Random Structures and Algorithms最新文献_第2页

Greedy maximal independent sets via local limits 通过局部极限实现贪婪的最大独立集

Random Structures and Algorithms

Pub Date : 2023-12-18 DOI: 10.1002/rsa.21200

Michael Krivelevich, Tamás Mészáros, Peleg Michaeli, Clara Shikhelman

The random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent to any previously added vertex. In this paper, we present a general framework for computing the asymptotic density of the random greedy independent set for sequences of (possibly random) graphs by employing a notion of local convergence. We use this framework to give straightforward proofs for results on previously studied families of graphs, like paths and binomial random graphs, and to study new ones, like random trees and sparse random planar graphs. We conclude by analysing the random greedy algorithm more closely when the base graph is a tree.

在图中寻找最大独立集的随机贪婪算法通过以随机顺序检查图的顶点来构建最大独立集，如果当前顶点与之前添加的顶点不相邻，则将其添加到独立集中。在本文中，我们提出了一个通用框架，通过使用局部收敛的概念来计算（可能是随机的）图序列的随机贪婪独立集的渐近密度。我们利用这个框架对以前研究过的图族（如路径和二项式随机图）的结果进行了直接证明，并对新的图族（如随机树和稀疏随机平面图）进行了研究。最后，我们将更仔细地分析基图为树时的随机贪婪算法。

引用次数: 0

Random graphs embeddable in order-dependent surfaces 可嵌入顺序相关曲面的随机图

Random Structures and Algorithms

Pub Date : 2023-12-05 DOI: 10.1002/rsa.21199

Colin McDiarmid, Sophia Saller

Given a ‘genus function’

� � � g � = � g � (� n �) � �$$ g=g(n) $$�

, we let

� � � �^{� E � � � g � �} � �$$ {mathcal{E}}^g $$�

be the class of all graphs

� � � G � �$$ G $$�

such that if

� � � G � �$$ G $$�

has order

� � � n � �$$ n $$�

(i.e., has

� � � n � �$$ n $$�

vertices) then it is embeddable in a surface of Euler genus at most

� � � g � (� n �) � �$$ g(n) $$�

. Let the random graph

� � � �_{� R � � � n � �} � �$$ {R}_n $$�

be sampled uniformly from the graphs in

� � � �^{� E � � � g � �} � �$$ {mathcal{E}}^g $$�

on vertex set

� � � [� n �] � = � {� 1 �, � \dots �, � n �} � �$

给定一个“格函数”g=g(n) $$ g=g(n) $$，我们设Eg $$ {mathcal{E}}^g $$为所有图g $$ G $$的类，使得如果g $$ G $$有n阶$$ n $$(即，有n $$ n $$个顶点)，那么它最多可嵌入到g(n) $$ g(n) $$的欧拉格曲面中。设随机图Rn $$ {R}_n $$在顶点集[n]=，n $$ left[nright]=left{1,dots, nright} $$上从Eg {}$$ {mathcal{E}}^g $$中的图中均匀采样。观察到，如果g(n) $$ g(n) $$为0，则Rn $$ {R}_n $$是一个随机平面图，如果g(n) $$ g(n) $$足够大，则Rn $$ {R}_n $$是一个二项随机图g(n,12) $$ Gleft(n,frac{1}{2}right) $$。我们研究了Rn $$ {R}_n $$的典型性质。我们发现，对于每一个格函数g $$ g $$，在高概率下Rn $$ {R}_n $$最多有一个分量是非平面的。相反，我们发现了一个过渡，例如连通性:如果g(n) $$ g(n) $$是O(n/logn) $$ Oleft(n/log nright) $$并且g $$ g $$是非递减的，那么lim infn→∞(Rnis connected)＆lt;1 $$ lim {operatorname{inf}}_{nto infty}mathbb{P}left({R}_nkern0.3em mathrm{is} mathrm{connected}right)<1 $$，如果g(n) n $$ g(n)gg n $$那么高概率Rn $$ {R}_n $$是连通的。当我们分别考虑可定向和不可定向表面时，这些结果也成立。我们还研究了从Eg $$ {mathcal{E}}^g $$的“遗传部分”或“小闭部分”均匀抽样的随机图，并简要考虑了未标记图的相应结果。

引用次数: 3

Small cycle structure for words in conjugation invariant random permutations 共轭不变随机排列中单词的小循环结构

Random Structures and Algorithms

Pub Date : 2023-11-29 DOI: 10.1002/rsa.21203

Mohamed Slim Kammoun, Mylène Maïda

We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive cyclic simplifications, the word

� � � w � �$$ w $$�

still contains at least two different letters, then we get a universal limiting joint law for short cycles for the word in these permutations. These results can be seen as an extension of our previous work (Kammoun and Maïda. Electron. Commun. Probab. 2020;25:1-14.) from the product of permutations to any non-trivial word in the permutations and also as an extension of the results of Nica (Random Struct. Algorithms1994;5:703-730.) from uniform permutations to general conjugation invariant random permutations. In particular, we get optimal assumptions in the case of the commutator of two such random permutations.

我们研究了几种随机排列中单词的循环结构。我们假设这些置换是独立的，它们的分布是共轭不变的，它们的短周期得到了很好的控制。如果在连续循环化简之后，单词w $$ w $$仍然包含至少两个不同的字母，那么我们就得到了单词在这些排列中的短循环的通用极限联合律。这些结果可以看作是我们之前工作的延伸(Kammoun和Maïda)。电子。普通的。Probab. 2020;25:1-14.)，从排列的乘积到排列中的任何非平凡单词，也作为Nica (Random Struct)结果的扩展。从一致排列到一般共轭不变随机排列。特别地，我们得到了两个这样的随机排列的对易子的最优假设。

引用次数: 0

On random irregular subgraphs 在随机不规则子图上

Random Structures and Algorithms

Pub Date : 2023-11-28 DOI: 10.1002/rsa.21204

Jacob Fox, Sammy Luo, Huy Tuan Pham

Let

� � � G � �$$ G $$�

be a

� � � d � �$$ d $$�

-regular graph on

� � � n � �$$ n $$�

vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model

� � � H � = � H � (� G �) � �$$ H=H(G) $$�

. Assign independently to each vertex

� � � v � �$$ v $$�

of

� � � G � �$$ G $$�

a uniform random number

� � � x � (� v �) � \in � [� 0 �, � 1 �] � �$$ x(v)in left[0,1right] $$�

, and an edge

� � � (� u �, � v �) � �$$ left(u,vright) $$�

of

� � � G � �$$ G $$�

is an edge of

� � � H � �$$ H $$�

if and only if

� � �

设G $$ G $$是一个有n $$ n $$个顶点的d $$ d $$正则图。Frieze, Gould, Karoński和Pfender开始研究以下随机生成子图模型H=H(G) $$ H=H(G) $$。为G $$ G $$的每个顶点v $$ v $$独立分配一个均匀随机数x(v)∈[0,1]$$ x(v)in left[0,1right] $$，并且当且仅当x(u)+x(v)≥1 $$ x(u)+x(v)ge 1 $$时，G $$ G $$的边(u,v) $$ left(u,vright) $$就是H $$ H $$的边。针对Alon和Wei的问题，我们证明了如果d=o(n/(logn)12) $$ d=oleft(n/{left(log nright)}^{12}right) $$，那么对于每一个k≤d $$ kle d $$的非负整数，H $$ H $$中有(1+o(1))n/(d+1) $$ left(1+o(1)right)n/left(d+1right) $$个k $$ k $$度的顶点。

引用次数: 0

Rainbow Hamilton cycles in random geometric graphs 彩虹汉密尔顿循环随机几何图形

Random Structures and Algorithms

Pub Date : 2023-11-27 DOI: 10.1002/rsa.21201

Alan Frieze, Xavier Pérez-Giménez

Let

� � � �_{� X � � � 1 � �} �, � �_{� X � � � 2 � �} �, � \dots �, � �_{� X � � � n � �} � �$$ {X}_1,{X}_2,dots, {X}_n $$�

be chosen independently and uniformly at random from the unit

� � � d � �$$ d $$�

-dimensional cube

� � � �^{� [� 0 �, � 1 �] � � � d � �} � �$$ {left[0,1right]}^d $$�

. Let

� � � r � �$$ r $$�

be given and let

� � 𝒳 � = � \{� � �_{� X � � � 1 � �} �, � �_{� X � � � 2 � �} �, � \dots �, � �_{� X � � � n � �} � �\} �

. The random geometric graph

� � G � = � �_{� G � � � 𝒳 �, � r � �} �

has vertex set

� � 𝒳 �

and an edge

� � � �_{� X � � � i � �} � �_{� X � � � j � �} � �$$ {X}_i{X}_j $$�

whenever

� � � ‖ � �_{� X � � � i �}

设X1 X2…Xn$$ {X}_1,{X}_2,dots, {X}_n $$ 从单位d中独立均匀随机选取$$ d $$-维立方体[0,1$$ {left[0,1right]}^d $$．设r$$ r $$ 设，f =X1,X2，…，Xn。随机几何图G=G∈，r具有顶点集∈∈和一条边∈$$ {X}_i{X}_j $$ $$ leftVert {X}_i-{X}_jrightVert le r $$．如果G的每条边$$ G $$ 与n+o(n)中的一个无关$$ n+o(n) $$ 颜色和r$$ r $$ 的值最小，使得G$$ G $$ 最小度至少为2，那么G$$ G $$ 几乎肯定包含一个渐近的彩虹汉密尔顿环。

引用次数: 1

Frozen 1-RSB structure of the symmetric Ising perceptron 对称Ising感知器的冻结1-RSB结构

Random Structures and Algorithms

Pub Date : 2023-11-23 DOI: 10.1002/rsa.21202

Will Perkins, Changji Xu

We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1-RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.

在实值函数临界点的假设下，证明了对称Ising感知器呈现出物理文献中kraauth和massaard猜想的“冻结1-RSB”结构;即模型的典型解存在于熵密度消失的聚类中。此外，我们用Huang, Wong和Kabashima猜想的一个非常强的形式证明了这一点:模型的一个典型解是高概率孤立的，并且到所有其他解的Hamming距离在维度上是线性的。冻结的1-RSB场景是Baldassi, Ingrosso, Lucibello, Saglietti和Zecchina最近对学习算法性能的有趣解释的一部分。我们通过比较对称的伊辛感知器模型和种植模型，并证明了两个模型之间的比较结果，证明了这一结构结果。我们进行这种比较的主要技术工具是对模型中解的对数的浓度的归纳论证。

引用次数: 0

Wireless random-access networks with bipartite interference graphs 具有二部干扰图的无线随机存取网络

Random Structures and Algorithms

Pub Date : 2023-11-23 DOI: 10.1002/rsa.21198

Sem C. Borst, Frank den Hollander, Francesca R. Nardi, Matteo Sfragara

We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

我们考虑随机访问网络，其中节点代表具有队列的服务器，可以是活动的也可以是不活动的。一个节点以单位速率去激活，而它的激活速率取决于它的队列长度，前提是它的邻居都不是活动的。我们考虑了任意二部图在初始队列长度变大时的极限，并确定了网络的一半是活动的，另一半是不活动的两种状态之间的过渡时间。将过渡路径分解为完全二部子图上的一系列过渡。我们制定了一个随机贪婪算法，将图作为输入，并给出网络最有可能遵循的过渡路径集作为输出。沿着每条路径，我们确定了平均过渡时间及其在其均值尺度上的规律。根据激活率，我们确定了三种行为模式。

引用次数: 3

Note on down-set thresholds 注意下限阈值

Random Structures and Algorithms

Pub Date : 2023-11-22 DOI: 10.1002/rsa.21194

Lutz Warnke

Gunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up‐sets do not apply to down‐sets. In particular, for the down‐set of triangle‐free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down‐sets of ‐free graphs.

Gunby-He-Narayanan表明，Kahn-Kalai和Talagrand(由Park-Pham和Frankston-Kahn-Narayanan-Park证明)关于上升阈值的对数间隙预测不适用于下降阈值。特别地，对于无三角图的下集，他们证明了阈值与分式期望阈值之间存在多项式间隙。在这篇简短的笔记中，我们给出了这个结果的一个更简单的证明，并将多项式阈值间隙扩展到F $$ F $$自由图的下集。

引用次数: 0

Random perfect matchings in regular graphs 正则图中的随机完美匹配

Random Structures and Algorithms

Pub Date : 2023-07-12 DOI: 10.1002/rsa.21172

Bertille Granet, Felix Joos

We prove that in all regular robust expanders

� � � G � �$$ G $$�

, every edge is asymptotically equally likely contained in a uniformly chosen perfect matching

� � � M � �$$ M $$�

. We also show that given any fixed matching or spanning regular graph

� � � N � �$$ N $$�

in

� � � G � �$$ G $$�

, the random variable

� � � | � M � \cap � E � (� N �) � | � �$$ mid Mcap E(N)mid $$�

is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.

我们证明了在所有正则鲁棒展开G $$ G $$中，每条边都是渐近等可能包含在一致选择的完美匹配M $$ M $$中。我们还证明了给定任意固定匹配或生成正则图N $$ N $$在G $$ G $$中，随机变量|M∩E(N)| $$ mid Mcap E(N)mid $$近似泊松分布。这特别证实了Spiro和Surya的一个猜想和问题，并补充了Kahn和Kim的结果，他们证明了在正则图中，每个顶点都是渐近等可能包含在一致选择的匹配中。我们的证明依赖于切换方法和简单随机漫步在鲁棒扩展器中快速混合的事实。

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引用次数: 0

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