Finding a Maximum Matching in a Sparse Random Graph in O(n) Expected Time

Chebolu, Prasad; Frieze, Alan; Melsted, Páll

doi:10.1007/978-3-540-70575-8_14

Cited by 18 publications

(43 citation statements)

References 8 publications

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“…Thus, it is of great interest to find specific graphs for which the maximum matching problem can be solved exactly [3]. In the past decades, the problems related maximum matchings have attracted considerable attention from the community of mathematics and theoretical computer science [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

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Maximum matchings in scale-free networks with identical degree distribution

Zhang

2017

Theoretical Computer Science

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The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size of maximum matchings in scale-free networks, where the size of maximum matchings is small and a perfect matching often does not exist. In this paper, we study analytically the maximum matchings in two scale-free networks with identical degree sequence, and show that the first network has no perfect matchings, while the second one has many. For the first network, we determine explicitly the size of maximum matchings, and provide an exact recursive solution for the number of maximum matchings. For the second one, we design an orientation and prove that it is Pfaffian, based on which we derive a closed-form expression for the number of perfect matchings. Moreover, we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended Sierpiński graph with the same average degree as both studied scale-free networks. Our results indicate that power-law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale-free networks.

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Section: Introductionmentioning

confidence: 99%

“…A vast majority of previous works about maximum matchings focused on regular graphs or random graphs [9,21,24,27], which cannot well describe realistic networks. Extensive empirical works [29] indicated that most real networks exhibit the prominent scale-free property [30], with their degree distribution following a power law form.…”

Section: Introductionmentioning

confidence: 99%

Maximum matchings in scale-free networks with identical degree distribution

Zhang

2017

Theoretical Computer Science

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“…It has been shown that these trees will meet if their lengths are of size O(log n). In [7], Chebolu et al take the output of the jump-start routine KSM (discussed in the next section), try to augment it by searching paths only between a subset of the unmatched vertices, and then use the algorithm of Bast et al if these searches have not found a maximum matching. It has been shown that the overall algorithm runs in linear expected time for sparse random graphs.…”

Section: Other Approachesmentioning

confidence: 99%

“…This version, referred to as KSM, is called the Karp-Sipser heuristic in the literature, and its performance is analysed by Aronson et al [2] and Chebolu et al [7]. In our implementation of Algorithm 2, we maintain the degrees dynamically, i.e., when a matching (u, v) occurs we decrease the degrees of vertices adjacent to u or v by 1.…”

Section: Karp-sipser Greedy Matching (Ksm)mentioning

confidence: 99%

“…The first heuristic, SGM, was previously used with some of the matching algorithms described in this paper [12,23,29,31]. The KSM heuristic was experimented and analysed extensively on its own and used as a part of some randomized algorithms [21,7]. The MDM heuristic is one of the best, but its effect on the matching algorithms have not been thoroughly investigated.…”

Section: Performances Of the Greedy Heuristicsmentioning

confidence: 99%

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Design, implementation, and analysis of maximum transversal algorithms

Duff¹,

Kaya

Uçcar

2011

ACM Trans. Math. Softw.

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We report on careful implementations of seven algorithms for solving the problem of finding a maximum transversal of a sparse matrix. We analyse the algorithms and discuss the design choices. To the best of our knowledge, this is the most comprehensive comparison of maximum transversal algorithms based on augmenting paths. Previous papers with the same objective either do not have all the algorithms discussed in this paper or they used non-uniform implementations from different researchers. We use a common base to implement all of the algorithms and compare their relative performance on a wide range of graphs and matrices. We systematize, develop and use several ideas for enhancing performance. One of these ideas improves the performance of one of the existing algorithms in most cases, sometimes significantly. So much so that we use this as the eighth algorithm in comparisons.

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On a greedy 2‐matching algorithm and Hamilton cycles in random graphs with minimum degree at least three

Frieze

2012

Random Struct Algorithms

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We describe and analyse a simple greedy algorithm 2greedy that finds a good 2-matching M in the random graph G = G δ≥3 n,cn when c ≥ 10. A 2-matching is a spanning subgraph of maximum degree two and G is drawn uniformly from graphs with vertex set [n], cn edges and minimum degree at least three. By good we mean that M has O(log n) components. We then use this 2-matching to build a Hamilton cycle in O(n 1.5+o(1) ) time w.h.p..

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Finding a Maximum Matching in a Sparse Random Graph in O(n) Expected Time

Abstract: We present a linear expected time algorithm for finding maximum cardinality matchings in sparse random graphs. This is optimal and improves on previous results by a logarithmic factor.

Cited by 18 publications

References 8 publications

Maximum matchings in scale-free networks with identical degree distribution

Maximum matchings in scale-free networks with identical degree distribution

Design, implementation, and analysis of maximum transversal algorithms

On a greedy 2‐matching algorithm and Hamilton cycles in random graphs with minimum degree at least three

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