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Rapid Mixing of [math]-Class Biased Permutations 数学]类偏置排列的快速混合

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-02-06 DOI: 10.1137/22m148063x

Sarah Miracle, Amanda Pascoe Streib

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 702-725, March 2024.
Abstract. In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements [math] and [math] are placed in order [math] with probability [math]. Our goal is to identify the class of parameter sets [math] for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill [Background on the Gap Problem (2003) and An Interesting Spectral Gap Problem (2003)] that all monotone, positively biased distributions are rapidly mixing. We resolve Fill’s conjecture in the affirmative for distributions arising from [math]-class particle processes, where the elements are divided into [math] classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that [math] is a constant and that all probabilities between elements in different classes are bounded away from [math]. These particle processes arise in the context of self-organizing lists, and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Our work generalizes recent work by Haddadan and Winkler [Mixing of permutations by biased transposition (2017)] studying 3-class particle processes. Additionally, we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et al. [Mixing times of Markov chains for self-organizing lists and biased permutations (2013)]. Our proof involves analyzing a generalized biased exclusion process, which is a nearest-neighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. [Sampling biased lattice configurations using exponential metrics (2009)] and Benjamini et al. [Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)] on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.

SIAM 离散数学杂志》，第 38 卷第 1 期，第 702-725 页，2024 年 3 月。摘要在本文中，我们研究了最近邻换位马尔可夫链在排列集合上的偏置版本，其中相邻元素[math]和[math]以概率[math]按顺序[math]排列。我们的目标是找出该马尔可夫链快速混合的参数集 [math] 类别。具体来说，我们考虑了吉姆-菲尔（Jim Fill）的公开猜想[《差距问题背景》（2003）和《有趣的谱差距问题》（2003）]，即所有单调、正偏分布都是快速混合的。对于[math]类粒子过程产生的分布，我们以肯定的态度解决了菲尔的猜想，在[math]类粒子过程中，元素被划分为[math]类，相邻元素交换的概率取决于元素所处的特定类。我们进一步要求[math]是一个常数，不同类别元素之间的所有概率都远离[math]。这些粒子过程是在自组织列表的背景下产生的，我们的结果也超越了排列，适用于一类中所有粒子都无法区分的情况。我们的工作概括了 Haddadan 和 Winkler [Mixing of permutations by biased transposition (2017)]最近研究 3 类粒子过程的工作。此外，我们还证明了基于树的一类更广泛的分布也是快速混合的，这也概括了 Bhakta 等人【自组织列表和偏置排列的马尔可夫链的混合时间（2013 年）】所分析的一类分布。我们的证明涉及对广义偏置排除过程的分析，该过程是应用于双粒子系统的近邻转置链。偏置排除过程具有独立的意义，可应用于自组装。我们推广了格林伯格等人（Greenberg et al. [Sampling biased lattice configurations using exponential metrics (2009)]）和本杰明尼等人（Benjamini et al. [Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)]）关于偏置排除过程的研究成果，只要最小偏置离 1 有界，就允许交换相邻元素的概率取决于整个系统。

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引用次数: 0

Some Cubic Time Regularity Algorithms for Triple Systems 三重系统的一些立方时间正则算法

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-02-05 DOI: 10.1137/21m145046x

Brendan Nagle, John Theado

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 668-701, March 2024.
Abstract. Szemerédi’s regularity lemma guarantees that, for fixed [math], every graph [math] admits an [math]-regular and [math]-equitable partition [math], where [math]. These partitions are constructed by Kohayakawa, Rödl, and Thoma in time [math]. Analogous partitions of [math]-graphs [math] are constructed by Czygrinow and Rödl in time [math]. For [math], we construct these partitions (and others with slightly stronger regularity) in time [math]. We also discuss some applications.

SIAM 离散数学杂志》，第 38 卷第 1 期，第 668-701 页，2024 年 3 月。摘要。Szemerédi 的正则性 Lemma 保证，对于固定的 [math]，每个图 [math] 都有一个 [math] 正则且 [math] 公平的分割 [math]，其中 [math]。这些分区由小早川、罗德尔和索马在[math]时间内构造。类似的[math]图[math]分区是由 Czygrinow 和 Rödl 在[math]时间内构造的。对于[math]，我们在[math]时间内构造了这些分区（以及其他规律性稍强的分区）。我们还讨论了一些应用。

引用次数: 0

Graphs without a Rainbow Path of Length 3 没有长度为 3 的彩虹路径的图形

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-02-02 DOI: 10.1137/22m1535048

Sebastian Babiński, Andrzej Grzesik

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 629-644, March 2024.
Abstract. In 1959, Erdős and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here, we study a rainbow version of their theorem, in which one considers [math] graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any [math].

SIAM 离散数学杂志》，第 38 卷第 1 期，第 629-644 页，2024 年 3 月。摘要1959 年，Erdős 和 Gallai 证明了图中不包含固定长度路径的最大边数的渐近最优约束。在这里，我们研究的是他们定理的彩虹版本，其中我们考虑的是共同顶点集上的 [math] 图，这些图不包含来自不同图的边，并求解每个图中的最大边数。我们证明了在三条边和任意 [math] 路径情况下的渐近最优约束。

引用次数: 0

Pure Pairs. IX. Transversal Trees 纯对IX.横向树

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-02-02 DOI: 10.1137/21m1456509

Alex Scott, Paul Seymour, Sophie T. Spirkl

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 645-667, March 2024.
Abstract. Fix [math], and let [math] be a graph, with vertex set partitioned into [math] subsets (“blocks”) of approximately equal size. An induced subgraph of [math] is “transversal” (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly [math] vertices). A “pure pair” in [math] is a pair [math] of disjoint subsets of [math] such that either all edges between [math] are present or none are; and in the present context we are interested in pure pairs [math] where each of [math] is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.

SIAM 离散数学杂志》，第 38 卷第 1 期，第 645-667 页，2024 年 3 月。摘要固定[math]，设[math]是一个图，其顶点集被分割成大小大致相同的[math]子集（"块"）。如果[math]的一个诱导子图在每个块中正好有一个顶点（因此它正好有[math]个顶点），那么这个诱导子图就是 "横向的"（关于这个分割）。[math]中的 "纯对 "是[math]的一对互不相交的子集[math]，使得[math]之间要么存在所有边，要么没有边；在本文中，我们感兴趣的是纯对[math]，其中每个[math]都是其中一个块的子集，而不是同一个块。本文收集了一些结果和悬而未决的问题，涉及如果排除各种类型的横向子图，纯对必须有多大。

引用次数: 0

Canonical Theorems for Colored Integers with Respect to Some Linear Combinations 关于某些线性组合的有色整数典范定理

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-02-01 DOI: 10.1137/21m1454195

Maria Axenovich, Hanno Lefmann

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 609-628, March 2024.
Abstract. Hindman proved in 1979 that no matter how natural numbers are colored in [math] colors, for a fixed positive integer [math], there is an infinite subset [math] of numbers and a color [math] such that for any finite nonempty subset [math] of [math], the color of the sum of elements from [math] is [math]. Later, Taylor extended this result to colorings with an unrestricted number of colors and five unavoidable color patterns on finite sums. This result is referred to as a canonization of Hindman’s theorem and parallels the canonical Ramsey theorem of Erdős and Rado. We extend Taylor’s result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and [math]. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.

SIAM 离散数学杂志》，第 38 卷第 1 期，第 609-628 页，2024 年 3 月。摘要。欣德曼在 1979 年证明，无论自然数如何用 [math] 颜色着色，对于一个固定的正整数 [math]，存在一个无限的数子集 [math] 和一种颜色 [math]，使得对于 [math] 的任何有限非空子集 [math]，来自 [math] 的元素之和的颜色是 [math]。后来，泰勒将这一结果扩展到具有不受限制的颜色数和有限和上五个不可避免的颜色模式的着色。这一结果被称为辛德曼定理的典范化，与厄多斯和拉多的典范拉姆齐定理相似。我们将泰勒的结果从系数为 1 的线性组合和扩展到系数为 1 和 [math] 的多个线性组合。这些结果反过来又可以解释为无限系统解的典型定理。

引用次数: 0

Graph Limits and Spectral Extremal Problems for Graphs 图形极限和图形谱极值问题

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-01-31 DOI: 10.1137/22m1508807

Lele Liu

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 590-608, March 2024.
Abstract. We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let [math] be the largest eigenvalue of the adjacency matrix of a graph [math] and [math] be the complement of [math]. A nice conjecture states that the graph on [math] vertices maximizing [math] is the join of a clique and an independent set with [math] and [math] (also [math] and [math] if [math]) vertices, respectively. We resolve this conjecture for sufficiently large [math] using analytic methods. Our second result concerns the [math]-spread of a graph [math], which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of [math]. It was conjectured by Cvetković, Rowlinson, and Simić [Publ. Inst. Math., 81 (2007), pp. 11–27] that the unique [math]-vertex connected graph of maximum [math]-spread is the graph formed by adding a pendant edge to [math]. We confirm this conjecture for sufficiently large [math].

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 590-608 页，2024 年 3 月。摘要。我们证明了谱极值图论中涉及图特征值线性组合的两个猜想。设[math]是图[math]邻接矩阵的最大特征值，[math]是[math]的补集。一个不错的猜想指出，[math] 顶点上[math] 最大的图是一个小群和一个独立集的连接，小群和独立集的顶点分别是[math]和[math]（如果是[math]，则也是[math]和[math]）。对于足够大的 [math]，我们用分析方法解决了这个猜想。我们的第二个结果涉及图[math]的[math]-spread，它被定义为[math]的无符号拉普拉奇的最大特征值和最小特征值之差。根据 Cvetković、Rowlinson 和 Simić 的猜想[Publ. Inst. Math., 81 (2007), pp.对于足够大的 [math]，我们证实了这一猜想。

引用次数: 0

The Power of Filling in Balanced Allocations 平衡分配的力量

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-01-29 DOI: 10.1137/23m1552231

Dimitrios Los, Thomas Sauerwald, John Sylvester

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 529-565, March 2024.
Abstract. We introduce a new class of balanced allocation processes which are primarily characterized by “filling” underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, and if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is [math] w.h.p. for any number of balls [math]. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of [math] on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar, and Shah [43rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 2002, pp. 799–808].

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 529-565 页，2024 年 3 月。摘要我们引入了一类新的平衡分配过程，其主要特征是 "填充 "负载不足的箱。打包过程就是一个典型的例子：在每一轮中，我们只取一个仓的样本，如果负载低于平均负载，我们就放置尽可能多的球，直到达到平均负载为止；否则，我们只放置一个球。我们证明，对于本类中的任何一个过程，对于任何数量的球，最大负荷和平均负荷之间的差距都是 [math]w.h.p.[math]。对于打包过程，我们还提供了一个匹配的下限。此外，我们还证明了打包过程的样本效率，即每次样本分配的预期球数严格大于 1。最后，我们还证明了 [math] 关于间隙的上界可以扩展到 Mitzenmacher、Prabhakar 和 Shah 所研究的记忆过程[第 43 届 IEEE 计算机科学基础年度研讨会，加拿大不列颠哥伦比亚省温哥华，2002 年，第 799-808 页]。

引用次数: 0

[math]-Modules in Graphs [数学]-图中的模块

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-01-29 DOI: 10.1137/21m1443534

Michel Habib, Lalla Mouatadid, Éric Sopena, Mengchuan Zou

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 566-589, March 2024.
Abstract. Modular decomposition focuses on repeatedly identifying a module [math] (a collection of vertices that shares exactly the same neighborhood outside of [math]) and collapsing it into a single vertex. This notion of exactitude of neighborhood is very strict, especially when dealing with real-world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an [math]-module, a relaxation that maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal [math]-modules can be computed in polynomial time, and we generalize series and parallel operation between graphs. This leads to [math]-cographs which have interesting properties. We study how to generalize Gallai’s theorem corresponding to the case for [math], but unfortunately we give evidence that computing such a decomposition tree can be difficult.

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 566-589 页，2024 年 3 月。摘要。模块分解的重点是反复识别模块[math]（[math]之外共享完全相同邻域的顶点集合）并将其折叠为单个顶点。这种邻域精确性的概念非常严格，尤其是在处理现实世界的图形时。我们研究了放宽这一精确性条件的新方法。然而，推广模块分解远非易事。之前的大多数提议都失去了模块的代数特性，因此也失去了大多数好的算法结果。我们引入了[math]模块的概念，这种放宽保留了部分代数结构。它带来了一种具有有趣特性的新组合分解。在这项工作的主要成果中，我们证明了最小[math]模块可以在多项式时间内计算，并对图之间的串联和并联操作进行了概括。这导致了具有有趣性质的[math]图。我们研究了如何推广与[math]情况相对应的伽来定理，但不幸的是，我们给出的证据表明，计算这样的分解树可能很困难。

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引用次数: 0

Square Coloring Planar Graphs with Automatic Discharging 自动放电的方形着色平面图形

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-01-23 DOI: 10.1137/22m1492623

Nicolas Bousquet, Quentin Deschamps, Lucas De Meyer, Théo Pierron

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 504-528, March 2024.
Abstract. The discharging method is a powerful proof technique, especially for graph coloring problems. Its major downside is that it often requires lengthy case analyses, which are sometimes given to a computer for verification. However, it is much less common to use a computer to actively look for a discharging proof. In this paper, we use a linear programming approach to automatically look for a discharging proof. While our system is not entirely autonomous, we manage to make some progress toward Wegner’s conjecture for distance-2 coloring of planar graphs by showing that 12 colors are sufficient to color at distance 2 every planar graph with maximum degree 4.

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 504-528 页，2024 年 3 月。摘要放电法是一种强大的证明技术，尤其适用于图着色问题。它的主要缺点是经常需要冗长的案例分析，有时需要交给计算机进行验证。然而，利用计算机主动寻找放电证明的情况却很少见。在本文中，我们使用线性规划方法来自动寻找放电证明。虽然我们的系统并非完全自主，但我们设法在实现韦格纳关于平面图距离-2着色的猜想方面取得了一些进展，证明了 12 种颜色足以在距离-2 处为最大度数为 4 的每个平面图着色。

引用次数: 0

On the Combinatorial Diameters of Parallel and Series Connections 论并联和串联的组合直径

IF 0.8 3区数学 Q2 Mathematics

SIAM Journal on Discrete Mathematics

Pub Date : 2024-01-22 DOI: 10.1137/22m1490508

Steffen Borgwardt, Weston Grewe, Jon Lee

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 485-503, March 2024.
Abstract. The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids. Oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyhedra whose combinatorial diameter satisfies the Hirsch-conjecture bound regardless of the right-hand sides in a standard-form description, the diameters of their parallel or series connections remain small in the Hirsch-conjecture bound. These results are a substantial step toward devising a diameter bound for all polyhedra defined through totally unimodular matrices based on Seymour’s famous decomposition theorem. Our proof techniques and results exhibit a number of interesting features. While the parallel connection leads to a bound that adds just a constant, for the series connection one has to linearly take into account the maximal value in a specific coordinate of any vertex. Our proofs also require a careful treatment of non-revisiting edge walks in degenerate polyhedra as well as the construction of edge walks that may take a “detour" to facets that satisfy the non-revisiting conjecture when the underlying polyhedron may not.

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 485-503 页，2024 年 3 月。摘要。研究多面体的组合直径是线性规划中的一个经典课题，因为它与为单纯形法提供有效枢轴规则的可能性有关。我们感兴趣的是由定向矩阵的所谓平行或串联连接形成的多面体的直径。定向矩阵是将可表示矩阵理论与线性规划组合学联系起来的自然方法，这些联系是由基本矩阵块构造更复杂矩阵的基本操作。我们证明，对于组合直径满足赫希猜想约束的多面体，无论其标准形式描述中的右手边是什么，其平行或串联连接的直径在赫希猜想约束中仍然很小。这些结果是基于西摩著名的分解定理，为所有通过完全单模矩阵定义的多面体设计直径约束的重要一步。我们的证明技术和结果呈现出许多有趣的特点。平行连接只需添加一个常数就能得到一个约束，而对于串联连接，则必须线性地考虑任意顶点特定坐标的最大值。我们的证明还需要仔细处理退化多面体中的非重访边走行，以及构建边走行，这些边走行可能会 "绕道 "到满足非重访猜想的面，而底层多面体可能不满足该猜想。

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