Pub Date : 2024-10-01DOI: 10.1016/j.disopt.2024.100862
Shmuel Onn
We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.
{"title":"Circuit and Graver walks and linear and integer programming","authors":"Shmuel Onn","doi":"10.1016/j.disopt.2024.100862","DOIUrl":"10.1016/j.disopt.2024.100862","url":null,"abstract":"<div><div>We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100862"},"PeriodicalIF":0.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425831","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 : 2024-09-17DOI: 10.1016/j.disopt.2024.100860
Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour
We consider the Min-Sum -Clustering (-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph and a parameter , the goal is to partition the points into clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.
The -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 in polynomial time for general metrics and an approximation ratio in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for -MSC (when 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 -MSC to the balanced -median problem and design approximation algorithms for the latter to obtain an approximation for -MSC.
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 -MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced -median and is a trade off between center-based clustering problems (such as balanced -median) and pair-wise clustering (such as Min-Sum -clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.
我们考虑的是最小和 k 聚类（k-MSC）问题。给定一个度量中的点集（由边加权图 G=(V,E) 表示）和一个参数 k，目标是将点 V 划分为 k 个聚类，使得同一聚类中所有点对之间的距离之和最小。Behsaz 等人（2019）针对该问题获得的已知最佳近似算法在一般度量条件下的多项式时间内达到了 O(log|V|)的近似率，在具有约束倍维度的度量条件下的准多项式时间内达到了 2+ϵ 的近似率。在我们的研究之前，还没有针对任何非三维度量的 k-MSC 近似方案（当 k 是输入的一部分时）。事实上，之前的大部分研究都依赖于一个简单的事实，即从 k-MSC 到平衡 k-median 问题有一个 2 近似的还原，并为后者设计近似算法，从而得到 k-MSC 的近似值。在本文中，我们首次获得了该问题的准多项式时间近似方案（QPTAS），该方案适用于有界树宽、有界公路维数、有界倍维数（包括固定维数欧几里得度量）的图以及平面图和无次要图所诱导的度量。我们通过引入一个新的聚类问题（我们称之为 min-hub 聚类），绕过了 k-MSC 的 2 的障碍，它是平衡 k-median 的广义化，是基于中心的聚类问题（如平衡 k-median）和成对聚类问题（如 Min-Sum k-clustering）之间的权衡。然后，我们展示了如何在某些度量类别上找到 Min-hub 聚类的近似方案。
{"title":"Approximation schemes for Min-Sum k-Clustering","authors":"Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour","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":"54 ","pages":"Article 100860"},"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}
Pub Date : 2024-09-14DOI: 10.1016/j.disopt.2024.100861
Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner
Došlić et al. defined the Mostar index of a graph as , where, for an edge of , the term denotes the number of vertices of that have a smaller distance in to than to . For a graph of order and maximum degree at most , we show where only depends on and the term only depends on . Furthermore, for integers and at least 3, we show the existence of a -regular graph of order at least with where
Došlić 等人将图 G 的莫斯塔尔指数定义为 Mo(G)=∑uv∈E(G)|nG(u,v)-nG(v,u)| 其中，对于 G 的边 uv，nG(u,v) 表示 G 中与 u 的距离小于与 v 的距离的顶点数。对于阶数为 n、最大度数最多为 Δ 的图 G，我们证明了 Mo(G)≤Δ2n2-(1-o(1))cΔnlog(log(n)) ，其中 cΔ>0 只取决于 Δ，而 o(1) 项只取决于 n。此外，对于 n0 和 Δ 至少为 3 的整数，我们证明存在阶数至少为 n0 的 Δ 不规则图，其 Mo(G)≥Δ2n2-cΔ′nlog(n) ，其中 cΔ′>0 只取决于 Δ。
{"title":"Mostar index and bounded maximum degree","authors":"Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner","doi":"10.1016/j.disopt.2024.100861","DOIUrl":"10.1016/j.disopt.2024.100861","url":null,"abstract":"<div><p>Došlić et al. defined the Mostar index of a graph <span><math><mi>G</mi></math></span> as <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder><mrow><mo>|</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, where, for an edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>, the term <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of vertices of <span><math><mi>G</mi></math></span> that have a smaller distance in <span><math><mi>G</mi></math></span> to <span><math><mi>u</mi></math></span> than to <span><math><mi>v</mi></math></span>. For a graph <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span> and maximum degree at most <span><math><mi>Δ</mi></math></span>, we show <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> only depends on <span><math><mi>Δ</mi></math></span> and the <span><math><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> term only depends on <span><math><mi>n</mi></math></span>. Furthermore, for integers <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><mi>Δ</mi></math></span> at least 3, we show the existence of a <span><math><mi>Δ</mi></math></span>-regular graph of order <span><math><mi>n</mi></math></span> at least <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> with <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><mat","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100861"},"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}
Pub Date : 2024-09-14DOI: 10.1016/j.disopt.2024.100849
Brady Hunsaker, Craig Tovey
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. ) 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 , it is NP-complete to separate over the class of constraints on odd sets of size or .
我们研究了介于分数匹配和完全匹配多边形之间的多边形，方法是施加严格的奇数集（开花）约束子集。对于稀疏约束，如果只存在以给定大小（如 ≤9+|V|/6）为边界的所有奇数集的约束，我们会给出一种多项式时间分离算法。我们的算法还能解决更普遍的问题，即根据定义节点集的万有引力和成本的上限找到 T 切。相反，关于密集约束，我们证明了对于每一个 0<α≤12，在大小为 2⌊(1+α|V|)/2⌋-1或≥α|V|的奇数集合上分离一类约束是 NP-完全的。
{"title":"Easy and hard separation of sparse and dense odd-set constraints in matching","authors":"Brady Hunsaker, Craig Tovey","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><</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":"54 ","pages":"Article 100849"},"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}
Pub Date : 2024-09-12DOI: 10.1016/j.disopt.2024.100859
Todd Easton , Jennifer Tryon , Fabio Vitor
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.
{"title":"Two-set inequalities for the binary knapsack polyhedra","authors":"Todd Easton , Jennifer Tryon , Fabio Vitor","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":"54 ","pages":"Article 100859"},"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}
Pub Date : 2024-08-19DOI: 10.1016/j.disopt.2024.100858
Abraham P. Punnen, Navpreet Kaur Dhanda
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.
{"title":"Revisiting some classical linearizations of the quadratic binary optimization problem and linkages with constraint aggregations","authors":"Abraham P. Punnen, Navpreet Kaur Dhanda","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":"54 ","pages":"Article 100858"},"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}
Pub Date : 2024-08-13DOI: 10.1016/j.disopt.2024.100857
Victoria Kaial, Hervé Kerivin , Annegret K. Wagler
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.
{"title":"On non-superperfection of edge intersection graphs of paths","authors":"Victoria Kaial, Hervé Kerivin , Annegret K. Wagler","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":"54 ","pages":"Article 100857"},"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}
Pub Date : 2024-08-01DOI: 10.1016/j.disopt.2024.100850
Wayne Goddard , Michael A. Henning
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 has a packing of size at least .
{"title":"The packing number of cubic graphs","authors":"Wayne Goddard , Michael A. Henning","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":"53 ","pages":"Article 100850"},"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}
Pub Date : 2024-08-01DOI: 10.1016/j.disopt.2024.100848
Jannis Blauth , Stephan Held , Dirk Müller , Niklas Schlomberg , Vera Traub , Thorben Tröbst , Jens Vygen
We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results.
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.
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.
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.
{"title":"Vehicle routing with time-dependent travel times: Theory, practice, and benchmarks","authors":"Jannis Blauth , Stephan Held , Dirk Müller , Niklas Schlomberg , Vera Traub , Thorben Tröbst , Jens Vygen","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":"53 ","pages":"Article 100848"},"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}
Pub Date : 2024-05-28DOI: 10.1016/j.disopt.2024.100845
Nadia Babou , Mourad Boudhar , Djamal Rebaine
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 -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.
{"title":"Two-machine job shop problem with a single server and sequence-independent non-anticipatory set-up times","authors":"Nadia Babou , Mourad Boudhar , 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":"53 ","pages":"Article 100845"},"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}
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