Discrete Optimization最新文献

{"title":"Spectral radius and size conditions for fractional (a,b,l)-critical graphs","authors":"Zengzhao Xu,&nbsp;Ligong Wang,&nbsp;Weige Xi","doi":"10.1016/j.disopt.2026.100958","DOIUrl":"10.1016/j.disopt.2026.100958","url":null,"abstract":"<div><div>In recent years, research on the spectral extremal problems of fractional <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span>-factors has attracted the attention of many scholars. The main content of our research is the generalization of fractional <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span>-factors, namely fractional <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></math></span>-critical graph. Let <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span> be two positive integers. A graph <span><math><mi>G</mi></math></span> is called a fractional <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></math></span>-critical graph if after deleting any <span><math><mi>l</mi></math></span> vertices of <span><math><mi>G</mi></math></span> the remaining graph of <span><math><mi>G</mi></math></span> has a fractional <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span>-factor. In this paper, we present spectral radius and size conditions for a graph to be fractional <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></math></span>-critical.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"61 ","pages":"Article 100958"},"PeriodicalIF":0.8,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148736405","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 isolation number of graphs with minimum degree four","authors":"Wayne Goddard,&nbsp;Michael A. Henning","doi":"10.1016/j.disopt.2026.100959","DOIUrl":"10.1016/j.disopt.2026.100959","url":null,"abstract":"<div><div>An isolating set in a graph <span><math><mi>G</mi></math></span> is a set <span><math><mi>S</mi></math></span> of vertices such that removing <span><math><mi>S</mi></math></span> and its neighborhood leaves no edge. The isolation number of <span><math><mi>G</mi></math></span> (also known as the vertex-edge domination number), denoted <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum size among all isolating sets of <span><math><mi>G</mi></math></span>. We provide a technique for proving upper bounds on this parameter for graphs with a given minimum degree. For example, we show that if <span><math><mi>G</mi></math></span> has order <span><math><mi>n</mi></math></span> and minimum degree at least 4, then <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>13</mn><mi>n</mi><mo>/</mo><mn>41</mn></mrow></math></span>, and if <span><math><mi>G</mi></math></span> is also triangle-free, then <span><math><mrow><mi>ι</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn><mi>n</mi><mo>/</mo><mn>10</mn></mrow></math></span>.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"61 ","pages":"Article 100959"},"PeriodicalIF":0.8,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148736404","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 inverse 1-median problem on median graphs under l2-norm","authors":"Kien Trung Nguyen,&nbsp;Nguyen Thanh Hung,&nbsp;Nguyen Thanh Luan,&nbsp;Cao Minh Quang","doi":"10.1016/j.disopt.2026.100957","DOIUrl":"10.1016/j.disopt.2026.100957","url":null,"abstract":"<div><div>This paper addresses the inverse 1-median problem on median graphs, a broad class of graphs that includes trees, grid graphs, and hypercubes. The objective is to find the smallest perturbation to vertex weights, under the weighted <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-norm, that renders a prespecified target vertex optimal. Our approach begins by establishing a novel optimality criterion for the forward 1-median problem. This criterion provides the theoretical foundation to formulate the inverse problem as a convex quadratic program (QP). The main contribution of this work is an efficient algorithm that solves this QP by tackling its duality. This dual formulation reduces the problem’s complexity, as validated through computational experiments.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"61 ","pages":"Article 100957"},"PeriodicalIF":0.8,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148641224","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 parametric matroid ℓ-interdiction problem","authors":"Nils Hausbrandt,&nbsp;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":"2026-05-01","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}

{"title":"On the number of independent sets in Halin graphs","authors":"Kexiang Xu,&nbsp;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":"2026-02-01","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}

{"title":"Exploiting symmetries in optimal quantum circuit design","authors":"Frank de Meijer ,&nbsp;Dion Gijswijt ,&nbsp;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":"2026-02-01","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}

{"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":"2026-02-01","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}

{"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":"2026-02-01","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}

{"title":"The Aα spectral radius of graphs with given independence number n−4","authors":"Xichan Liu,&nbsp;Ligong Wang","doi":"10.1016/j.disopt.2026.100930","DOIUrl":"10.1016/j.disopt.2026.100930","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a graph with adjacency matrix &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and degree diagonal matrix &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In 2017, Nikiforov (2017) defined the matrix &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&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;α&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for any real &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The largest eigenvalue of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is called the spectral radius of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, while the largest eigenvalue of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is called the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; spectral radius of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the set of graphs of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with independence number &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Recently, for all graphs in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; having the minimum or the maximum of &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; spectral radius where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;⌊&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌋&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;⌈&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌉&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; there are some results given by Xu, Li and Sun et al., respectively. In 2022, Luo and Guo (2022) determined all graphs in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; having the minimum spectral radius. In this paper, we characterize the graphs in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; having the minimum and the maximum &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; spectr","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"59 ","pages":"Article 100930"},"PeriodicalIF":1.6,"publicationDate":"2026-02-01","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}

{"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":"2026-02-01","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}