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Random Matchings in Regular Graphs

Jeff Kahn, Jeong Han Kim
1998 Combinatorica  
For a simple d-regular graph G, let M be chosen uniformly at random from the set of all matchings of G, and for x ∈ V (G) let p(x) be the probability that M does not cover x.  ...  We show that for large d, the p(x)'s and the mean µ and variance σ 2 of |M | are determined to within small tolerances just by d and (in the case of µ and σ 2 ) |V (G)|: Theorem For any d-regular graph  ...  We use "graph" to mean simple graph.) In this paper we are concerned with the behavior of M , and in particular of the random variable ξ = ξ G = |M |, when G is regular of large degree.  ... 
doi:10.1007/pl00009817 fatcat:es5pd2s64fbt3he6hzxb3eqdhi

Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs [article]

Roman Glebov, Zur Luria, Michael Simkin
2020 arXiv   pre-print
Consider the random process in which the edges of a graph G are added one by one in a random order.  ...  Using a construction due to Goel, Kapralov and Khanna, we show that there exist bipartite k-regular graphs in which the last isolated vertex disappears long before a perfect matching appears.  ...  This is the case, for example, in random regular graphs. However, in an arbitrary graph this need not hold.  ... 
arXiv:1805.06944v5 fatcat:ras6da3wifeb3lkmmfp7g5ts4a

Randomized Online Matching in Regular Graphs [chapter]

Ilan Reuven Cohen, David Wajc
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs.  ...  In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order.  ...  Online Matching in Regular Graphs. In this work, we study the classic problem of online matching, in the class of d-regular graphs.  ... 
doi:10.1137/1.9781611975031.62 dblp:conf/soda/CohenW18 fatcat:drza5fz4fzcflhcmragkckfqcm

The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs [article]

Deepak Bal, Patrick Bennett
2021 arXiv   pre-print
In this note, we analyze two random greedy processes on sparse random graphs and hypergraphs with a given degree sequence.  ...  Our main contribution is to significantly reduce the associated systems of differential equations and simplify the expression for the final size of the matching or independent set.  ...  Indeed, Bohman and Frieze [3] showed that the Karp-Sipser algorithm produces a matching covering all but o(n) vertices in random regular graphs.  ... 
arXiv:1603.09232v2 fatcat:3u7hu2sykbb6riecyib4mvxssi

Finding perfect matchings in random regular graphs in linear time [article]

Michael Anastos
2018 arXiv   pre-print
We show that the Reduce-Construct algorithm finds a maximum matching in random k=O(1)-regular graphs in linear time in expectation, as opposed to O(n^3/2) time for the worst-case.  ...  In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task.  ...  In this paper we analyze the Reduce-Construct algorithm for finding perfect matchings in random regular graphs.  ... 
arXiv:1811.11035v1 fatcat:v6uhqrqtynhuxioh4a4utni7ba

The number of matchings in random regular graphs and bipartite graphs

B Bollobás, B.D McKay
1986 Journal of combinatorial theory. Series B (Print)  
Our main aim in this paper is to determine the asymptotic values of the first two moments of the number of l-factors in random regular graphs and bipartite graphs.  ...  The systematic study of random regular graphs was started only recently, when Bender and Canfield [l] determined the asymptotic number of k-regular graphs.  ... 
doi:10.1016/0095-8956(86)90029-8 fatcat:zxztigkyjzcwzjqvvkknueabpu

Online Matching in Regular Bipartite Graphs with Randomized Adversary [chapter]

Josep Fàbrega, Xavier Muñoz
2018 Lecture Notes in Computer Science  
This paper deals with the problem of finding matchings in bipartite regular graphs in the online model with randomized adversary.  ...  We determine the competitive ratio for 2-regular graphs for any number of random decisions made by the adversary. We also give upper and lower bounds for other values of the degree.  ...  Online Matching in Regular Graphs with Randomized Adversary In the online bipartite matching problem, we are given a bipartite graph, G = (U ∪ W, E).  ... 
doi:10.1007/978-3-319-98355-4_17 fatcat:nwymopujkrgyzmi73aaopwotqi

Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections [article]

David Gamarnik, David Goldberg
2008 arXiv   pre-print
We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs.  ...  A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings.  ...  Acknowledgements The authors would like to thank Alan Frieze and Nick Wormald for a very helpful correspondence explaining the state of the art results in the area.  ... 
arXiv:0807.1277v1 fatcat:ut5xforrerhl5faivaixnv4zfq

Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

DAVID GAMARNIK, DAVID A. GOLDBERG
2009 Combinatorics, probability & computing  
We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs.  ...  A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings.  ...  Acknowledgements The authors would like to thank Alan Frieze and Nick Wormald for a very helpful correspondence explaining the state of the art results in the area.  ... 
doi:10.1017/s0963548309990186 fatcat:smhwkggubfhl3h3ctu73hhdz2i

Virtual path network topology optimization using random graphs

A. Farago, I. Chlamtac, S. Basagni
1999 IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320)  
This result is made possible by applying the theory of random graphs to this type of networks.  ...  Therefore, the paper exemplifies that the theory of random graphs, beyond supporting analysis purposes, may serve as a useful tool in the design of algorithms that overcome the "scalability bottleneck,  ...  Now, taking into account that the complement of a random i -1 regular graph is a random k-regular graph with k = ni , and by ( A ) a random k-regular graph almost surely has a perfect matching, we have  ... 
doi:10.1109/infcom.1999.751382 dblp:conf/infocom/FaragoCB99 fatcat:tlqicvwsknhdtirkoa7rgo5s6y

Perfect Matchings in O(n n) Time in Regular Bipartite Graphs [article]

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2010 arXiv   pre-print
In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n n) time (both in expectation and with high probability).  ...  In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost and Schirra).  ...  Matchings in d-Regular Bipartite Graphs The Basic Algorithm Let G = (P, Q, E) denote the input d-regular graph and let M be a partial matching in G.  ... 
arXiv:0909.3346v3 fatcat:rh4z6n5dvzbshmwyhl7scsluaq

Perfect Matchings in $O(n\log n)$ Time in Regular Bipartite Graphs

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2013 SIAM journal on computing (Print)  
In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n log n) time (both in expectation and with high probability).  ...  In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost, and Schirra).  ...  MATCHINGS IN d-REGULAR BIPARTITE GRAPHS The Basic Algorithm Let G = (P, Q, E) denote the input d-regular graph and let M be a partial matching in G.  ... 
doi:10.1137/100812513 fatcat:2fsrhiuj4jgptou63nhqonxqs4

Perfect matchings in o(nlogn) time in regular bipartite graphs

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n log n) time (both in expectation and with high probability).  ...  In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost, and Schirra).  ...  MATCHINGS IN d-REGULAR BIPARTITE GRAPHS The Basic Algorithm Let G = (P, Q, E) denote the input d-regular graph and let M be a partial matching in G.  ... 
doi:10.1145/1806689.1806697 dblp:conf/stoc/GoelKK10 fatcat:wx4wlsgsmvhxliw7otjsezac54

Page 7571 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
More generally, it is shown here that various ways of building a random r-regular graph as the union of a number of random regular graphs of smaller degree are equivalent in the sense of contiguity.  ...  matchings on 2n vertices and conditioning on the result being an r-regular graph (i.e., conditioning on the event that all matchings are dis- joint).  ... 

On the total domatic number of regular graphs

H. Aram, S.M. Sheikholeslami, L. Volkmann
2012 Transactions on Combinatorics  
We show that the total domatic number of a random r-regular graph is almost surely at most r  ...  A set S of vertices of a graph G = (V;E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S.  ...  Thus if an event is almost surely true in a random graph constructed from a random Hamilton cycle plus a random matching, then it is almost surely true in a random 3-regular graph.  ... 
doaj:1226c8fd2aa74158906b6c8ef4bf4644 fatcat:2qy3meqhljarjaqsdputqpkwui
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