Maximum matchings in sparse random graphs: Karp–Sipser revisited

Aronson, Jonathan; Frieze, Alan; Pittel, Boris

doi:10.1002/(sici)1098-2418(199803)12:2<111::aid-rsa1>3.3.co;2-4

Cited by 3 publications

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“…The result was strengthened later by Aronson, Frieze and Pittel[APF98].Karp and Sipser proved that w.h.p.…”

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confidence: 91%

Linear phase transition in random linear constraint satisfaction problems

Gamarnik

2004

Probab. Theory Relat. Fields

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Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from C also uniformly at random. This procedure is repeated m times independently. We are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition, when n → ∞ and m = cn for a constant c. Namely, there exists a critical value c * such that, when c < c * , the problem is feasible or is asymptotically almost feasible, as n → ∞, but, when c > c * , the "distance" to feasibility is at least a positive constant independent of n. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [Ald92], [Ald01], Aldous and Steele [AS03], Steele [Ste02] and martingale techniques.By exploiting a linear programming duality, our theorem implies the following result in the context of sparse random graphs G(n, cn) on n nodes with cn edges, where edges are equipped with randomly generated weights. Let M(n, c) denote maximum weight matching in G(n, cn). We prove that when c is a constant and n → ∞, the limit lim n→∞ M(n, c)/n, exists, with high probability. We further extend this result to maximum weight b-matchings also in G(n, cn).

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“…The result was strengthened later by Aronson, Frieze and Pittel[APF98].Karp and Sipser proved that w.h.p.…”

mentioning

confidence: 91%

Linear phase transition in random linear constraint satisfaction problems

Gamarnik

2004

Probab. Theory Relat. Fields

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“…Aronson, Frieze and Pittel [6] prove that this small modification yields a matching that is within n 1/5+o (1) of maximum size on random graph instances with edge probability c/n.…”

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confidence: 99%

Randomized Greedy Algorithms for the Maximum Matching Problem with New Analysis

Poloczek

Szegedy

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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It is a long-standing problem to lower bound the performance of randomized greedy algorithms for maximum matching. Aronson, Dyer, Frieze and Suen [1] studied the modified randomized greedy (MRG) algorithm and proved that it approximates the maximum matching within a factor of at least 1 2 + 1/400, 000. They use heavy combinatorial methods in their analysis. We introduce a new technique we call Contrast Analysis, and show a 1 2 + 1/256 performance lower bound for the MRG algorithm. The technique seems to be useful not only for the MRG, but also for other related algorithms.

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Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs

Goldberg

Hollinger

Magdon‐Ismail

2005

Experimental and Efficient Algorithms

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Abstract. This work is motivated by the long-standing open problem of designing a polynomial-time algorithm that with high probability constructs an asymptotically maximum independent set in a random graph. We present the results of an experimental investigation of the comparative performance of several efficient heuristics for constructing maximal independent sets. Among the algorithms that we evaluate are the well known randomized heuristic, the greedy heuristic, and a modification of the latter which breaks ties in a novel way. All algorithms deliver online upper bounds on the size of the maximum independent set for the specific input-graph. In our experiments, we consider random graphs parameterized by the number of vertices n and the average vertex degree d. Our results provide strong experimental evidence in support of the following conjectures:1. for d = c · n (c is a constant), the greedy and random algorithms are asymptotically equivalent; 2. for fixed d, the greedy algorithms are asymptotically superior to the random algorithm; 3. for graphs with d ≤ 3, the approximation ratio of the modified greedy algorithm is asymptotically < 1.005. We also consider random 3-regular graphs, for which non-trivial lower and upper bounds on the size of a maximum independent set are known. Our experiments suggest that the lower bound is asymptotically tight.

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Maximum matchings in sparse random graphs: Karp–Sipser revisited

Cited by 3 publications

References 0 publications

Linear phase transition in random linear constraint satisfaction problems

Linear phase transition in random linear constraint satisfaction problems

Randomized Greedy Algorithms for the Maximum Matching Problem with New Analysis

Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs

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