Discrete OptimizationPub Date : 2023-11-01Epub Date: 2023-09-13DOI: 10.1016/j.disopt.2023.100803
Leah Epstein
{"title":"More on online weighted edge coloring","authors":"Leah Epstein","doi":"10.1016/j.disopt.2023.100803","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100803","url":null,"abstract":"<div><p><span><span>We revisit online weighted edge coloring. In this problem, weighted edges of a graph are presented one by one, to be colored with positive integers. It is required that for every vertex, all its edges of every common color will have a total weight not exceeding 1. We provide an improved upper bound on the performance of a </span>greedy algorithm First Fit for the case of arbitrary weights, and for the case of weights not exceeding </span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Here, the meaning of First Fit is that every edge is colored with a color of the smallest index that will keep the coloring valid. This improves the state-of-the-art with respect to online algorithms for this variant of edge coloring. We also show new lower bounds on the performance of any online algorithm with weights in <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>t</mi></mrow></mfrac><mo>]</mo></mrow></math></span>, for any integer <span><math><mrow><mi>t</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"50 ","pages":"Article 100803"},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49733922","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}
Discrete OptimizationPub Date : 2023-11-01Epub Date: 2023-10-14DOI: 10.1016/j.disopt.2023.100806
K. Subramani, Piotr Wojciechowski
{"title":"Optimal length cutting plane refutations of integer programs","authors":"K. Subramani, Piotr Wojciechowski","doi":"10.1016/j.disopt.2023.100806","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100806","url":null,"abstract":"<div><p><span>In this paper, we discuss the computational complexities of determining optimal length refutations of infeasible integer programs (IPs). We focus on three different types of refutations, namely read-once refutations, tree-like refutations, and dag-like refutations. For each refutation type, we are interested in finding the length of the shortest possible refutation of that type. In the case of this paper, the length of a refutation is equal to the number of inferences in that refutation. The refutations in this paper are also defined by the types of inferences that can be used to derive new constraints. We are interested in refutations with two inference rules. The first rule corresponds to the summation of two constraints and is called the ADD rule. The second rule is the DIV rule which divides a constraint by a positive integer. For integer programs, we study the complexity of approximating the length of the shortest refutation of each type (read-once, tree-like, and dag-like). In this paper, we show that the problem of finding the shortest read-once refutation is </span><strong>NPO PB-complete</strong>. Additionally, we show that the problem of finding the shortest tree-like refutation is <strong>NPO-hard</strong> for IPs. We also show that the problem of finding the shortest dag-like refutation is <strong>NPO-hard</strong> for IPs. Finally, we show that the problems of finding the shortest tree-like and dag-like refutations are in <strong>FPSPACE</strong>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"50 ","pages":"Article 100806"},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49712009","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}
Discrete OptimizationPub Date : 2023-11-01Epub Date: 2023-11-03DOI: 10.1016/j.disopt.2023.100809
Shunhai He , Huiqing Liu
{"title":"The maximum number of short paths in a Halin graph","authors":"Shunhai He , Huiqing Liu","doi":"10.1016/j.disopt.2023.100809","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100809","url":null,"abstract":"<div><p>A Halin graph <span><math><mi>G</mi></math></span> is a plane graph consisting of a plane embedding of a tree <span><math><mi>T</mi></math></span> of order at least 4 containing no vertex of degree 2, and of a cycle <span><math><mi>C</mi></math></span> connecting all leaves of <span><math><mi>T</mi></math></span>. Let <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the maximum number of copies of <span><math><mi>G</mi></math></span> in a Halin graph on <span><math><mi>n</mi></math></span> vertices. In this paper, we give exact values of <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is a path on <span><math><mi>k</mi></math></span> vertices for <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>5</mn></mrow></math></span>. Moreover, we develop a new graph transformation preserving the number of vertices, so that the resulting graph has a monotone behavior with respect to the number of short paths.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"50 ","pages":"Article 100809"},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91729833","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}
Discrete OptimizationPub Date : 2023-11-01Epub Date: 2023-11-08DOI: 10.1016/j.disopt.2023.100810
Patrick Chervet , Roland Grappe , Mathieu Lacroix , Francesco Pisanu , Roberto Wolfler Calvo
{"title":"Hard problems on box-totally dual integral polyhedra","authors":"Patrick Chervet , Roland Grappe , Mathieu Lacroix , Francesco Pisanu , Roberto Wolfler Calvo","doi":"10.1016/j.disopt.2023.100810","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100810","url":null,"abstract":"<div><p>In this paper, we study the complexity of some fundamental questions regarding box-totally dual integral (box-TDI) polyhedra. First, although box-TDI polyhedra have strong integrality properties, we prove that Integer Programming over box-TDI polyhedra is NP-complete, that is, finding an integer point optimizing a linear function over a box-TDI polyhedron is hard. Second, we complement the result of Ding et al. (2008) who proved that deciding whether a given system is box-TDI is co-NP-complete: we prove that recognizing whether a polyhedron is box-TDI is co-NP-complete.</p><p>To derive these complexity results, we exhibit new classes of totally equimodular matrices – a generalization of totally unimodular matrices – by characterizing the total equimodularity of incidence matrices of graphs.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"50 ","pages":"Article 100810"},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92046424","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 k-aggregation closure for covering sets","authors":"Haoran Zhu","doi":"10.1016/j.disopt.2023.100805","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100805","url":null,"abstract":"<div><p>In this paper, we will answer a more general version of one of the questions proposed by Bodur et al. (2017). Specifically, we show that the <span><math><mi>k</mi></math></span><span>-aggregation closure of a covering set is a polyhedron.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"50 ","pages":"Article 100805"},"PeriodicalIF":1.1,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49712363","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}
Daniel Bienstock , Gonzalo Muñoz , Sebastian Pokutta
{"title":"Principled deep neural network training through linear programming","authors":"Daniel Bienstock , Gonzalo Muñoz , Sebastian Pokutta","doi":"10.1016/j.disopt.2023.100795","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100795","url":null,"abstract":"<div><p><span>Deep learning<span> has received much attention lately due to the impressive empirical performance achieved by training algorithms. Consequently, a need for a better theoretical understanding of these problems has become more evident and multiple works in recent years have focused on this task. In this work, using a unified framework, we show that there exists a polyhedron that simultaneously encodes, in its facial structure, all possible </span></span>deep neural network<span> training problems that can arise from a given architecture, activation functions, loss function, and sample size. Notably, the size of the polyhedral representation depends only linearly on the sample size, and a better dependency on several other network parameters is unlikely. Using this general result, we compute the size of the polyhedral encoding for commonly used neural network architectures. Our results provide a new perspective on training problems through the lens of polyhedral theory and reveal strong structure arising from these problems.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"49 ","pages":"Article 100795"},"PeriodicalIF":1.1,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49715745","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}
Discrete OptimizationPub Date : 2023-05-01Epub Date: 2023-05-11DOI: 10.1016/j.disopt.2023.100776
Christoph Buchheim, Maja Hügging
{"title":"The polytope of binary sequences with bounded variation","authors":"Christoph Buchheim, Maja Hügging","doi":"10.1016/j.disopt.2023.100776","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100776","url":null,"abstract":"<div><p><span>We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length </span><span><math><mi>n</mi></math></span><span> with bounded variation<span>, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary optimal control problems<span> subject to a bounded total variation. We study two variants of the problem. In the first one, the variation of the binary vector is penalized in the objective function, while in the second one, it is bounded by a hard constraint. We show that the first variant is easy to deal with while the second variant turns out to be more complex, but still tractable. For the latter case, we present a complete polyhedral description of the convex hull of feasible solutions by facet-inducing inequalities and devise an exact linear-time separation algorithm. The proof of completeness also yields a new exact primal algorithm with a running time of </span></span></span><span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, which is significantly faster than the straightforward dynamic programming approach. Finally, we devise a compact extended formulation.</p><p>A preliminary version of this article has been published in the Proceedings of the 7th International Symposium on Combinatorial Optimization (ISCO 2022) (Buchheim and Hügging, 2022).</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100776"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809022","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}
Discrete OptimizationPub Date : 2023-05-01Epub Date: 2023-04-01DOI: 10.1016/j.disopt.2023.100772
Ádám X. Fraknói , Dávid Á. Márton , Dániel G. Simon , Dániel A. Lenger
{"title":"On the Rényi–Ulam game with restricted size queries","authors":"Ádám X. Fraknói , Dávid Á. Márton , Dániel G. Simon , Dániel A. Lenger","doi":"10.1016/j.disopt.2023.100772","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100772","url":null,"abstract":"<div><p>We investigate the following version of the well-known Rényi–Ulam game. Two players – the Questioner and the Responder – play against each other. The Responder thinks of a number from the set <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></math></span>, and the Questioner has to find this number. To do this, he can ask whether a chosen set of at most <span><math><mi>k</mi></math></span> elements contains the thought number. The Responder answers with YES or NO immediately, but during the game, he may lie at most <span><math><mi>ℓ</mi></math></span> times. The minimum number of queries needed for the Questioner to surely find the unknown element is denoted by <span><math><mrow><mi>R</mi><msubsup><mrow><mi>U</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of <span><math><mrow><mi>R</mi><msubsup><mrow><mi>U</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and an upper bound which differs from the lower one by at most <span><math><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span>. We also give its exact value when <span><math><mi>n</mi></math></span> is sufficiently large compared to <span><math><mi>k</mi></math></span>. With these, we managed to improve and generalize the results obtained by Meng, Lin, and Yang in a 2013 paper about the case <span><math><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100772"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716348","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}
Natalia de Castro , María A. Garrido-Vizuete , Rafael Robles , María Trinidad Villar-Liñán
{"title":"Minimum gradation in greyscales of graphs","authors":"Natalia de Castro , María A. Garrido-Vizuete , Rafael Robles , María Trinidad Villar-Liñán","doi":"10.1016/j.disopt.2023.100773","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100773","url":null,"abstract":"<div><p>In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100773"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716349","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}
Discrete OptimizationPub Date : 2023-05-01Epub Date: 2023-02-25DOI: 10.1016/j.disopt.2023.100768
Štefko Miklavič , Johannes Pardey , Dieter Rautenbach , Florian Werner
{"title":"Maximizing the Mostar index for bipartite graphs and split graphs","authors":"Štefko Miklavič , Johannes Pardey , Dieter Rautenbach , Florian Werner","doi":"10.1016/j.disopt.2023.100768","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100768","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><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><mspace></mspace><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><span>. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order </span><span><math><mi>n</mi></math></span> is at most <span><math><mrow><mfrac><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow><mrow><mn>18</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>, and that the Mostar index of split graphs of order <span><math><mi>n</mi></math></span> is at most <span><math><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>27</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100768"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809016","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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