valid inequalities for this problem and they either dominate or are equivalent to some of these existing valid inequalities. The computational performance of these inequalities, when integrated into a cutting-plane algorithm, shows significant promise.
{"title":"Valid inequalities for the Time-Indexed Non-Preemptive Single Machine Scheduling Problem","authors":"Ankit Bansal","doi":"10.1016/j.disopt.2025.100926","DOIUrl":"10.1016/j.disopt.2025.100926","url":null,"abstract":"<div><div>This paper introduces a novel class of valid inequalities for the Time-Indexed Non-Preemptive Single Machine Scheduling Problem (<span>T-SMSP</span>). Under some assumptions on the length of the planning horizon, these inequalities are proven to define facets for the convex hull of a relaxation of <span>T-SMSP</span> that is tighter than the one considered in the literature. Furthermore, it is shown that the proposed set of valid inequalities are not dominated by certain existing valid inequalities for this problem and they either dominate or are equivalent to some of these existing valid inequalities. The computational performance of these inequalities, when integrated into a cutting-plane algorithm, shows significant promise.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100926"},"PeriodicalIF":1.6,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927102","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 : 2026-01-04DOI: 10.1016/j.disopt.2026.100928
Kexiang Xu, Ximei Chen
For a graph , we denote by the number of independent sets, including the empty set, in . A Halin graph is a plane graph which consists of a plane embedding of a tree of order at least 4 without a vertex of degree 2 and a cycle connecting all leaves of . In this paper we characterize the maximum general Halin graphs and maximum cubic Halin graphs, respectively, of order with respect to the number of independent sets. Moreover, the existence of asymptotic lower bounds is provided on the number of independent sets of general and cubic Halin graphs, respectively. Also two open problems are provided for future research.
{"title":"On the number of independent sets in Halin graphs","authors":"Kexiang Xu, Ximei Chen","doi":"10.1016/j.disopt.2026.100928","DOIUrl":"10.1016/j.disopt.2026.100928","url":null,"abstract":"<div><div>For a graph <span><math><mi>G</mi></math></span>, we denote by <span><math><mrow><mi>σ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> the number of independent sets, including the empty set, in <span><math><mi>G</mi></math></span>. A Halin graph is a plane graph which consists of a plane embedding of a tree <span><math><mi>T</mi></math></span> of order at least 4 without a vertex of degree 2 and a cycle <span><math><mi>C</mi></math></span> connecting all leaves of <span><math><mi>T</mi></math></span>. In this paper we characterize the maximum general Halin graphs and maximum cubic Halin graphs, respectively, of order <span><math><mi>n</mi></math></span> with respect to the number of independent sets. Moreover, the existence of asymptotic lower bounds is provided on the number of independent sets of general and cubic Halin graphs, respectively. Also two open problems are provided for future research.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100928"},"PeriodicalIF":1.6,"publicationDate":"2026-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927101","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 : 2025-12-26DOI: 10.1016/j.disopt.2025.100927
Javier Marenco
The manufacturer’s pallet loading problem asks for a maximum-sized axis-parallel packing of identical rectangles within an enclosing axis-parallel rectangular pallet. This problem has been widely studied in the literature, and most instances coming from practical settings have been solved with optimality. Contrary to the situation in other classical combinatorial optimization problems, integer programming techniques are not at the core of the most successful exact procedures for this problem, which are based on combinatorial exhaustive search coupled with sophisticated heuristics. In this work we are interested in evaluating whether it is possible to extend the reach of integer programming techniques at solving this problem. To this end, we evaluate two different column generation procedures for this problem, one of them based on a clustering idea by Ribeiro and Lorena, and the other based on a simultaneous row and column generation procedure proposed by Feillet et al. We show that this second procedure is effective and allows to solve with optimality many instances that were, until now, open.
{"title":"Column (and row) generation algorithms for the pallet loading problem","authors":"Javier Marenco","doi":"10.1016/j.disopt.2025.100927","DOIUrl":"10.1016/j.disopt.2025.100927","url":null,"abstract":"<div><div>The manufacturer’s pallet loading problem asks for a maximum-sized axis-parallel packing of identical rectangles within an enclosing axis-parallel rectangular pallet. This problem has been widely studied in the literature, and most instances coming from practical settings have been solved with optimality. Contrary to the situation in other classical combinatorial optimization problems, integer programming techniques are not at the core of the most successful exact procedures for this problem, which are based on combinatorial exhaustive search coupled with sophisticated heuristics. In this work we are interested in evaluating whether it is possible to extend the reach of integer programming techniques at solving this problem. To this end, we evaluate two different column generation procedures for this problem, one of them based on a clustering idea by Ribeiro and Lorena, and the other based on a simultaneous row and column generation procedure proposed by Feillet et al. We show that this second procedure is effective and allows to solve with optimality many instances that were, until now, open.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100927"},"PeriodicalIF":1.6,"publicationDate":"2025-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841827","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 : 2025-12-16DOI: 10.1016/j.disopt.2025.100925
Frank de Meijer , Dion Gijswijt , Renata Sotirov
A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding of qubits in a given architecture such that the total number of SWAP gates to be inserted is minimized. In this paper we study the NNCP on general quantum architectures. Building upon an existing linear programming formulation, we show how the model can be reduced by exploiting the symmetries of the graph underlying the formulation. The resulting model is equivalent to a generalized network flow problem and follows from an in-depth analysis of the automorphism group of specific Cayley graphs. As a byproduct of our approach, we show that the NNCP is polynomial time solvable for several classes of symmetric quantum architectures. Numerical tests on various architectures indicate that the reduction in the number of variables and constraints is on average at least 90%. In particular, NNCP instances on the star architecture can be solved for quantum circuits up to 100 qubits and more than 1000 quantum gates within a very short computation time. These results are far beyond the computational capacity when solving the instances without the exploitation of symmetries.
{"title":"Exploiting symmetries in optimal quantum circuit design","authors":"Frank de Meijer , Dion Gijswijt , Renata Sotirov","doi":"10.1016/j.disopt.2025.100925","DOIUrl":"10.1016/j.disopt.2025.100925","url":null,"abstract":"<div><div>A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding of qubits in a given architecture such that the total number of SWAP gates to be inserted is minimized. In this paper we study the NNCP on general quantum architectures. Building upon an existing linear programming formulation, we show how the model can be reduced by exploiting the symmetries of the graph underlying the formulation. The resulting model is equivalent to a generalized network flow problem and follows from an in-depth analysis of the automorphism group of specific Cayley graphs. As a byproduct of our approach, we show that the NNCP is polynomial time solvable for several classes of symmetric quantum architectures. Numerical tests on various architectures indicate that the reduction in the number of variables and constraints is on average at least 90%. In particular, NNCP instances on the star architecture can be solved for quantum circuits up to 100 qubits and more than 1000 quantum gates within a very short computation time. These results are far beyond the computational capacity when solving the instances without the exploitation of symmetries.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100925"},"PeriodicalIF":1.6,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760627","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 : 2025-11-01DOI: 10.1016/j.disopt.2025.100915
Nanlan Zhang , Fanrong Xie
Capacitated bilevel time minimizing transportation problem (CBTMTP) is a generalization of bilevel time minimizing transportation problem (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.
{"title":"An efficient solution approach to capacitated bilevel time minimizing transportation problem","authors":"Nanlan Zhang , 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}
Pub Date : 2025-10-24DOI: 10.1016/j.disopt.2025.100914
S. Gueye , P. Michelon
To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of (integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches.
{"title":"A preprocessing technique for quadratic unconstrained binary optimization","authors":"S. Gueye , P. Michelon","doi":"10.1016/j.disopt.2025.100914","DOIUrl":"10.1016/j.disopt.2025.100914","url":null,"abstract":"<div><div>To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of <span><math><mi>p</mi></math></span> (integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100914"},"PeriodicalIF":1.6,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363123","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 : 2025-10-03DOI: 10.1016/j.disopt.2025.100913
Nina Chiarelli , Clément Dallard , Andreas Darmann , Stefan Lendl , Martin Milanič , Peter Muršič , Ulrich Pferschy
We consider the task of allocating indivisible items to agents, when the agents’ preferences over the items are identical. The preferences are captured by means of a directed acyclic graph, with vertices representing items and an edge , meaning that each of the agents prefers item over item . The dissatisfaction of an agent is measured by the number of items that the agent does not receive and for which it also does not receive any more preferred item. The aim is to allocate the items to the agents in a fair way, i.e., to minimize the maximum dissatisfaction among the agents. We study the status of computational complexity of that problem and establish the following dichotomy: the problem is NP-hard for the case of at least three agents, even on fairly restricted graphs, but polynomially solvable for two agents. We also provide several polynomial-time results with respect to different underlying graph structures, such as graphs of width at most two and tree-like structures such as stars and matchings. These findings are complemented with fixed parameter tractability results related to path modules and independent set modules. Techniques employed in the paper include bottleneck assignment problem, greedy algorithm, dynamic programming, maximum network flow, and integer linear programming.
{"title":"Minimizing maximum dissatisfaction in the allocation of indivisible items under a common preference graph","authors":"Nina Chiarelli , Clément Dallard , Andreas Darmann , Stefan Lendl , Martin Milanič , Peter Muršič , Ulrich Pferschy","doi":"10.1016/j.disopt.2025.100913","DOIUrl":"10.1016/j.disopt.2025.100913","url":null,"abstract":"<div><div>We consider the task of allocating indivisible items to agents, when the agents’ preferences over the items are identical. The preferences are captured by means of a directed acyclic graph, with vertices representing items and an edge <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span>, meaning that each of the agents prefers item <span><math><mi>a</mi></math></span> over item <span><math><mi>b</mi></math></span>. The dissatisfaction of an agent is measured by the number of items that the agent does not receive and for which it also does not receive any more preferred item. The aim is to allocate the items to the agents in a fair way, i.e., to minimize the maximum dissatisfaction among the agents. We study the status of computational complexity of that problem and establish the following dichotomy: the problem is <span>NP</span>-hard for the case of at least three agents, even on fairly restricted graphs, but polynomially solvable for two agents. We also provide several polynomial-time results with respect to different underlying graph structures, such as graphs of width at most two and tree-like structures such as stars and matchings. These findings are complemented with fixed parameter tractability results related to path modules and independent set modules. Techniques employed in the paper include bottleneck assignment problem, greedy algorithm, dynamic programming, maximum network flow, and integer linear programming.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100913"},"PeriodicalIF":1.6,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222444","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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