Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T13:55:13.476Z Has data issue: false hasContentIssue false
  • English
  • Français

A Graph Polynomial for Independent Sets of Bipartite Graphs

Published online by Cambridge University Press:  10 July 2012

Q. GE  and
D. ŠTEFANKOVIČ
Q. GE
Affiliation:
Department of Computer Science, University of Rochester, Rochester, NY 14627–0226, USA (e-mail: [email protected] and [email protected])
D. ŠTEFANKOVIČ
Affiliation:
Department of Computer Science, University of Rochester, Rochester, NY 14627–0226, USA (e-mail: [email protected] and [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).

We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.

Keywords

Primary 05C31Secondary 03D15
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 5 , September 2012 , pp. 695 - 714
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arratia, R., Bollobás, B. and Sorkin, G. B. (2004) A two-variable interlace polynomial. Combinatorica 24 567–584.Google Scholar
[2]Bach, E. and Shallit, J. (1996) Algorithmic Number Theory, Vol. 1, Efficient Algorithms, Foundations of Computing Series, MIT Press.Google Scholar
[3]Baxter, R. J. (1989) Exactly Solved Models in Statistical Mechanics, Academic Press. Reprint of the 1982 original.Google Scholar
[4]Bläser, M. and Hoffmann, C. (2008) On the complexity of the interlace polynomial. In STACS (Albers, S. and Weil, P., eds), Vol. 1 of LIPIcs, Leibniz-Zentrum für Informatik, pp. 97–108.Google Scholar
[5]Bläser, M. and Hoffmann, C. (2011) Fast evaluation of interlace polynomials on graphs of bounded treewidth. Algorithmica 61 3–35.Google Scholar
[6]Bordewich, M. and Kang, R. J. (2011) Rapid mixing of subset Glauber dynamics on graphs of bounded tree-width. In ICALP (1), pp. 533–544.Google Scholar
[7]Courcelle, B. (2008) A multivariate interlace polynomial and its computation for graphs of bounded clique-width. Electron. J. Combin. 15 #R69.Google Scholar
[8]Courcelle, B., Makowsky, J. A. and Rotics, U. (2001) On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math. 108 23–52.CrossRefGoogle Scholar
[9]Dyer, M. E., Frieze, A. M. and Jerrum, M. R. (2002) On counting independent sets in sparse graphs. SIAM J. Comput. 31 1527–1541.CrossRefGoogle Scholar
[10]Dyer, M. E., Goldberg, L. A., Greenhill, C. and Jerrum, M. R. (2004) The relative complexity of approximate counting problems. Algorithmica 38 471–500.Google Scholar
[11]Dyer, M. E., Goldberg, L. A. and Jerrum, M. R. (2010) An approximation trichotomy for boolean #CSP. Journal of Computer and System Sciences 76 267–277.CrossRefGoogle Scholar
[12]Dyer, M. E. and Greenhill, C. (2000) On Markov chains for independent sets. J. Algorithms 35 17–49.Google Scholar
[13]Ellis-Monaghan, J. and Merino, C. (2011) Graph polynomials and their applications I: The Tutte polynomial. Structural Analysis of Complex Networks (Dehmer, M., ed.), Birkhuser Boston, pp. 219–255.Google Scholar
[14]Ellis-Monaghan, J. and Merino, C. (2011) Graph polynomials and their applications II: Interrelations and interpretations. Structural Analysis of Complex Networks (Dehmer, M., ed.), Birkhuser Boston, pp. 257–292.Google Scholar
[15]Frandsen, G. S. and Frandsen, P. F. (2006) Dynamic matrix rank. In Automata, Languages and Programming, Part I, Vol. 4051 of Lecture Notes in Computer Science, Springer, pp. 395–406.Google Scholar
[16]Ge, Q. and Štefankovič, D. (2010) A graph polynomial for independent sets of bipartite graphs. In FSTTCS, pp. 240–250.Google Scholar
[17]Goldberg, L. A. and Jerrum, M. R. (2007) The complexity of ferromagnetic Ising with local fields. Combin. Probab. Comput. 16 43–61.CrossRefGoogle Scholar
[18]Goldberg, L. A. and Jerrum, M. R. (2010) Personal communication.Google Scholar
[19]Goldberg, L. A., Jerrum, M. R. and Paterson, M. (2003) The computational complexity of two-state spin systems. Random Struct. Alg. 23 133–154.CrossRefGoogle Scholar
[20]Jaeger, F., Vertigan, D. and Welsh, D. J. A. (1990) On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108 35–53.Google Scholar
[21]Jerrum, M. R. and Sinclair, A. (1993) Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22 1087–1116.Google Scholar
[22]Jerrum, M. R., Valiant, L. G. and Vazirani, V. V. (1986) Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43 169–188.CrossRefGoogle Scholar
[23]Luby, M. and Vigoda, E. (1999) Fast convergence of the Glauber dynamics for sampling independent sets. Random Struct. Alg. 15 229–241.Google Scholar
[24]Makowsky, J. A. (2008) From a zoo to a zoology: Towards a general theory of graph polynomials. Theory Comput. Syst. 43 542–562.Google Scholar
[25]Moree, P. and Stevenhagen, P. (2000) A two-variable Artin conjecture. J. Number Theory 85 291–304.Google Scholar
[26]Moree, P. and Stevenhagen, P. (2001) Prime divisors of the Lagarias sequence. J. Théor. Nombres Bordeaux 13 241–251.Google Scholar
[27]Provan, J. S. and Ball, M. O. (1983) The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12 777–788.CrossRefGoogle Scholar
[28]Sly, A. (2010) Computational transition at the uniqueness threshold. In Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296.CrossRefGoogle Scholar
[29]Stevenhagen, P. (2000) Prime densities for second order torsion sequences. Preprint.Google Scholar
[30]Tutte, W. T. (1947) A ring in graph theory. Proc. Cambridge Philos. Soc. 43 26–40.Google Scholar
[31]Tutte, W. T. (1954) A contribution to the theory of chromatic polynomials. Canadian J. Math. 6 80–91.CrossRefGoogle Scholar
[32]Vadhan, S. P. (2001) The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31 398–427.Google Scholar
[33]Weitz, D. (2006) Counting independent sets up to the tree threshold. In STOC'06: Proc. 38th Annual ACM Symposium on Theory of Computing, ACM, pp. 140–149.Google Scholar
[34]Welsh, D. J. A. (1993) Complexity: Knots, Colourings and Counting, Vol. 186 of London Mathematical Society Lecture Note Series, Cambridge University Press.CrossRefGoogle Scholar
[35]Xia, M., Zhang, P. and Zhao, W. (2007) Computational complexity of counting problems on 3-regular planar graphs. Theoret. Comput. Sci. 384 111–125.CrossRefGoogle Scholar