On Markov Chains for Independent Sets

Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. In 1997, Luby and Vigoda described a rapidly mixing Markov chain for independent sets, which we refer to as the Luby?Vigoda chain. A new rapidly mixing Markov chain for independent sets is defined in this paper. Using path coupling, we obtain a polynomial upper bound for the mixing time of the new chain for a certain range of values of the parameter ?. This range is wider than the range for which the mixing time of the Luby?Vigoda chain is known to be polynomially bounded. Moreover, the upper bound on the mixing time of the new chain is always smaller than the best known upper bound on the mixing time of the Luby?Vigoda chain for larger values of ? (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov chains for independent sets, a simple insert/delete chain, and the new chain, to show that the insert/delete chain is rapidly mixing for a wider range of values of ? than was previously known.

[1]  J. A. Fill.  An interruptible algorithm for perfect sampling via Markov chains , 1998 .

[2]  Dan Roth,et al.  On the Hardness of Approximate Reasoning , 1993, IJCAI.

[3]  Martin E. Dyer,et al.  On counting independent sets in sparse graphs , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[4]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[5]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[6]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[7]  James Allen Fill,et al.  An interruptible algorithm for perfect sampling via Markov chains , 1997, STOC '97.

[8]  Eric Vigoda,et al.  Fast convergence of the Glauber dynamics for sampling independent sets , 1999, Random Struct. Algorithms.

[9]  Dana Randall,et al.  Markov Chain Algorithms for Planar Lattice Structures (Extended Abstract). , 1995, FOCS 1995.

[10]  David S. Johnson,et al.  Computers and Inrracrobiliry: A Guide ro the Theory of NP-Completeness , 1979 .

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  Dana Randall,et al.  Analyzing Glauber Dynamics by Comparison of Markov Chains , 1998, LATIN.

[13]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[14]  Fan Chung Graham,et al.  Random walks on generating sets for finite groups , 1996, Electron. J. Comb..

[15]  J. Propp,et al.  Exact sampling with coupled Markov chains and applications to statistical mechanics , 1996 .

[16]  S. Caracciolo,et al.  Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints , 1990 .

[17]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.

[18]  Martin Dyer,et al.  A more rapidly mixing Markov chain for graph colorings , 1998 .

[19]  Martin Dyer,et al.  Some #P-completeness Proofs for Colourings and Independent Sets , 1997 .

[20]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[21]  Catherine S. Greenhill.  The complexity of counting colourings and independent sets in sparse graphs and hypergraphs , 2000, computational complexity.

[22]  Martin E. Dyer,et al.  A more rapidly mixing Markov chain for graph colorings , 1998, Random Struct. Algorithms.

[23]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[24]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[25]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

[26]  N. S. Barnett,et al.  Private communication , 1969 .

[27]  Martin E. Dyer,et al.  On Approximately Counting Colorings of Small Degree Graphs , 1999, SIAM J. Comput..

[28]  M. Luby,et al.  Fast convergence of the Glauber dynamics for sampling independent sets , 1999 .

[29]  Martin E. Dyer,et al.  Path coupling: A technique for proving rapid mixing in Markov chains , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[30]  Salil P. Vadhan.  The Complexity of Counting , 1995 .

[31]  A. Sokal,et al.  Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem , 1996, cond-mat/9603068.

[32]  Martin E. Dyer,et al.  Graph orientations with no sink and an approximation for a hard case of #SAT , 1997, SODA '97.

[33]  P. Diaconis,et al.  COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS , 1993 .

[34]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[35]  Martin E. Dyer,et al.  On Counting Independent Sets in Sparse Graphs , 2002, SIAM J. Comput..