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Linear phase transition in random linear constraint satisfaction problems
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  • Published: 29 April 2004

Linear phase transition in random linear constraint satisfaction problems

  • David Gamarnik1 

Probability Theory and Related Fields volume 129, pages 410–440 (2004)Cite this article

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Abstract.

Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from 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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Authors and Affiliations

  1. IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA

    David Gamarnik

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  1. David Gamarnik
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Correspondence to David Gamarnik.

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Gamarnik, D. Linear phase transition in random linear constraint satisfaction problems. Probab. Theory Relat. Fields 129, 410–440 (2004). https://doi.org/10.1007/s00440-004-0345-z

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  • Received: 14 May 2003

  • Revised: 17 January 2004

  • Published: 29 April 2004

  • Issue Date: July 2004

  • DOI: https://doi.org/10.1007/s00440-004-0345-z

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Key words or phrases:

  • Random K-SAT
  • Satisfiability Threshold
  • Linear Programming
  • Random Graphs
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