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Don’t Roll the Dice, Ask Twice: The Two-Query Distortion of Matching Problems and Beyond 不要掷骰子，要问两次：匹配问题及其他问题的两次查询失真
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-03-12 DOI: 10.1137/23m1545677
Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, Alexandros A. Voudouris
{"title":"Don’t Roll the Dice, Ask Twice: The Two-Query Distortion of Matching Problems and Beyond","authors":"Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, Alexandros A. Voudouris","doi":"10.1137/23m1545677","DOIUrl":"https://doi.org/10.1137/23m1545677","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1007-1029, March 2024. <br/> Abstract. In most social choice settings, the participating agents express their preferences over the different alternatives in the form of linear orderings. While this clearly simplifies preference elicitation, it inevitably leads to poor performance with respect to optimizing a cardinal objective, such as the social welfare, since the values of the agents remain virtually unknown. This loss in performance because of lack of information is measured by the notion of distortion. A recent array of works put forward the agenda of designing mechanisms that learn the values of the agents for a small number of alternatives via queries, and use this limited extra information to make better-informed decisions, thus improving distortion. Following this agenda, in this work we focus on a class of combinatorial problems that includes most well-known matching problems and several of their generalizations. For problems such as One-Sided Matching, Two-Sided Matching, General Graph Matching, and Short Cycle Packing, we design two-query mechanisms that achieve the best-possible worst-case distortion in terms of social welfare, and outperform the best-possible expected distortion achieved by randomized ordinal mechanisms. Our results extend to problems like [math]-Constrained Resource Allocation, General Graph [math]-Matching, and [math]-Clique Packing, when [math] is restricted to be any constant.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124938","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}
引用次数: 0
Online Spanners in Metric Spaces 公制空间中的在线施展器
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-03-12 DOI: 10.1137/22m1534572
Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, Csaba D. Tóth
{"title":"Online Spanners in Metric Spaces","authors":"Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, Csaba D. Tóth","doi":"10.1137/22m1534572","DOIUrl":"https://doi.org/10.1137/22m1534572","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1030-1056, March 2024. <br/> Abstract. Given a metric space [math], a weighted graph [math] over [math] is a metric [math]-spanner of [math] if for every [math], [math], where [math] is the shortest path metric in [math]. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points [math], where the points are presented one at a time (i.e., after [math] steps, we see [math]). The algorithm is allowed to add edges to the spanner when a new point arrives; however, it is not allowed to remove any edge from the spanner. The goal is to maintain a [math]-spanner [math] for [math] for all [math], while minimizing the number of edges, and their total weight. We construct online [math]-spanners in the Euclidean [math]-space, [math]-spanners for general metrics, and [math]-spanners for ultrametrics. Most notably, in the Euclidean plane, we construct a [math]-spanner with competitive ratio [math], bypassing the classic lower bound [math] for lightness, which compares the weight of the spanner to that of the minimum spanning tree.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"76 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125215","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}
引用次数: 0
On the Concentration of the Maximum Degree in the Duplication-Divergence Models 论重复-发散模型中最大度数的集中问题
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-03-07 DOI: 10.1137/23m1592766
Alan M. Frieze, Krzysztof Turowski, Wojciech Szpankowski
{"title":"On the Concentration of the Maximum Degree in the Duplication-Divergence Models","authors":"Alan M. Frieze, Krzysztof Turowski, Wojciech Szpankowski","doi":"10.1137/23m1592766","DOIUrl":"https://doi.org/10.1137/23m1592766","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 988-1006, March 2024. <br/> Abstract. We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Solé et al. [Adv. Complex Syst., 5 (2002), pp. 43–54] in which the graph grows according to a duplication-divergence mechanism, i.e., by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability [math]. This model captures the growth of some real-world processes, e.g., biological or social networks. In this paper, we prove that for some [math], the maximum degree and the average degree of a duplication-divergence graph on [math] vertices are asymptotically concentrated with high probability around [math] and [math], respectively, i.e., they are within at most a polylogarithmic factor from these values with probability at least [math] for any constant [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140057685","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}
引用次数: 0
Treewidth, Circle Graphs, and Circular Drawings 树宽、圆图和圆形图画
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-03-06 DOI: 10.1137/22m1542854
Robert Hickingbotham, Freddie Illingworth, Bojan Mohar, David R. Wood
{"title":"Treewidth, Circle Graphs, and Circular Drawings","authors":"Robert Hickingbotham, Freddie Illingworth, Bojan Mohar, David R. Wood","doi":"10.1137/22m1542854","DOIUrl":"https://doi.org/10.1137/22m1542854","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 965-987, March 2024. <br/> Abstract. A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the “usual suspects.” Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. Using the same tools, we also study the treewidth of graphs [math] that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, we show that if the crossing graph is [math]-minor-free, then [math] has treewidth at most [math] and has no [math]-topological minor. On the other hand, we show that there are graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are 2-degenerate.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"127 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140044392","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}
引用次数: 0
On [math]-Counting of Noncrossing Chains and Parking Functions 论非交叉链的[数学]计数和停车函数
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-03-05 DOI: 10.1137/23m1572386
Yen-Jen Cheng, Sen-Peng Eu, Tung-Shan Fu, Jyun-Cheng Yao
{"title":"On [math]-Counting of Noncrossing Chains and Parking Functions","authors":"Yen-Jen Cheng, Sen-Peng Eu, Tung-Shan Fu, Jyun-Cheng Yao","doi":"10.1137/23m1572386","DOIUrl":"https://doi.org/10.1137/23m1572386","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 917-946, March 2024. <br/> Abstract. For a finite Coxeter group [math], Josuat-Vergès derived a [math]-polynomial counting the maximal chains in the lattice of noncrossing partitions of [math] by weighting some of the covering relations, which we call bad edges, in these chains with a parameter [math]. We study the connection of these weighted chains with parking functions of type [math] ([math], respectively) from the perspective of the [math]-polynomial. The [math]-polynomial turns out to be the generating function for parking functions (of either type) with respect to the number of cars that do not park in their preferred spaces. In either case, we present a bijective result that carries bad edges to unlucky cars while preserving their relative order. Using this, we give an interpretation of the [math]-positivity of the [math]-polynomial in the case when [math] is the hyperoctahedral group.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140037687","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}
引用次数: 0
Eigenpolytope Universality and Graphical Designs 特征多面体普遍性和图形设计
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-03-05 DOI: 10.1137/22m1528768
Catherine Babecki, David Shiroma
{"title":"Eigenpolytope Universality and Graphical Designs","authors":"Catherine Babecki, David Shiroma","doi":"10.1137/22m1528768","DOIUrl":"https://doi.org/10.1137/22m1528768","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 947-964, March 2024. <br/> Abstract. We show that the eigenpolytopes of graphs are universal in the sense that every polytope, up to affine equivalence, appears as the eigenpolytope of some positively weighted graph. We next extend the theory of graphical designs, which are quadrature rules for graphs, to positively weighted graphs. Through Gale duality for polytopes, we show a bijection between graphical designs and the faces of eigenpolytopes. This bijection proves the existence of graphical designs with positive quadrature weights and upper bounds the size of a minimal graphical design. Connecting this bijection with the universality of eigenpolytopes, we establish three complexity results: It is strongly NP-complete to determine if there is a graphical design smaller than the mentioned upper bound, it is NP-hard to find a smallest graphical design, and it is #P-complete to count the number of minimal graphical designs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140037685","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}
引用次数: 0
Robust Factorizations and Colorings of Tensor Graphs 张量图的稳健因式分解和着色
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-02-28 DOI: 10.1137/23m1552474
Joshua Brakensiek, Sami Davies
{"title":"Robust Factorizations and Colorings of Tensor Graphs","authors":"Joshua Brakensiek, Sami Davies","doi":"10.1137/23m1552474","DOIUrl":"https://doi.org/10.1137/23m1552474","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 883-916, March 2024. <br/> Abstract. Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around rounding the solution to a Semidefinite Program. However, it is likely that important combinatorial or algebraic insights are needed in order to break the [math] threshold. One way to develop new understanding in graph coloring is to study special subclasses of graphs. For instance, Blum studied the 3-coloring of random graphs, and Arora and Ge studied the 3-coloring of graphs with low threshold-rank. In this work, we study graphs that arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form [math] with [math] and [math], where [math] is any edge set such that no vertex has more than an [math]-fraction of its edges in [math]. We show that one can construct [math] with [math] that is close to [math]. For arbitrary [math], [math] satisfies [math]. Additionally, when [math] is a mild expander, we provide a 3-coloring for [math] in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on [math], we show that it is NP-hard to 3-color [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"109 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009282","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}
引用次数: 0
Rainbow Spanning Trees in Randomly Colored [math] 随机彩色彩虹生成树[数学］
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-02-27 DOI: 10.1137/22m1537497
Deepak Bal, Alan Frieze, Paweł Prałat
{"title":"Rainbow Spanning Trees in Randomly Colored [math]","authors":"Deepak Bal, Alan Frieze, Paweł Prałat","doi":"10.1137/22m1537497","DOIUrl":"https://doi.org/10.1137/22m1537497","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 867-882, March 2024. <br/> Abstract. Given a graph [math] on [math] vertices and an assignment of colors to its edges, a set of edges [math] is said to be rainbow if edges from [math] have pairwise different colors assigned to them. In this paper, we investigate rainbow spanning trees in randomly colored random [math] graphs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009473","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}
引用次数: 0
Pathwidth Versus Cocircumference 路径宽度与头围
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-02-26 DOI: 10.1137/23m158663x
Marcin Briański, Gwenaël Joret, Michał T. Seweryn
{"title":"Pathwidth Versus Cocircumference","authors":"Marcin Briański, Gwenaël Joret, Michał T. Seweryn","doi":"10.1137/23m158663x","DOIUrl":"https://doi.org/10.1137/23m158663x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 857-866, March 2024. <br/> Abstract. The circumference of a graph [math] with at least one cycle is the length of a longest cycle in [math]. A classic result of Birmelé [J. Graph Theory, 43 (2003), pp. 24–25] states that the treewidth of [math] is at most its circumference minus 1. In case [math] is 2-connected, this upper bound also holds for the pathwidth of [math]; in fact, even the treedepth of [math] is upper bounded by its circumference (Briański et al. [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659–664]). In this paper, we study whether similar bounds hold when replacing the circumference of [math] by its cocircumference, defined as the largest size of a bond in [math], an inclusionwise minimal set of edges [math] such that [math] has more components than [math]. In matroidal terms, the cocircumference of [math] is the circumference of the bond matroid of [math]. Our first result is the following “dual” version of Birmelé’s theorem: The treewidth of a graph [math] is at most its cocircumference. Our second and main result is an upper bound of [math] on the pathwidth of a 2-connected graph [math] with cocircumference [math]. Contrary to circumference, no such bound holds for the treedepth of [math]. Our two upper bounds are best possible up to a constant factor.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979291","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}
引用次数: 0
Cops and Robbers on [math]-Free Graphs 无[数学]图形上的警察与强盗
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-02-26 DOI: 10.1137/23m1549912
Maria Chudnovsky, Sergey Norin, Paul D. Seymour, Jérémie Turcotte
{"title":"Cops and Robbers on [math]-Free Graphs","authors":"Maria Chudnovsky, Sergey Norin, Paul D. Seymour, Jérémie Turcotte","doi":"10.1137/23m1549912","DOIUrl":"https://doi.org/10.1137/23m1549912","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 845-856, March 2024. <br/> Abstract. We prove that every connected [math]-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected [math]-free graph [math] with independence number at least three contains a three-vertex induced path with vertices [math] in order, such that every neighbor of [math] is also adjacent to one of [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979285","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}
引用次数: 0
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