Discrete Optimization最新文献_第10页

A column generation approach to the discrete barycenter problem 离散重心问题的一种列生成方法

IF 1.1 4区数学

Discrete Optimization Pub Date : 2022-02-01 DOI: 10.1016/j.disopt.2021.100674

Steffen Borgwardt , Stephan Patterson

{"title":"A column generation approach to the discrete barycenter problem","authors":"Steffen Borgwardt , Stephan Patterson","doi":"10.1016/j.disopt.2021.100674","DOIUrl":"https://doi.org/10.1016/j.disopt.2021.100674","url":null,"abstract":"<div>The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation.In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig–Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91757809","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}

引用次数: 5

Node-based valid inequalities for the optimal transmission switching problem 最优传输交换问题的基于节点的有效不等式

IF 1.1 4区数学

Discrete Optimization Pub Date : 2022-02-01 DOI: 10.1016/j.disopt.2021.100683

Santanu S. Dey , Burak Kocuk , Nicole Redder

引用次数: 3

Exact values of defective Ramsey numbers in graph classes 图类中有缺陷Ramsey数的精确值

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-11-01 DOI: 10.1016/j.disopt.2021.100673

Yunus Emre Demirci , Tınaz Ekim , John Gimbel , Mehmet Akif Yıldız

{"title":"Exact values of defective Ramsey numbers in graph classes","authors":"Yunus Emre Demirci , Tınaz Ekim , John Gimbel , Mehmet Akif Yıldız","doi":"10.1016/j.disopt.2021.100673","DOIUrl":"10.1016/j.disopt.2021.100673","url":null,"abstract":"<div>Given a graph <math><mi>G</mi></math>, a <math><mi>k</mi></math>-sparse <math><mi>j</mi></math>-set is a set of <math><mi>j</mi></math> vertices inducing a subgraph with maximum degree at most <math><mi>k</mi></math>. A <math><mi>k</mi></math>-dense <math><mi>i</mi></math>-set is a set of <math><mi>i</mi></math> vertices that is <math><mi>k</mi></math>-sparse in the complement of <math><mi>G</mi></math>. As a generalization of Ramsey numbers, the <math><mi>k</mi></math>-defective Ramsey number <math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></math> for the graph class <math><mi>G</mi></math> is defined as the smallest natural number <math><mi>n</mi></math> such that all graphs on <math><mi>n</mi></math> vertices in the class <math><mi>G</mi></math> have either a <math><mi>k</mi></math>-dense <math><mi>i</mi></math>-set or a <math><mi>k</mi></math>-sparse <math><mi>j</mi></math>-set. In this paper, we examine <math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></math> where <math><mi>G</mi></math> represents various graph classes. For forests and cographs, we give exact formulas for all defective Ramsey numbers. For cacti, bipartite graphs and split graphs, we derive defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54146644","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}

引用次数: 1

Efficient constructions of convex combinations for 2-edge-connected subgraphs on fundamental classes 基本类上2边连通子图凸组合的有效构造

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-11-01 DOI: 10.1016/j.disopt.2021.100659

Arash Haddadan , Alantha Newman

{"title":"Efficient constructions of convex combinations for 2-edge-connected subgraphs on fundamental classes","authors":"Arash Haddadan , Alantha Newman","doi":"10.1016/j.disopt.2021.100659","DOIUrl":"10.1016/j.disopt.2021.100659","url":null,"abstract":"<div>We present coloring-based algorithms for tree augmentation and use them to construct convex combinations of 2-edge-connected subgraphs. This classic tool has been applied previously to the problem, but our algorithms illustrate its flexibility, which – in coordination with the choice of spanning tree – can be used to obtain various properties (e.g., 2-vertex connectivity) that are useful in our applications.We use these coloring algorithms to design approximation algorithms for the 2-edge-connected multigraph problem (2ECM) and the 2-edge-connected spanning subgraph problem (2ECS) on two well-studied types of LP solutions. The first type of points, half-integer square points, belong to a class of fundamental extreme points, which exhibit the same integrality gap as the general case. For half-integer square points, the integrality gap for 2ECM is known to be between <math><mfrac><mrow><mn>6</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math> and <math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math>. We improve the upper bound to <math><mfrac><mrow><mn>9</mn></mrow><mrow><mn>7</mn></mrow></mfrac></math>. The second type of points we study are uniform points whose support is a 3-edge-connected graph and each entry is <math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math>. Although the best-known upper bound on the integrality gap of 2ECS for these points is less than <math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math>, previous results do not yield an efficient algorithm. We give the first approximation algorithm for 2ECS with ratio below <math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math> for this class of points.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2021.100659","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127663909","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}

引用次数: 6

An exact cutting plane method for k-submodular function maximization k次模函数最大化的精确切割平面方法

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-11-01 DOI: 10.1016/j.disopt.2021.100670

Qimeng Yu, Simge Küçükyavuz

{"title":"An exact cutting plane method for k-submodular function maximization","authors":"Qimeng Yu, Simge Küçükyavuz","doi":"10.1016/j.disopt.2021.100670","DOIUrl":"10.1016/j.disopt.2021.100670","url":null,"abstract":"<div>A natural and important generalization of submodularity – <math><mi>k</mi></math>-submodularity – applies to set functions with <math><mi>k</mi></math> arguments and appears in a broad range of applications, such as infrastructure design, machine learning, and healthcare. In this paper, we study maximization problems with <math><mi>k</mi></math>-submodular objective functions. We propose valid linear inequalities, namely the <math><mi>k</mi></math>-submodular inequalities, for the hypograph of any <math><mi>k</mi></math>-submodular function. This class of inequalities serves as a novel generalization of the well-known submodular inequalities. We show that maximizing a <math><mi>k</mi></math>-submodular function is equivalent to solving a mixed-integer linear program with exponentially many <math><mi>k</mi></math>-submodular inequalities. Using this representation in a delayed constraint generation framework, we design the first exact algorithm, that is not a complete enumeration method, to solve general <math><mi>k</mi></math>-submodular maximization problems. Our computational experiments on the multi-type sensor placement problems demonstrate the efficiency of our algorithm in constrained nonlinear <math><mi>k</mi></math>-submodular maximization problems for which no alternative compact mixed-integer linear formulations are available. The computational experiments show that our algorithm significantly outperforms the only available exact solution method—exhaustive search. Problems that would require over 13 years to solve by exhaustive search can be solved within ten minutes using our method.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2021.100670","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54146616","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}

引用次数: 7

Tight bounds on the relative performances of pricing optimization mechanisms in storable good markets 仓储商品市场中定价优化机制相对性能的严格限制

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-11-01 DOI: 10.1016/j.disopt.2021.100671

Gerardo Berbeglia , Shant Boodaghians , Adrian Vetta

{"title":"Tight bounds on the relative performances of pricing optimization mechanisms in storable good markets","authors":"Gerardo Berbeglia , Shant Boodaghians , Adrian Vetta","doi":"10.1016/j.disopt.2021.100671","DOIUrl":"10.1016/j.disopt.2021.100671","url":null,"abstract":"<div>In the storable good monopoly problem, a monopolist sells a storable good by announcing a price in each time period. Each consumer has a unitary demand per time period with an arbitrary valuation. In each period, consumers may buy none, one, or more than one good (in which case the extra goods are stored for future consumption incurring in a linear storage cost). We compare the performance of two important monopoly pricing optimization mechanisms: price optimization using pre-announced prices and price optimization without commitments (contingent mechanism). In pre-announced pricing the prices in each time period are stated in advance; in a price contingent mechanism each price is stated at the start of the time period, and these prices are dependent upon past purchases. We prove that monopolist can earn up to <math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mi>T</mi><mo>+</mo><mo>log</mo><mi>N</mi><mo>)</mo></mrow></mrow></math> times more profit by using a pre-announced pricing mechanism rather than a price contingent mechanism. Here <math><mi>T</mi></math> denotes the number of time periods and <math><mi>N</mi></math> denotes the number of consumers. This bound is tight; examples exist where the monopolist would earn a factor <math><mrow><mi>Ω</mi><mrow><mo>(</mo><mo>log</mo><mi>T</mi><mo>+</mo><mo>log</mo><mi>N</mi><mo>)</mo></mrow></mrow></math> more by using a pre-announced pricing mechanism.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54146628","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}

引用次数: 0

Hardness results for Multimarginal Optimal Transport problems 多边际最优运输问题的硬度结果

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-11-01 DOI: 10.1016/j.disopt.2021.100669

Jason M. Altschuler, Enric Boix-Adserà

{"title":"Hardness results for Multimarginal Optimal Transport problems","authors":"Jason M. Altschuler, Enric Boix-Adserà","doi":"10.1016/j.disopt.2021.100669","DOIUrl":"10.1016/j.disopt.2021.100669","url":null,"abstract":"<div>Multimarginal Optimal Transport (<math><mi>MOT</mi></math>) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving <math><mi>MOT</mi></math>: the linear program has exponential size in the number of marginals <math><mi>k</mi></math> and their support sizes <math><mi>n</mi></math>. A recent line of work has shown that <math><mi>MOT</mi></math> is <math><mrow><mi>poly</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math>-time solvable for certain families of costs that have <math><mrow><mi>poly</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math>-size implicit representations. However, it is unclear what further families of costs this line of algorithmic research can encompass. In order to understand these fundamental limitations, this paper initiates the study of intractability results for <math><mi>MOT</mi></math>.Our main technical contribution is developing a toolkit for proving <math><mi>NP</mi></math>-hardness and inapproximability results for <math><mi>MOT</mi></math> problems. This toolkit reduces proving intractability of <math><mi>MOT</mi></math> problems to proving intractability of more amenable discrete optimization problems. We demonstrate this toolkit by using it to establish the intractability of a number of <math><mi>MOT</mi></math> problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make <math><mi>MOT</mi></math> intractable by showing that several such problems of interest are <math><mi>NP</mi></math>-hard to solve—even approximately.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122910312","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}

引用次数: 23

Arc-routing for winter road maintenance 冬季道路养护的弧形路线

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-08-01 DOI: 10.1016/j.disopt.2021.100644

Jiří Fink , Martin Loebl , Petra Pelikánová

{"title":"Arc-routing for winter road maintenance","authors":"Jiří Fink , Martin Loebl , Petra Pelikánová","doi":"10.1016/j.disopt.2021.100644","DOIUrl":"https://doi.org/10.1016/j.disopt.2021.100644","url":null,"abstract":"<div>The winter road maintenance arc-routing is recognized as a notoriously hard problem not only from the algorithmic point of view. This paper lays down foundations of theoretical understanding of our new winter road maintenance optimization for the Plzeň region of the Czech Republic which has been implemented by the regional authorities since the winter of 2019–20. Our approach is not, contrary to most of existing work, based on the integer and linear programming machinery. We concentrate on studying arc-routing on trees. This is practical since routes of single vehicles can be well represented by trees, and allows algorithms and complementary hardness results. We then extend the approach to the bounded tree width graphs. This leads to considering planar graphs which well abstract the realistic road networks. We formalize important aspects of the winter road maintenance problem which were not formalized before, e.g., public complaints. The number of complaints from public against the winter road maintenance is a quantitative measure of the quality of the service which is focused on, e.g., in media or in election campaigns. A fear of ’complaints’ is a fact every optimizer must deal with. Hence, a formal model of public complaints and its inclusion in the optimization is vital. Our formalization of the winter road maintenance is robust in the sense that it relates to well-known extensively studied concepts of discrete mathematics like graph cutting and splitting of necklaces.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2021.100644","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92219145","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}

引用次数: 1

Parallelization of a branch-and-bound algorithm for the maximum weight clique problem 最大权团问题的分支定界并行化算法

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-08-01 DOI: 10.1016/j.disopt.2021.100646

Satoshi Shimizu, Kazuaki Yamaguchi, Sumio Masuda

引用次数: 2

The optimal proper connection number of a graph with given independence number 给定独立数的图的最优固有连接数

IF 1.1 4区数学

Discrete Optimization Pub Date : 2021-08-01 DOI: 10.1016/j.disopt.2021.100660

Shinya Fujita , Boram Park

{"title":"The optimal proper connection number of a graph with given independence number","authors":"Shinya Fujita , Boram Park","doi":"10.1016/j.disopt.2021.100660","DOIUrl":"https://doi.org/10.1016/j.disopt.2021.100660","url":null,"abstract":"<div>An edge-colored connected graph <math><mi>G</mi></math> is properly connected if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced the optimal proper connection number pcopt(G) for a monochromatic connected graph <math><mi>G</mi></math>, to make a connected graph properly connected efficiently. More precisely, pcopt\u0000(<math><mi>G</mi></math>) is the smallest integer <math><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></math> when one converts a given monochromatic graph <math><mi>G</mi></math> into a properly connected graph by recoloring <math><mi>p</mi></math> edges with <math><mi>q</mi></math> colors. In this paper, we show that pcopt\u0000(<math><mi>G</mi></math>) has an upper bound in terms of the independence number <math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. Namely, we prove that for a connected graph <math><mi>G</mi></math>, pcopt\u0000(<math><mi>G</mi></math>)<math><mrow><mo>≤</mo><mfrac><mrow><mn>5</mn><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math>. Moreover, for the case <math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math>, we improve the upper bound to 4, which is tight.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2021.100660","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92219034","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}

引用次数: 0