Discrete Optimization最新文献

{"title":"On inequalities with bounded coefficients and pitch for the min knapsack polytope","authors":"Daniel Bienstock ,&nbsp;Yuri Faenza ,&nbsp;Igor Malinović ,&nbsp;Monaldo Mastrolilli ,&nbsp;Ola Svensson ,&nbsp;Mark Zuckerberg","doi":"10.1016/j.disopt.2020.100567","DOIUrl":"10.1016/j.disopt.2020.100567","url":null,"abstract":"<div><p>The min knapsack problem appears as a major component in the structure of capacitated covering problems. Its polyhedral relaxations have been extensively studied, leading to strong relaxations for networking, scheduling and facility location problems.</p><p>A valid inequality <span><math><mrow><msup><mrow><mi>α</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>x</mi><mo>≥</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> with <span><math><mrow><mi>α</mi><mo>≥</mo><mn>0</mn></mrow></math></span> for a min knapsack instance is said to have pitch <span><math><mrow><mo>≤</mo><mi>π</mi></mrow></math></span>\u0000(<span><math><mi>π</mi></math></span> a positive integer) if the <span><math><mi>π</mi></math></span> smallest strictly positive <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> sum to at least <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. An inequality with coefficients and right-hand side in <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>π</mi><mo>}</mo></mrow></math></span> has pitch <span><math><mrow><mo>≤</mo><mi>π</mi></mrow></math></span><span>. The notion of pitch has been used for measuring the complexity of valid inequalities for the min knapsack polytope. Separating inequalities of pitch-1 is already NP-Hard. In this paper, we show an algorithm for efficiently separating inequalities with coefficients in </span><span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>π</mi><mo>}</mo></mrow></math></span> for any fixed <span><math><mi>π</mi></math></span><span> up to an arbitrarily small additive error. As a special case, this allows for approximate separation of inequalities with pitch at most 2. We moreover investigate the integrality gap of minimum knapsack instances when bounded pitch inequalities (possibly in conjunction with other inequalities) are added. Among other results, we show that the CG closure of minimum knapsack has unbounded integrality gap even after a constant number of rounds.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2020.100567","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115276159","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":"Integer programming in parameterized complexity: Five miniatures","authors":"Tomáš Gavenčiak ,&nbsp;Martin Koutecký ,&nbsp;Dušan Knop","doi":"10.1016/j.disopt.2020.100596","DOIUrl":"10.1016/j.disopt.2020.100596","url":null,"abstract":"<div><p>Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra’s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between <span><math><mi>FPT</mi></math></span> and <span><math><mi>XP</mi></math></span> algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining <span><math><mi>FPT</mi></math></span> algorithms with runtime <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span>poly<span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>. We focus on: </p><ul><li><span>•</span><span><p><em>Modeling</em>: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.</p></span></li><li><span>•</span><span><p><em>Optimality program:</em> after giving an <span><math><mi>FPT</mi></math></span> algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.</p></span></li><li><span>•</span><span><p><em>Minding the</em> poly<span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>: reducing <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> often has the unintended consequence of increasing poly<span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>; so we highlight the common trade-offs and show how to get the best of both worlds.</p></span></li></ul> Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several <span><math><mi>FPT</mi></math></span> algorithms for <span>Capacitated Dominating Set</span>, <span>Sum Coloring</span>, <span>Max-</span>\u0000<span><math><mi>q</mi></math></span>\u0000<span>-Cut</span>, and certain other coloring problems by modeling them as convex programs in fixed dimension, <span><math><mi>n</mi></math></span>-fold integer programs, bounded dual treewidth programs, indefinite quadratic programs in fixed dimension, parametric integer programs in fixed dimension, and 2-stage stochastic integer programs.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2020.100596","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130068932","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 the intrinsic volumes of intersections of congruent balls","authors":"Károly Bezdek","doi":"10.1016/j.disopt.2019.03.002","DOIUrl":"10.1016/j.disopt.2019.03.002","url":null,"abstract":"<div><p>Let <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> denote the <span><math><mi>d</mi></math></span>-dimensional Euclidean space. The <span><math><mi>r</mi></math></span>-ball body generated by a given set in <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is the intersection of balls of radius <span><math><mi>r</mi></math></span> centered at the points of the given set. In this paper we prove the following Blaschke–Santaló-type inequalities for <span><math><mi>r</mi></math></span>-ball bodies: for all <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi></math></span> and for any set of given volume in <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> the <span><math><mi>k</mi></math></span>th intrinsic volume of the <span><math><mi>r</mi></math></span>-ball body generated by the set becomes maximal if the set is a ball. As an application we investigate the Gromov–Klee–Wagon problem for congruent balls in <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, which is a question on proving or disproving that if the centers of a family of <span><math><mi>N</mi></math></span> congruent balls in <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span><span> are contracted, then the volume of the intersection does not decrease. In particular, we investigate this problem for uniform contractions, which are contractions where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. Bezdek and Naszódi (2018), proved that the intrinsic volumes of the intersection of </span><span><math><mi>N</mi></math></span> congruent balls in <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn></math></span> increase under any uniform contraction of the center points when <span><math><mi>N</mi><mo>≥</mo><msup><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></mfenced></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We give a short proof of this result using the Blaschke–Santaló-type inequalities of <span><math><mi>r</mi></math></span>-ball bodies and improve it for <span><math><mi>d</mi><mo>≥</mo><mn>42</mn></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2019.03.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116315316","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":"Minimum energy configurations on a toric lattice as a quadratic assignment problem","authors":"Daniel Brosch,&nbsp;Etienne de Klerk","doi":"10.1016/j.disopt.2020.100612","DOIUrl":"10.1016/j.disopt.2020.100612","url":null,"abstract":"<div><p>We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman et al. (2013). The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Permenter and Parrilo (2020) to make computation of the SDP bounds possible.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2020.100612","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114210427","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":"Circuit walks in integral polyhedra","authors":"Steffen Borgwardt,&nbsp;Charles Viss","doi":"10.1016/j.disopt.2019.100566","DOIUrl":"10.1016/j.disopt.2019.100566","url":null,"abstract":"<div><p><span><span>Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method<span>. We are interested in better understanding the properties of circuit walks in integral polyhedra. In this paper, we introduce a hierarchy for integral polyhedra based on different types of behavior exhibited by their circuit walks. Many problems in combinatorial optimization fall into the most interesting categories of this hierarchy — steps of circuit walks only stop at </span></span>integer points, at vertices, or follow actual edges. We classify several classical families of polyhedra within the hierarchy, including </span><span><math><mrow><mn>0</mn><mo>/</mo><mn>1</mn></mrow></math></span><span>-polytopes, polyhedra defined by totally unimodular matrices, and more specifically matroid polytopes, transportation polytopes, and partition polytopes. Finally, we prove three characterizations of the simple polytopes that appear in the bottom level of the hierarchy where all circuit walks are edge walks, showing that such polytopes constitute a generalization of simplices and parallelotopes.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2019.100566","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129798260","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":"Complexity of training ReLU neural network","authors":"Digvijay Boob,&nbsp;Santanu S. Dey,&nbsp;Guanghui Lan","doi":"10.1016/j.disopt.2020.100620","DOIUrl":"10.1016/j.disopt.2020.100620","url":null,"abstract":"<div><p>In this paper, we explore some basic questions on the complexity of training neural networks<span> with ReLU activation function. We show that it is NP-hard to train a two-hidden layer feedforward ReLU neural network. If dimension of the input data and the network topology is fixed, then we show that there exists a polynomial time algorithm for the same training problem. We also show that if sufficient over-parameterization is provided in the first hidden layer of ReLU neural network, then there is a polynomial time algorithm which finds weights such that output of the over-parameterized ReLU neural network matches with the output of the given data.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2020.100620","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115398189","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":"Submodular reassignment problem for reallocating agents to tasks with synergy effects","authors":"Naonori Kakimura ,&nbsp;Naoyuki Kamiyama ,&nbsp;Yusuke Kobayashi ,&nbsp;Yoshio Okamoto","doi":"10.1016/j.disopt.2021.100631","DOIUrl":"10.1016/j.disopt.2021.100631","url":null,"abstract":"<div><p><span>We propose a new combinatorial optimization problem that we call the submodular reassignment problem. We are given </span><span><math><mi>k</mi></math></span><span><span> submodular functions over the same ground set, and we want to find a set that minimizes the sum of the distances to the sets of minimizers of all functions. The problem is motivated by a two-stage stochastic optimization problem with recourse summarized as follows. We are given two tasks to be processed and want to assign a set of workers to maximize the sum of profits. However, we do not know the value functions exactly, but only know a finite number of possible scenarios. Our goal is to determine the first-stage allocation of workers to minimize the expected number of reallocated workers after a scenario is realized at the second stage. This problem can be modeled by the submodular reassignment problem. We prove that the submodular reassignment problem can be solved in strongly polynomial time via submodular function minimization. We further provide a maximum-flow formulation of the problem that enables us to solve the problem without using a general submodular function minimization algorithm, and more efficiently both in theory and in practice. In our algorithm, we make use of Birkhoff’s </span>representation theorem<span> for distributive lattices.</span></span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2021.100631","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127431038","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":"An improved zig zag approach for competitive group testing","authors":"Jun Wu ,&nbsp;Yongxi Cheng ,&nbsp;Ding-Zhu Du","doi":"10.1016/j.disopt.2022.100687","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100687","url":null,"abstract":"<div><p><span>In many fault detection problems, we want to identify defective items from a set of </span><span><math><mi>n</mi></math></span><span> items using the minimum number of tests. Group testing is for the scenario where each test is on a subset of items, and tells whether the subset contains at least one defective item or not. In practice, the number </span><span><math><mi>d</mi></math></span> of defective items is often unknown in advance. In this paper, we improve the previously best algorithm for a central problem in combinatorial group testing with unknown number of defectives (Cheng et al., 2014), and prove that the number of tests used by our new algorithm is no more than <span><math><mrow><mi>d</mi><mo>log</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>+</mo><mrow><mo>(</mo><mn>5</mn><mo>−</mo><mo>log</mo><mn>5</mn><mo>)</mo></mrow><mi>d</mi><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mo>log</mo></math></span> is of base 2.</p></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":"91757811","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":"GO-MOCE: Greedy Order Method of Conditional Expectations for Max Sat","authors":"Daniel Berend ,&nbsp;Shahar Golan ,&nbsp;Yochai Twitto","doi":"10.1016/j.disopt.2022.100685","DOIUrl":"10.1016/j.disopt.2022.100685","url":null,"abstract":"<div><p>In this paper we present and study a new algorithm for the Maximum Satisfiability (Max Sat) problem. The algorithm is based on the Method of Conditional Expectations (MOCE, also known as Johnson’s Algorithm) and applies a greedy variable ordering to MOCE. Thus, we name it Greedy Order MOCE (GO-MOCE). We also suggest a combination of GO-MOCE with CCLS, a state-of-the-art solver. We refer to this combined solver as GO-MOCE-CCLS.</p><p>We conduct a comprehensive comparative evaluation of GO-MOCE versus MOCE on random instances and on public competition benchmark instances. We show that GO-MOCE reduces the number of unsatisfied clauses by tens of percents, while keeping the runtime almost the same. The worst case time complexity of GO-MOCE is linear. We also show that GO-MOCE-CCLS improves on CCLS consistently by up to about 80%.</p><p>We study the asymptotic performance of GO-MOCE. To this end, we introduce three measures for evaluating the asymptotic performance of algorithms for Max Sat. We point out to further possible improvements of GO-MOCE, based on an empirical study of the main quantities managed by GO-MOCE during its execution.</p></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":"125991195","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":"Mathematical formulations and solution methods for the uncapacitated r-allocation p-hub maximal covering problem","authors":"Olivera Stančić ,&nbsp;Zorica Stanimirović ,&nbsp;Raca Todosijević ,&nbsp;Stefan Mišković","doi":"10.1016/j.disopt.2021.100672","DOIUrl":"10.1016/j.disopt.2021.100672","url":null,"abstract":"<div><p>This paper considers the uncapacitated <span><math><mi>r</mi></math></span>-allocation <span><math><mi>p</mi></math></span><span>-hub maximal covering problem (UrApHMCP), which represents a generalization of the well-known </span><span><math><mi>p</mi></math></span>-hub maximal covering problem, as it allows each non-hub node to send and receive flow via at most <span><math><mi>r</mi></math></span> hubs, <span><math><mrow><mi>r</mi><mo>≤</mo><mi>p</mi></mrow></math></span><span>. Two coverage criteria are considered in UrApHMCP — binary and, for the first time in the literature, partial coverage. Novel mathematical formulations of UrApHMCP for both coverage criteria are proposed. As the considered UrApHMCP is an NP-hard optimization problem, two efficient heuristic methods are proposed as solution approaches. The first one is a variant of General Variable Neighborhood Search (GVNS), and the second one is based on combining a Greedy Randomized Adaptive Search Procedure (GRASP) with Variable Neighborhood Descent (VND), resulting in a hybrid GRASP-VND method. Computational study is performed over the set of CAB and AP benchmark instances with up to 25 and 200 nodes, respectively, on TR instances including 81 nodes, as well as on the challenging USA423 and URAND hub instances with up 423 and 1000 nodes, respectively. Optimal or feasible solutions are obtained by CPLEX solver for instances with up to 50 nodes, while instances of larger dimensions are out of reach for CPLEX solver. On the other hand, both GVNS and GRASP-VND reach optimal solutions or improve lower bounds provided by CPLEX in short CPU times. In addition, both heuristics quickly return solutions on problem instances of large dimensions, thus indicating their potential to solve effectively large, realistic sized problem instances. The conducted non-parametric statistical tests confirm robustness of the proposed GVNS and GRASP-VND and demonstrate that the these two metaheuristics outperform other tested algorithms for UrApHMCP.</span></p></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":"121816212","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}