Discrete Optimization最新文献

{"title":"Approximation schemes for Min-Sum k-Clustering","authors":"","doi":"10.1016/j.disopt.2024.100860","DOIUrl":"10.1016/j.disopt.2024.100860","url":null,"abstract":"<div><p>We consider the Min-Sum <span><math><mi>k</mi></math></span>-Clustering (<span><math><mi>k</mi></math></span>-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted 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> and a parameter <span><math><mi>k</mi></math></span>, the goal is to partition the points <span><math><mi>V</mi></math></span> into <span><math><mi>k</mi></math></span> clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.</p><p>The <span><math><mi>k</mi></math></span>-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow></math></span> in polynomial time for general metrics and an approximation ratio <span><math><mrow><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></math></span> in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for <span><math><mi>k</mi></math></span>-MSC (when <span><math><mi>k</mi></math></span> is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from <span><math><mi>k</mi></math></span>-MSC to the balanced <span><math><mi>k</mi></math></span>-median problem and design approximation algorithms for the latter to obtain an approximation for <span><math><mi>k</mi></math></span>-MSC.</p><p>In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for <span><math><mi>k</mi></math></span>-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced <span><math><mi>k</mi></math></span>-median and is a trade off between center-based clustering problems (such as balanced <span><math><mi>k</mi></math></span>-median) and pair-wise clustering (such as Min-Sum <span><math><mi>k</mi></math></span>-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000392/pdfft?md5=0298fb9c3c75e407870e412a1aae1a26&pid=1-s2.0-S1572528624000392-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242680","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}

{"title":"Mostar index and bounded maximum degree","authors":"","doi":"10.1016/j.disopt.2024.100861","DOIUrl":"10.1016/j.disopt.2024.100861","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Došlić et al. defined the Mostar index of a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;o&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;munder&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;E&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;/munder&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where, for an edge &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, the term &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denotes the number of vertices of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; that have a smaller distance in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; than to &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. For a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and maximum degree at most &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, we show &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;o&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;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&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;/msup&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;o&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; only depends on &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and the &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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; term only depends on &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Furthermore, for integers &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; at least 3, we show the existence of a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-regular graph of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; at least &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;o&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;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&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;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;mat","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000409/pdfft?md5=e34071dea61722ee4baab21c7039f3bf&pid=1-s2.0-S1572528624000409-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228672","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}

{"title":"Easy and hard separation of sparse and dense odd-set constraints in matching","authors":"","doi":"10.1016/j.disopt.2024.100849","DOIUrl":"10.1016/j.disopt.2024.100849","url":null,"abstract":"<div><p>We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. <span><math><mrow><mo>≤</mo><mn>9</mn><mo>+</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>/</mo><mn>6</mn></mrow></math></span>) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, it is NP-complete to separate over the class of constraints on odd sets of size <span><math><mrow><mn>2</mn><mrow><mo>⌊</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> or <span><math><mrow><mo>≥</mo><mi>α</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228671","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":"Two-set inequalities for the binary knapsack polyhedra","authors":"","doi":"10.1016/j.disopt.2024.100859","DOIUrl":"10.1016/j.disopt.2024.100859","url":null,"abstract":"<div><p>This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172518","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":"Revisiting some classical linearizations of the quadratic binary optimization problem and linkages with constraint aggregations","authors":"","doi":"10.1016/j.disopt.2024.100858","DOIUrl":"10.1016/j.disopt.2024.100858","url":null,"abstract":"<div><p>In this paper, we examine explicit linearization models associated with the quadratic unconstrained binary optimization problem (QUBO). After reviewing some well-known basic linearization techniques, we present a new explicit linearization technique (PK) for QUBO with interesting properties. In particular, PK yields the same LP relaxation value as that of the standard linearization of QUBO while employing less number of constraints. We then show that using weighted aggregations of selected constraints of the basic linearization models, several new and effective linearization models of QUBO can be developed. Although aggregation of constraints has been studied in the past for solving systems of Diophantine equations in non-negative variables, none of them resulted in practical models, particularly due to the large size of the associated multipliers. For our aggregation based models, the multipliers can be of any positive real number. Moreover, we show that by choosing the multipliers appropriately, the LP relaxations of the resulting linearizations have optimal objective function values identical to that of the corresponding non-aggregated models. Theoretical and experimental comparisons of the new and existing explicit linearization models are also provided. Although our discussions were focused primarily on QUBO, the models and results we obtained extend naturally to the quadratic binary optimization problem with linear constraints.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000379/pdfft?md5=88f5ed7ffa74e8ecf9bb69fd52845013&pid=1-s2.0-S1572528624000379-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012067","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}

{"title":"On non-superperfection of edge intersection graphs of paths","authors":"","doi":"10.1016/j.disopt.2024.100857","DOIUrl":"10.1016/j.disopt.2024.100857","url":null,"abstract":"<div><p>The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141979335","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 packing number of cubic graphs","authors":"","doi":"10.1016/j.disopt.2024.100850","DOIUrl":"10.1016/j.disopt.2024.100850","url":null,"abstract":"<div><p>A packing in a graph is a set of vertices that are mutually distance at least 3 apart. By using optimization and linear programming to help analyze the greedy algorithm, we improve on a result of Favaron and show that every connected cubic graph of order <span><math><mi>n</mi></math></span> has a packing of size at least <span><math><mrow><mfrac><mrow><mn>17</mn></mrow><mrow><mn>132</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938261","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":"Vehicle routing with time-dependent travel times: Theory, practice, and benchmarks","authors":"","doi":"10.1016/j.disopt.2024.100848","DOIUrl":"10.1016/j.disopt.2024.100848","url":null,"abstract":"<div><p>We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results.</p><p>First, we study basic operations on piecewise linear arrival time functions. In particular, we devise a faster algorithm to compute the pointwise minimum of a set of piecewise linear functions and a monotonicity-preserving variant of the Imai–Iri algorithm to approximate an arrival time function with fewer breakpoints.</p><p>Next, we show how to evaluate insertion and deletion operations in tours efficiently and update the underlying data structure faster than previously known when a tour changes. Evaluating a tour also requires a scheduling step which is non-trivial in the presence of time windows and time-dependent travel times. We show how to perform this in linear time.</p><p>Based on these results, we develop a local search heuristic to solve real-world vehicle routing problems with various constraints efficiently and report experimental results on classical benchmarks. Since most of these do not have time-dependent travel times, we generate and publish new benchmark instances that are based on real-world data. This data also demonstrates the importance of considering time-dependent travel times in instances with tight time windows.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000276/pdfft?md5=22a97d8c380cf8927372907e85523ccf&pid=1-s2.0-S1572528624000276-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938257","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}

{"title":"Two-machine job shop problem with a single server and sequence-independent non-anticipatory set-up times","authors":"Nadia Babou ,&nbsp;Mourad Boudhar ,&nbsp;Djamal Rebaine","doi":"10.1016/j.disopt.2024.100845","DOIUrl":"10.1016/j.disopt.2024.100845","url":null,"abstract":"<div><p>We address in this paper the two-machine job shop scheduling problem with a single server that sets up the jobs before they get processed on the machines. The server is only needed during the set-up and becomes free at the end of this phase. Moreover, the set-ups are non-anticipatory and the set-up times are sequence-independent. We seek a schedule that minimizes the overall completion time, also called the makespan. We propose several lower bounds to the problem and prove the <span><math><mrow><mi>N</mi><mi>P</mi></mrow></math></span>-hardness in the strong sense of two restricted cases. In addition, we present a linear time algorithm for a special case. In order to solve the general problem, we develop a genetic and simulated annealing algorithms that use feasibility guaranteed procedures. An experimental study is carried out to analyze the performance of these meta-heuristic methods.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189692","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 non-stop disjoint trajectories problem","authors":"Benno Hoch ,&nbsp;Frauke Liers ,&nbsp;Sarah Neumann ,&nbsp;Francisco Javier Zaragoza Martínez","doi":"10.1016/j.disopt.2024.100837","DOIUrl":"https://doi.org/10.1016/j.disopt.2024.100837","url":null,"abstract":"<div><p>Consider an undirected network with traversal times on its edges and a set of commodities with connection requests from sources to destinations and release dates. The non-stop disjoint trajectories problem is to find trajectories that fulfill all requests, such that the commodities never meet. In this extension to the <span><math><mi>NP</mi></math></span>-complete disjoint paths problem, trajectories must satisfy a non-stop condition, which disallows waiting at vertices or along arcs. This problem variant appears, for example, when disjoint aircraft trajectories shall be determined or in bufferless packet routing. We study the border of tractability for feasibility and optimization problems on three graph classes that are frequently used where space and time are discretized simultaneously: the path, the grid, and the mesh. We show that if all commodities have a common release date, feasibility can be decided in polynomial time on paths. For the unbounded mesh and unit-costs, we show how to construct optimal trajectories. In contrast, if commodities have individual release intervals and turns are forbidden, then even feasibility is <span><math><mi>NP</mi></math></span>-complete for the path. For the mesh and arbitrary edge costs, with individual release dates and turning abilities of commodities restricted to at most 90°, we show that optimization and approximation are not fixed-parameter tractable.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000161/pdfft?md5=58a07aefa7044d86caaa7f7f71ef6b5b&pid=1-s2.0-S1572528624000161-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140349866","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}