Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, Alexander Wolff
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1537-1565, June 2024. Abstract. An obstacle representation of a graph [math] consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of [math] to points such that two vertices are adjacent in [math] if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each [math]-vertex graph is [math] [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143–164] and that there are [math]-vertex graphs whose obstacle number is [math] [V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to [math] for simple polygons and to [math] for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of [math]-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some [math]-vertex graph is given as part of the input, then for some drawings [math] obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph [math] is fixed-parameter tractable in the vertex cover number of [math]. Second, we show that, given a graph [math] and a simple polygon [math], it is NP-hard to decide whether [math] admits an obstacle representation using [math] as the only obstacle.
SIAM 离散数学杂志》，第 38 卷第 2 期，第 1537-1565 页，2024 年 6 月。 摘要。一个图[math]的障碍表示由一组成对不相交的简单连接封闭区域和[math]顶点到点的一一映射组成，当且仅当连接两个对应点的线段不与任何障碍相交时，两个顶点在[math]中相邻。图形的障碍数是图形在平面上的障碍表示中，所有障碍都是简单多边形的最小障碍数。已知每个[math]-顶点图的障碍数为[math] [M. Balko, J. Cibibi, J. Cibibi, J. M.Balko, J. Cibulka, and P. Valtr, Discrete Comput.Geom., 59 (2018)，第 143-164 页]，并且存在障碍数为[math]的[math]-顶点图[V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015)，3.1]。对于简单多边形，我们将这一下界改进为[math]；对于凸多边形，我们将其改进为[math]。为了得到这些更强的下界，我们改进了对障碍数有界的[math]顶点图数量的已知估计，解决了杜伊莫维奇和莫林的一个猜想。我们还证明了，如果把某个[数学]顶点图的绘制作为输入的一部分，那么对于某些绘制来说，需要[数学]障碍才能把它们变成图的障碍表示。我们的边界在一些情况下是渐近紧密的。我们用两个复杂度结果来补充这些组合界限。首先，我们证明了计算一个图[math]的障碍数在[math]的顶点覆盖数中是固定参数可控的。其次，我们证明，给定一个图 [math] 和一个简单多边形 [math]，用 [math] 作为唯一的障碍来决定 [math] 是否允许障碍表示是 NP 难的。
{"title":"Bounding and Computing Obstacle Numbers of Graphs","authors":"Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, Alexander Wolff","doi":"10.1137/23m1585088","DOIUrl":"https://doi.org/10.1137/23m1585088","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1537-1565, June 2024. <br/> Abstract. An obstacle representation of a graph [math] consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of [math] to points such that two vertices are adjacent in [math] if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each [math]-vertex graph is [math] [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143–164] and that there are [math]-vertex graphs whose obstacle number is [math] [V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to [math] for simple polygons and to [math] for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of [math]-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some [math]-vertex graph is given as part of the input, then for some drawings [math] obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph [math] is fixed-parameter tractable in the vertex cover number of [math]. Second, we show that, given a graph [math] and a simple polygon [math], it is NP-hard to decide whether [math] admits an obstacle representation using [math] as the only obstacle.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146395","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1526-1536, June 2024. Abstract. Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and are still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which leads to new axiom systems of matroids and oriented matroids.
{"title":"An Axiomatization of Matroids and Oriented Matroids as Conditional Independence Models","authors":"Xiangying Chen","doi":"10.1137/23m1558653","DOIUrl":"https://doi.org/10.1137/23m1558653","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1526-1536, June 2024. <br/> Abstract. Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and are still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which leads to new axiom systems of matroids and oriented matroids.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062659","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1492-1525, June 2024. Abstract. We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs at criticality. Even though the main result is not new, we believe the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above-mentioned bounds.
{"title":"A Simple Path to Component Sizes in Critical Random Graphs","authors":"Umberto De Ambroggio","doi":"10.1137/22m151056x","DOIUrl":"https://doi.org/10.1137/22m151056x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1492-1525, June 2024. <br/> Abstract. We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs at criticality. Even though the main result is not new, we believe the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above-mentioned bounds.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889887","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1462-1471, June 2024. Abstract. We show that for every [math], the maximal running time of the [math]-bootstrap percolation in the complete [math]-uniform hypergraph on [math] vertices [math] is [math]. This answers a recent question of Noel and Ranganathan in the affirmative and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form [math] as [math].
{"title":"The Maximal Running Time of Hypergraph Bootstrap Percolation","authors":"Ivailo Hartarsky, Lyuben Lichev","doi":"10.1137/22m151995x","DOIUrl":"https://doi.org/10.1137/22m151995x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1462-1471, June 2024. <br/> Abstract. We show that for every [math], the maximal running time of the [math]-bootstrap percolation in the complete [math]-uniform hypergraph on [math] vertices [math] is [math]. This answers a recent question of Noel and Ranganathan in the affirmative and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form [math] as [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883085","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1472-1491, June 2024. Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function [math] such that every matroid that can be factorized into [math] bases for some [math] can be covered by [math] rainbow bases if every partition class contains at most 2 elements.
{"title":"Rainbow Bases in Matroids","authors":"Florian Hörsch, Tomáš Kaiser, Matthias Kriesell","doi":"10.1137/22m1516750","DOIUrl":"https://doi.org/10.1137/22m1516750","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1472-1491, June 2024. <br/> Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function [math] such that every matroid that can be factorized into [math] bases for some [math] can be covered by [math] rainbow bases if every partition class contains at most 2 elements.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883083","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}
Stijn Cambie, Penny Haxell, Ross J. Kang, Ronen Wdowinski
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1451-1461, June 2024. Abstract. Given a bipartite graph [math] in which any vertex in [math] (resp., [math]) has degree at most [math] (resp., [math]), suppose there is a partition of [math] that is a refinement of the bipartition [math] such that the parts in [math] (resp., [math]) have size at least [math] (resp., [math]). We prove that the condition [math] is sufficient for the existence of an independent set of vertices of [math] that is simultaneously transversal to the partition and show, moreover, that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szabó and Tardos.
{"title":"A Precise Condition for Independent Transversals in Bipartite Covers","authors":"Stijn Cambie, Penny Haxell, Ross J. Kang, Ronen Wdowinski","doi":"10.1137/23m1600384","DOIUrl":"https://doi.org/10.1137/23m1600384","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1451-1461, June 2024. <br/> Abstract. Given a bipartite graph [math] in which any vertex in [math] (resp., [math]) has degree at most [math] (resp., [math]), suppose there is a partition of [math] that is a refinement of the bipartition [math] such that the parts in [math] (resp., [math]) have size at least [math] (resp., [math]). We prove that the condition [math] is sufficient for the existence of an independent set of vertices of [math] that is simultaneously transversal to the partition and show, moreover, that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szabó and Tardos.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883090","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}
Sandra Albrechtsen, Tony Huynh, Raphael W. Jacobs, Paul Knappe, Paul Wollan
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1438-1450, June 2024. Abstract. We give an approximate Menger-type theorem for the case when a graph [math] contains two [math] paths [math] and [math] such that [math] is an induced subgraph of [math]. More generally, we prove that there exists a function [math], such that for every graph [math] and [math], either there exist two [math] paths [math] and [math] such that the distance between [math] and [math] is at least [math], or there exists [math] such that the ball of radius [math] centered at [math] intersects every [math] path.
{"title":"A Menger-Type Theorem for Two Induced Paths","authors":"Sandra Albrechtsen, Tony Huynh, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"10.1137/23m1573082","DOIUrl":"https://doi.org/10.1137/23m1573082","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1438-1450, June 2024. <br/>Abstract. We give an approximate Menger-type theorem for the case when a graph [math] contains two [math] paths [math] and [math] such that [math] is an induced subgraph of [math]. More generally, we prove that there exists a function [math], such that for every graph [math] and [math], either there exist two [math] paths [math] and [math] such that the distance between [math] and [math] is at least [math], or there exists [math] such that the ball of radius [math] centered at [math] intersects every [math] path.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833159","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1417-1437, June 2024. Abstract. Horn functions form a subclass of Boolean functions possessing interesting structural and computational properties. These functions play a fundamental role in algebra, artificial intelligence, combinatorics, computer science, database theory, and logic. In the present paper, we introduce the subclass of hypergraph Horn functions that generalizes matroids and equivalence relations. We provide multiple characterizations of hypergraph Horn functions in terms of implicate-duality and the closure operator, which are, respectively, regarded as generalizations of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure. We also study algorithmic issues on hypergraph Horn functions and show that the recognition problem (i.e., deciding if a given definite Horn CNF represents a hypergraph Horn function) and key realization (i.e., deciding if a given hypergraph is realized as a key set by a hypergraph Horn function) can be done in polynomial time, while implicate sets can be generated with polynomial delay.
{"title":"Hypergraph Horn Functions","authors":"Kristóf Bérczi, Endre Boros, Kazuhisa Makino","doi":"10.1137/23m1569162","DOIUrl":"https://doi.org/10.1137/23m1569162","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1417-1437, June 2024. <br/> Abstract. Horn functions form a subclass of Boolean functions possessing interesting structural and computational properties. These functions play a fundamental role in algebra, artificial intelligence, combinatorics, computer science, database theory, and logic. In the present paper, we introduce the subclass of hypergraph Horn functions that generalizes matroids and equivalence relations. We provide multiple characterizations of hypergraph Horn functions in terms of implicate-duality and the closure operator, which are, respectively, regarded as generalizations of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure. We also study algorithmic issues on hypergraph Horn functions and show that the recognition problem (i.e., deciding if a given definite Horn CNF represents a hypergraph Horn function) and key realization (i.e., deciding if a given hypergraph is realized as a key set by a hypergraph Horn function) can be done in polynomial time, while implicate sets can be generated with polynomial delay.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833156","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1409-1416, June 2024. Abstract. In this paper, we strengthen a result by Green about an analogue of Sárközy’s theorem in the setting of polynomial rings [math]. In the integer setting, for a given polynomial [math] with constant term zero, (a generalization of) Sárközy’s theorem gives an upper bound on the maximum size of a subset [math] that does not contain distinct [math] satisfying [math] for some [math]. Green proved an analogous result with much stronger bounds in the setting of subsets [math] of the polynomial ring [math], but this result required the additional condition that the number of roots of the polynomial [math] be coprime to [math]. We generalize Green’s result, removing this condition. As an application, we also obtain a version of Sárközy’s theorem with similar strong bounds for subsets [math] for [math] for a fixed prime [math] and large [math].
{"title":"Sárközy’s Theorem in Various Finite Field Settings","authors":"Anqi Li, Lisa Sauermann","doi":"10.1137/23m1563256","DOIUrl":"https://doi.org/10.1137/23m1563256","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1409-1416, June 2024. <br/>Abstract. In this paper, we strengthen a result by Green about an analogue of Sárközy’s theorem in the setting of polynomial rings [math]. In the integer setting, for a given polynomial [math] with constant term zero, (a generalization of) Sárközy’s theorem gives an upper bound on the maximum size of a subset [math] that does not contain distinct [math] satisfying [math] for some [math]. Green proved an analogous result with much stronger bounds in the setting of subsets [math] of the polynomial ring [math], but this result required the additional condition that the number of roots of the polynomial [math] be coprime to [math]. We generalize Green’s result, removing this condition. As an application, we also obtain a version of Sárközy’s theorem with similar strong bounds for subsets [math] for [math] for a fixed prime [math] and large [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833114","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}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1381-1408, June 2024. Abstract. It is known that any open necklace with beads of [math] types, in which the number of beads of each type is divisible by [math], can be partitioned by at most [math] cuts into intervals that can be distributed into [math] collections, each containing the same number of beads of each type. This is tight for all values of [math] and [math]. Here, we consider the case of random necklaces, where the number of beads of each type is [math]. Then the minimum number of cuts required for a “fair” partition with the above property is a random variable [math]. We prove that for fixed [math] and large [math], this random variable is at least [math] with high probability. For [math], fixed [math], and large [math], we determine the asymptotic behavior of the probability that [math] for all values of [math]. We show that this probability is polynomially small when [math], is bounded away from zero when [math], and decays like [math] when [math]. We also show that for large [math], [math] is at most [math] with high probability and that for large [math] and large ratio [math], [math] is [math] with high probability.
{"title":"Random Necklaces Require Fewer Cuts","authors":"Noga Alon, Dor Elboim, János Pach, Gábor Tardos","doi":"10.1137/22m1506699","DOIUrl":"https://doi.org/10.1137/22m1506699","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1381-1408, June 2024. <br/>Abstract. It is known that any open necklace with beads of [math] types, in which the number of beads of each type is divisible by [math], can be partitioned by at most [math] cuts into intervals that can be distributed into [math] collections, each containing the same number of beads of each type. This is tight for all values of [math] and [math]. Here, we consider the case of random necklaces, where the number of beads of each type is [math]. Then the minimum number of cuts required for a “fair” partition with the above property is a random variable [math]. We prove that for fixed [math] and large [math], this random variable is at least [math] with high probability. For [math], fixed [math], and large [math], we determine the asymptotic behavior of the probability that [math] for all values of [math]. We show that this probability is polynomially small when [math], is bounded away from zero when [math], and decays like [math] when [math]. We also show that for large [math], [math] is at most [math] with high probability and that for large [math] and large ratio [math], [math] is [math] with high probability.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799188","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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