Abstract
We examine the computational complexity of approximately counting the list H-colorings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph, then counting list H-colorings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph, then approximately counting list H-colorings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem that is believed to be of intermediate complexity—it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colorings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (1) it has a natural formulation in terms of hereditary graph classes, and (2) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.
- Jin-Yi Cai, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Mark Jerrum, Daniel Štefankovič, and Eric Vigoda. 2016. #BIS-Hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region. J. Comput. System Sci. 82, 5 (2016), 690--711. DOI:http://dx.doi.org/10.1016/j.jcss.2015.11.009 Google ScholarDigital Library
- Xi Chen, Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Pinyan Lu, Colin McQuillan, and David Richerby. 2015. The complexity of approximating conservative counting CSPs. J. Comput. System Sci. 81, 1 (2015), 311--329. DOI:http://dx.doi.org/10.1016/j.jcss.2014.06.006 Google ScholarDigital Library
- Martin Dyer, Leslie Ann Goldberg, Catherine Greenhill, and Mark Jerrum. 2003. The relative complexity of approximate counting problems. Algorithmica 38, 3 (2003), 471--500. DOI:http://dx.doi.org/10.1007/s00453-003-1073-y Google ScholarCross Ref
- Martin Dyer and Catherine Greenhill. 2000. The complexity of counting graph homomorphisms. Random Structures Algorithms 17, 3--4 (2000), 260--289. DOI:http://dx.doi.org/10.1002/1098-2418(200010/12)17:3/4<260::AID-RSA5>3.3.CO;2-N Google ScholarCross Ref
- Martin Dyer, Mark Jerrum, and Haiko Müller. 2016. On the switch markov chain for perfect matchings. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’16). 1972--1983. DOI:http://dx.doi.org/10.1137/1.9781611974331.ch138 Google ScholarCross Ref
- Tomás Feder and Pavol Hell. 1998. List homomorphisms to reflexive graphs. J. Combinatorial Theory B 72, 2 (1998), 236--250. DOI:http://dx.doi.org/10.1006/jctb.1997.1812 Google ScholarDigital Library
- Tomás Feder, Pavol Hell, and Jing Huang. 1999. List homomorphisms and circular arc graphs. Combinatorica 19, 4 (1999), 487--505. DOI:http://dx.doi.org/10.1007/s004939970003 Google ScholarCross Ref
- Tomás Feder, Pavol Hell, and Jing Huang. 2003. Bi-arc graphs and the complexity of list homomorphisms. J. Graph Theory 42, 1 (2003), 61--80. DOI:http://dx.doi.org/10.1002/jgt.10073 Google ScholarDigital Library
- Andreas Galanis, Leslie Ann Goldberg, and Mark Jerrum. 2016a. Approximately counting H-colourings is #BIS-Hard. SIAM J. Comput. 45, 3 (2016), 680--711. DOI:http://dx.doi.org/10.1137/15M1020551 Google ScholarCross Ref
- Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. 2016b. Inapproximability of the partition function for the antiferromagnetic ising and hard-core models. Combinatorics Probabil. Comput. 25, 4 (2016), 500--559. DOI:http://dx.doi.org/10.1017/S0963548315000401 Google ScholarCross Ref
- T. Gallai. 1967. Transitiv orientierbare graphen. Acta Math. Acad. Sci. Hungar 18 (1967), 25--66. Google ScholarCross Ref
- Leslie Ann Goldberg and Mark Jerrum. 2015. A complexity classification of spin systems with an external field. Proc. Natl. Acad. Sci. 112, 43 (2015), 13161--13166. DOI:http://dx.doi.org/10.1073/pnas.1505664112 Google ScholarCross Ref
- Pavol Hell and Jing Huang. 2004. Interval bigraphs and circular arc graphs. J. Graph Theory 46, 4 (2004), 313--327. DOI:http://dx.doi.org/10.1002/jgt.20006 Google ScholarDigital Library
- Mark Jerrum and Alistair Sinclair. 1993. Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22, 5 (1993), 1087--1116. DOI:http://dx.doi.org/10.1137/0222066 Google ScholarDigital Library
- Steven Kelk. 2003. On the Relative Complexity of Approximately Counting H-colourings. Ph.D. Dissertation. Warwick University.Google Scholar
- Ekkehard G. Köhler. 1999. Graphs Without Asteroidal Triples. Ph.D. Dissertation. Technische Universität Berlin.Google Scholar
- Liang Li, Pinyan Lu, and Yitong Yin. 2012. Approximate counting via correlation decay in spin systems. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12). 922--940. Google ScholarCross Ref
- George B. Mertzios. 2008. A matrix characterization of interval and proper interval graphs. Appl. Math. Lett. 21, 4 (2008), 332--337. DOI:http://dx.doi.org/10.1016/j.aml.2007.04.001 Google ScholarCross Ref
- Fred S. Roberts. 1968. Representations of Indifference Relations. Ph.D. Dissertation. Stanford University, Stanford, CA.Google Scholar
- Fred S. Roberts. 1969. Indifference graphs. In Proof Techniques in Graph Theory (Proc. 2nd Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968). Academic Press, New York, 139--146.Google Scholar
- Sanjeev Saluja, K. V. Subrahmanyam, and Madhukar N. Thakur. 1995. Descriptive complexity of #P functions. J. Comput. Syst. Sci. 50, 3 (1995), 493--505. DOI:http://dx.doi.org/10.1006/jcss.1995.1039 Google ScholarDigital Library
- Allan Sly. 2010. Computational transition at the uniqueness threshold. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS’10). 287--296. DOI:http://dx.doi.org/10.1109/FOCS.2010.34 Google ScholarDigital Library
- Jeremy Spinrad, Andreas Brandstädt, and Lorna Stewart. 1987. Bipartite permutation graphs. Discrete Appl. Math. 18, 3 (1987), 279--292. DOI:http://dx.doi.org/10.1016/0166-218X(87)90064-3 Google ScholarDigital Library
- Leslie G. Valiant and Vijay V. Vazirani. 1986. NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 3 (1986), 85--93. DOI:http://dx.doi.org/10.1016/0304-3975(86)90135-0 Google ScholarDigital Library
- Gerd Wegner. 1967. Eigenschaften der Nerven Homologisch-einfacher Familien im Rn. Ph.D. Dissertation. Universität Göttingen, Göttingen, Germany.Google Scholar
- Dror Weitz. 2006. Counting independent sets up to the tree threshold. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing. 140--149. DOI:http://dx.doi.org/10.1145/1132516.1132538 Google ScholarDigital Library
Index Terms
- A Complexity Trichotomy for Approximately Counting List H-Colorings
Recommendations
The Complexity of Approximately Counting Tree Homomorphisms
We study two computational problems, parameterised by a fixed tree H. #HOMSTO(H) is the problem of counting homomorphisms from an input graph G to H. #WHOMSTO(H) is the problem of counting weighted homomorphisms to H, given an input graph G and a weight ...
Approximately Counting $H$-Colorings is $\#\mathrm{BIS}$-Hard
We consider the problem of counting $H$-colorings from an input graph $G$ to a target graph $H$. We show that if $H$ is any fixed graph without trivial components, then the problem is as hard as the well-known problem $\#\mathrm{BIS}$, which is the problem ...
The complexity of approximately counting stable roommate assignments
We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the k-attribute model, in which the preference lists are determined by dot products of ''preference vectors'' with ''attribute vectors'' and (ii) the ...
Comments