SIAM Journal on Discrete Mathematics最新文献

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Strong Cocomparability Graphs and Slash-Free Orderings of Matrices","authors":"Pavol Hell, Jing Huang, Jephian C.-H. Lin","doi":"10.1137/22m153238x","DOIUrl":"https://doi.org/10.1137/22m153238x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 828-844, March 2024. <br/> Abstract. We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01,10, which we call Slash. We provide an ordering characterization, a forbidden structure characterization, and a polynomial-time certifying recognition algorithm for the class. These results complete the picture in which in addition to, or instead of, the [math] matrix one forbids the [math] matrix (which has rows 11,10). It is well known that in these two cases one obtains the class of interval graphs and the class of strongly chordal graphs, respectively. By complementation, we obtain the class of strong comparability graphs, whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the two-by-two identity submatrix. Thus our results give characterizations and algorithms for this class of irreflexive graphs as well. In other words, our results may be interpreted as solving the following problem: given a symmetric 0,1-matrix with 0-diagonal, can the rows and columns of be simultaneously permuted to avoid the two-by-two identity submatrix?","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772952","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":"Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem","authors":"Mark de Berg, Arpan Sadhukhan, Frits Spieksma","doi":"10.1137/23m1545975","DOIUrl":"https://doi.org/10.1137/23m1545975","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 790-827, March 2024. <br/> Abstract. Let [math] be a set of points in [math], where each point [math] has an associated transmission range, denoted [math]. The range assignment [math] induces a directed communication graph [math] on [math], which contains an edge [math] iff [math]. In the broadcast range-assignment problem, the goal is to assign the ranges such that [math] contains an arborescence rooted at a designated root node and the cost [math] of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution—the number of ranges that are modified when a point is inserted into or deleted from [math]—and its approximation ratio. To this end we study [math]-stable algorithms, which are algorithms that modify the range of at most [math] points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm [math] that, for any given fixed parameter [math], is [math]-stable and that maintains a solution with approximation ratio [math], where the stability parameter [math] only depends on [math] and not on the size of [math]. We study such trade-offs in three settings. (1) For the problem in [math], we present a SAS with [math]. Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have [math]. We also present 1-, 2-, and 3-stable algorithms with constant approximation ratio. (2) For the problem in [math] (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in [math], we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in [math]. (3) For the problem in [math], we also prove that no SAS exists, and we present a [math]-stable [math]-approximation algorithm. Most results generalize to the setting where, for any given constant [math], the range-assignment cost is [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758631","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":"Bernoulli Factories for Flow-Based Polytopes","authors":"Rad Niazadeh, Renato Paes Leme, Jon Schneider","doi":"10.1137/23m1558343","DOIUrl":"https://doi.org/10.1137/23m1558343","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 726-742, March 2024. <br/> Abstract. We construct explicit combinatorial Bernoulli factories for the following class of flow-based polytopes: integral 0/1-polytopes defined by a set of network flow constraints. This generalizes the results of Niazadeh et al. (who constructed an explicit factory for the specific case of bipartite perfect matchings) and provides novel exact sampling procedures for sampling paths, circulations, and [math]-flows. In the process, we uncover new connections to algebraic combinatorics.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772968","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":"Fractionally Subadditive Maximization under an Incremental Knapsack Constraint with Applications to Incremental Flows","authors":"Yann Disser, Max Klimm, Annette Lutz, David Weckbecker","doi":"10.1137/23m1569265","DOIUrl":"https://doi.org/10.1137/23m1569265","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 764-789, March 2024. <br/> Abstract. We consider the problem of maximizing a fractionally subadditive function under an increasing knapsack constraint. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most [math], under the assumption that the values of singleton sets are in the range [math], and we give a lower bound of [math] on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of [math] and an upper bound of [math] for the incremental maximization of classical flows with capacities in [math] which is tight for the unit capacity case.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772969","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":"Global Rigidity of Line Constrained Frameworks","authors":"James Cruickshank, Fatemeh Mohammadi, Harshit J. Motwani, Anthony Nixon, Shin-ichi Tanigawa","doi":"10.1137/22m151707x","DOIUrl":"https://doi.org/10.1137/22m151707x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 743-763, March 2024. <br/> Abstract. We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in [math]. In our setting, we allow multiple vertices to be constrained to the same line. We give a combinatorial characterization of generic rigidity in this setting for arbitrary line sets. Further, under a mild assumption on the given set of lines, we give a complete combinatorial characterization of graphs that are generically globally rigid. This gives a [math]-dimensional extension of the well-known combinatorial characterization of two-dimensional global rigidity. In particular, our results imply that global rigidity is a generic property in this setting.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758544","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}