Some Problems on Approximate Counting in Graphs and Matroids

I shall discuss some of the more well known counting problems associated with graphs and matroids. Except in special cases all these problems have no exact counting algorithm which runs in polynomial time unless there is a remarkable collapse of some existing classes. Hence the focus is on obtaining fast algorithms which give good approximations. The problems studied include counting forests, trees and colourings of graphs and bases of matroids.

[1]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.

[2]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[3]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  A. Sinclair.  Randomised algorithms for counting and generating combinatorial structures , 1988 .

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

[6]  Anthony V. Fiacco,et al.  Mathematical programming study 21 , 1985, Mathematical programming.

[7]  Eric Vigoda.  Improved bounds for sampling colorings , 2000 .

[8]  Dominic Welsh,et al.  The Markov Chain of Colourings , 1995, IPCO.

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

[10]  Paul D. Seymour,et al.  Nowhere-zero 6-flows , 1981, J. Comb. Theory, Ser. B.

[11]  Claus-Peter Schnorr,et al.  Optimal Algorithms for Self-Reducible Problems , 1976, ICALP.

[12]  Ravi Kannan,et al.  Markov chains and polynomial time algorithms , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[13]  László Lovász,et al.  Random Walks on Graphs: A Survey , 1993 .

[14]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[15]  Mark Jerrum.  Counting Trees in a Graph is #P-Complete , 1994, Inf. Process. Lett..

[16]  Tomás Feder,et al.  Balanced matroids , 1992, STOC '92.

[17]  A. Sokal,et al.  Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality , 1988 .

[18]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

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

[20]  Mark Jerrum.  APPROXIMATING THE TUTTE POLYNOMIAL , 2007 .

[21]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..

[22]  D. Welsh,et al.  Combinatorial applications of an inequality from statistical mechanics , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  D. Welsh.  Complexity: Knots, Colourings and Counting: Link polynomials and the Tait conjectures , 1993 .

[24]  Samir Khuller,et al.  Planar Graph Coloring is not Self-Reducible, Assuming P != NP , 1991, Theor. Comput. Sci..

[25]  Thomas P. Hayes,et al.  A non-Markovian coupling for randomly sampling colorings , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[26]  G. Grimmett,et al.  Combinatorics, Complexity, and Chance , 2007 .

[27]  D. Welsh,et al.  The computational complexity of matroid properties , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[28]  Alan M. Frieze,et al.  On the Problem of Approximating the Number of Bases of a Matroid , 1994, Inf. Process. Lett..

[29]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[30]  M. Jerrum.  Counting, Sampling and Integrating: Algorithms and Complexity , 2003 .

[31]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[32]  Mark Jerrum,et al.  Fast Uniform Generation of Regular Graphs , 1990, Theor. Comput. Sci..

[33]  Martin E. Dyer,et al.  Random walks, totally unimodular matrices, and a randomised dual simplex algorithm , 1994, IPCO.

[34]  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.

[35]  Eric Vigoda,et al.  A survey on the use of Markov chains to randomly sample colorings , 2006 .

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

[37]  J. D. Annan,et al.  A Randomised Approximation Algorithm for Counting the Number of Forests in Dense Graphs , 1994, Combinatorics, Probability and Computing.

[38]  Bernhard Korte,et al.  Colouring criteria for adjacency on 0–1-polyhedra , 1978 .