{"title":"A Property on Monochromatic Copies of Graphs Containing a Triangle","authors":"Hao Chen, Jie Ma","doi":"10.1137/23m1564894","DOIUrl":"https://doi.org/10.1137/23m1564894","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 316-326, March 2024. <br/> Abstract. A graph [math] is called common and, respectively, strongly common if the number of monochromatic copies of [math] in a 2-edge-coloring [math] of a large clique is asymptotically minimized by the random coloring with an equal proportion of each color and, respectively, by the random coloring with the same proportion of each color as in [math]. A well-known theorem of Jagger, Št’ovíček, and Thomason states that every graph containing a [math] is not common. Here we prove an analogous result that every graph containing a [math] and with at least four edges is not strongly common.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Davide Bilò, Tobias Friedrich, Pascal Lenzner, Anna Melnichenko
{"title":"Geometric Network Creation Games","authors":"Davide Bilò, Tobias Friedrich, Pascal Lenzner, Anna Melnichenko","doi":"10.1137/20m1376662","DOIUrl":"https://doi.org/10.1137/20m1376662","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 277-315, March 2024. <br/> Abstract. Network creation games are a well-known approach for explaining and analyzing the structure, quality, and dynamics of real-world networks that evolved via the interaction of selfish agents without a central authority. In these games selfish agents corresponding to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games only considered the creation of networks with unit-weight edges. In practice, e.g., when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depend on the distance between the involved nodes, and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al. [Proceedings of PODC ’03, ACM, 2003, pp. 347–351] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state of the art for the unit-weight version, where the price of anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight nonconstant bound on the price of anarchy for the metric version and a slightly weaker upper bound for the nonmetric case. Moreover, we analyze the existence of equilibria, the computational hardness, and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of the classical network design problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"74 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bogdan Alecu, Vadim V. Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev
{"title":"The Treewidth and Pathwidth of Graph Unions","authors":"Bogdan Alecu, Vadim V. Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev","doi":"10.1137/22m1524047","DOIUrl":"https://doi.org/10.1137/22m1524047","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 261-276, March 2024. <br/> Abstract. Given two [math]-vertex graphs [math] and [math] of bounded treewidth, is there an [math]-vertex graph [math] of bounded treewidth having subgraphs isomorphic to [math] and [math]? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if [math] is a binary tree and [math] is a ternary tree. We also provide an extensive study of cases where such “gluing” is possible. In particular, we prove that if [math] has treewidth [math] and [math] has pathwidth [math], then there is an [math]-vertex graph of treewidth at most [math] containing both [math] and [math] as subgraphs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Ramsey Size-Linear Graphs and Related Questions","authors":"Domagoj Bradač, Lior Gishboliner, Benny Sudakov","doi":"10.1137/22m1481713","DOIUrl":"https://doi.org/10.1137/22m1481713","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 225-242, March 2024. <br/> Abstract. In this paper we prove several results on Ramsey numbers [math] for a fixed graph [math] and a large graph [math], in particular for [math]. These results extend earlier work of Erdős, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey size-linear graphs. Among other results, we show that if [math] is a subdivision of [math] with at least six vertices, then [math] for every graph [math]. We also conjecture that if [math] is a connected graph with [math], then [math]. The case [math] was proved by Erdős, Faudree, Rousseau, and Schelp. We prove the case [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Highly Connected Subgraphs with Large Chromatic Number","authors":"Tung H. Nguyen","doi":"10.1137/22m150040x","DOIUrl":"https://doi.org/10.1137/22m150040x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 243-260, March 2024. <br/> Abstract. For integers [math] and [math], let [math] be the least integer [math] such that every graph with chromatic number at least [math] contains a [math]-connected subgraph with chromatic number at least [math]. Refining the recent result of Girão and Narayanan [Bull. Lond. Math. Soc., 54 (2022), pp. 868–875] that [math] for all [math], we prove that [math] for all [math] and [math]. This sharpens earlier results of Alon et al. [J. Graph Theory, 11 (1987), pp. 367–371], of Chudnovsky [J. Combin. Theory Ser. B, 103 (2013), pp. 567–586], and of Penev, Thomassé, and Trotignon [SIAM J. Discrete Math., 30 (2016), pp. 592–619]. Our result implies that [math] for all [math], making a step closer towards a conjecture of Thomassen [J. Graph Theory, 7 (1983), pp. 261–271] that [math], which was originally a result with a false proof and was the starting point of this research area.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stochastic Probing with Increasing Precision","authors":"Martin Hoefer, Kevin Schewior, Daniel Schmand","doi":"10.1137/22m149466x","DOIUrl":"https://doi.org/10.1137/22m149466x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 148-169, March 2024. <br/> Abstract. We consider a selection problem with stochastic probing. There is a set of items whose values are drawn from independent distributions. The distributions are known in advance. Each item can be tested repeatedly. Each test reduces the uncertainty about the realization of its value. We study a testing model, where the first test reveals whether the realized value is smaller or larger than the [math]-quantile of the underlying distribution of some constant [math]. Subsequent tests allow us to further narrow down the interval in which the realization is located. There is a limited number of possible tests, and our goal is to design near-optimal testing strategies that allow us to maximize the expected value of the chosen item. We study both identical and nonidentical distributions and develop polynomial-time algorithms with constant approximation factors in both scenarios.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mikhael Carmona, Victor Chepoi, Guyslain Naves, Pascal Préa
{"title":"Modules in Robinson Spaces","authors":"Mikhael Carmona, Victor Chepoi, Guyslain Naves, Pascal Préa","doi":"10.1137/22m1494348","DOIUrl":"https://doi.org/10.1137/22m1494348","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 190-224, March 2024. <br/> Abstract. A Robinson space is a dissimilarity space [math] (i.e., a set [math] of size [math] and a dissimilarity [math] on [math]) for which there exists a total order [math] on [math] such that [math] implies that [math]. Recognizing if a dissimilarity space is Robinson has numerous applications in seriation and classification. An mmodule of [math] (generalizing the notion of a module in graph theory) is a subset [math] of [math] which is not distinguishable from the outside of [math]; i.e., the distance from any point of [math] to all points of [math] is the same. If [math] is any point of [math], then [math], and the maximal-by-inclusion mmodules of [math] not containing [math] define a partition of [math], called the copoint partition. In this paper, we investigate the structure of mmodules in Robinson spaces and use it and the copoint partition to design a simple and practical divide-and-conquer algorithm for recognition of Robinson spaces in optimal [math] time.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exchange Distance of Basis Pairs in Split Matroids","authors":"Kristóf Bérczi, Tamás Schwarcz","doi":"10.1137/23m1565115","DOIUrl":"https://doi.org/10.1137/23m1565115","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 132-147, March 2024. <br/> Abstract. The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a global understanding of the structure of matroid bases is a fundamental goal in matroid optimization. While studying the structure of symmetric exchanges, Gabow proposed the problem that any pair of bases admits a sequence of symmetric exchanges. A different extension of the exchange axiom was proposed by White, who investigated the equivalence of compatible basis sequences. These conjectures suggest that the family of bases of a matroid possesses much stronger structural properties than we are aware of. In the present paper, we study the distance of basis pairs of a matroid in terms of symmetric exchanges. In particular, we give a polynomial-time algorithm that determines a shortest possible exchange sequence that transforms a basis pair into another for split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. As a corollary, we verify the above-mentioned long-standing conjectures for this large class. As paving matroids form a subclass of split matroids, our result settles the conjectures for paving matroids as well.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eun Jung Kim, Tomáš Masařík, Marcin Pilipczuk, Roohani Sharma, Magnus Wahlström
{"title":"On Weighted Graph Separation Problems and Flow Augmentation","authors":"Eun Jung Kim, Tomáš Masařík, Marcin Pilipczuk, Roohani Sharma, Magnus Wahlström","doi":"10.1137/22m153118x","DOIUrl":"https://doi.org/10.1137/22m153118x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 170-189, March 2024. <br/> Abstract. One of the first applications of the recently introduced technique of flow augmentation [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of Directed Feedback Vertex Set, a landmark problem in parameterized complexity. In this article, we explore the applicability of flow augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following: In weighted undirected graphs, Multicut is fixed-parameter tractable (FPT) in both the edge- and the vertex-deletion version. The weighted version of Group Feedback Vertex Set is FPT, even with oracle access to group operations. The weighted version of Directed Subset Feedback Vertex Set is FPT. Our study reveals Directed Symmetric Multicut as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"58 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abdulmajeed Alqasem, Heshan Aravinda, Arnaud Marsiglietti, James Melbourne
{"title":"On a Conjecture of Feige for Discrete Log-Concave Distributions","authors":"Abdulmajeed Alqasem, Heshan Aravinda, Arnaud Marsiglietti, James Melbourne","doi":"10.1137/22m1539514","DOIUrl":"https://doi.org/10.1137/22m1539514","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 93-102, March 2024. <br/> Abstract. A remarkable conjecture of Feige [SIAM J. Comput., 35 (2006), pp. 964–984] asserts that for any collection of [math] independent nonnegative random variables [math], each with expectation at most 1, [math], where [math]. In this paper, we investigate this conjecture for the class of discrete log-concave probability distributions, and we prove a strengthened version. More specifically, we show that the conjectured bound [math] holds when [math]’s are independent discrete log-concave with arbitrary expectation.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139376180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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