Abstract
Let G be a graph that contains an induced subgraph H. A retraction from G to H is a homomorphism from G to H that is the identity function on H. Retractions are very well studied: Given H, the complexity of deciding whether there is a retraction from an input graph G to H is completely classified, in the sense that it is known for which H this problem is tractable (assuming P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #P). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs without short cycles. The result is as follows: (1) Approximately counting retractions to a graph H of girth at least 5 is in FP if every connected component of H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if every component is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph—a problem that is complete in the approximate counting complexity class RH Π 1. (3) Finally, if none of these hold, then approximately counting retractions to H is equivalent to approximately counting the satisfying assignments of a Boolean formula.
Our second contribution is to locate the retraction counting problem for each H in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms—whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems.
- M. Bíró, M. Hujter, and Zs. Tuza. 1992. Precoloring extension. I. Interval graphs. Discrete Math. 100, 1--3 (1992), 267--279. DOI:http://dx.doi.org/10.1016/0012-365X(92)90646-W Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regulären Graphs,” Part I.Google Scholar
Digital Library
- Manuel Bodirsky, Jan Kára, and Barnaby Martin. 2012. The complexity of surjective homomorphism problems—A survey. Discrete Appl. Math. 160, 12 (2012), 1680--1690. DOI:http://dx.doi.org/10.1016/j.dam.2012.03.029Google Scholar
Digital Library
- Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. 2006. Counting graph homomorphisms. In Topics in Discrete Mathematics. Algorithms Combin., Vol. 26. Springer, Berlin, 315--371. DOI:http://dx.doi.org/10.1007/3-540-33700-8_18Google Scholar
- Andrei A. Bulatov. 2017. A dichotomy theorem for nonuniform CSPs. In 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS’17). IEEE Computer Society, Los Alamitos, CA, 319--330.Google Scholar
Cross Ref
- Martin Dyer, Leslie Ann Goldberg, Catherine Greenhill, and Mark Jerrum. 2004. The relative complexity of approximate counting problems. Algorithmica 38, 3 (2004), 471--500. DOI:http://dx.doi.org/10.1007/s00453-003-1073-yGoogle Scholar
Cross Ref
- Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum. 2004. Counting and sampling H-colourings. Inform. Comput. 189, 1 (2004), 1--16. DOI:http://dx.doi.org/10.1016/j.ic.2003.09.001Google Scholar
Digital Library
- Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum. 2010. An approximation trichotomy for Boolean ♯CSP. J. Comput. System Sci. 76, 3--4 (2010), 267--277. DOI:http://dx.doi.org/10.1016/j.jcss.2009.08.003Google Scholar
Digital Library
- Martin Dyer and Catherine Greenhill. 1999. Random walks on combinatorial objects. In Surveys in Combinatorics, 1999 (Canterbury) (London Math. Soc. Lecture Note Ser.), Vol. 267. Cambridge Univ. Press, Cambridge, 101--136.Google Scholar
- Martin E. Dyer and Catherine S. Greenhill. 2000. The complexity of counting graph homomorphisms. Random Struct. Algorithms 17, 3--4 (2000), 260--289.Google Scholar
Digital Library
- Tomas Feder and Pavol Hell. 1998. List homomorphisms to reflexive graphs. J. Combin. Theory Ser. B 72, 2 (1998), 236--250. DOI:http://dx.doi.org/10.1006/jctb.1997.1812Google Scholar
Digital Library
- Tomas 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/s004939970003Google Scholar
- Tomas Feder, Pavol Hell, and Jing Huang. 2009. Extension problems with degree bounds. Discrete Appl. Math. 157, 7 (2009), 1592--1599. DOI:http://dx.doi.org/10.1016/j.dam.2008.04.006Google Scholar
Digital Library
- Tomás Feder, Pavol Hell, Peter Jonsson, Andrei Krokhin, and Gustav Nordh. 2010. Retractions to pseudoforests. SIAM J. Discrete Math. 24, 1 (2010), 101--112. DOI:http://dx.doi.org/10.1137/080738866Google Scholar
Digital Library
- Jacob Focke, Leslie Ann Goldberg, and Stanislav Živný. 2017. The complexity of counting surjective homomorphisms and compactions. CoRR abs/1706.08786 (2017). Retrieved from http://arxiv.org/abs/1706.08786. A preliminary version of this work appeared in the Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1772-1781.Google Scholar
- Andreas Galanis, Leslie Ann Goldberg, and Mark Jerrum. 2016. Approximately counting H-colorings is #BIS-hard. SIAM J. Comput. 45, 3 (2016), 680--711. DOI:http://dx.doi.org/10.1137/15M1020551Google Scholar
Cross Ref
- Andreas Galanis, Leslie Ann Goldberg, and Mark Jerrum. 2017. A complexity trichotomy for approximately counting list H-colorings. ACM Trans. Comput. Theory 9, 2 (2017), Art. 9, 22. DOI:http://dx.doi.org/10.1145/3037381Google Scholar
Digital Library
- Andreas Galanis, Daniel Štefankovič, Eric Vigoda, and Linji Yang. 2016. Ferromagnetic Potts model: Refined #BIS-hardness and related results. SIAM J. Comput. 45, 6 (2016), 2004--2065. DOI:http://dx.doi.org/10.1137/140997580Google Scholar
Cross Ref
- Leslie Ann Goldberg and Mark Jerrum. 2012. Approximating the partition function of the ferromagnetic Potts model. J. ACM 59, 5 (2012), Art. 25, 31. DOI:http://dx.doi.org/10.1145/2371656.2371660Google Scholar
Digital Library
- Leslie Ann Goldberg and Mark Jerrum. 2014. The complexity of approximately counting tree homomorphisms. ACM Trans. Comput. Theory 6, 2 (2014), Art. 8, 31. DOI:http://dx.doi.org/10.1145/2600917Google Scholar
Digital Library
- Leslie Ann Goldberg, Steven Kelk, and Mike Paterson. 2004. The complexity of choosing an H-coloring (nearly) uniformly at random. SIAM J. Comput. 33, 2 (2004), 416--432. DOI:http://dx.doi.org/10.1137/S0097539702408363Google Scholar
Digital Library
- Petr A. Golovach, Matthew Johnson, Barnaby Martin, Daniël Paulusma, and Anthony Stewart. 2017. Surjective H-colouring: New hardness results. In Unveiling Dynamics and Complexity. Lecture Notes in Comput. Sci., Vol. 10307. Springer, Cham, 270--281.Google Scholar
- Petr A. Golovach, Bernard Lidický, Barnaby Martin, and Daniël Paulusma. 2012a. Finding vertex-surjective graph homomorphisms. Acta Inform. 49, 6 (2012), 381--394. DOI:http://dx.doi.org/10.1007/s00236-012-0164-0Google Scholar
Digital Library
- Petr A. Golovach, Daniël Paulusma, and Jian Song. 2012b. Computing vertex-surjective homomorphisms to partially reflexive trees. Theoret. Comput. Sci. 457 (2012), 86--100. DOI:http://dx.doi.org/10.1016/j.tcs.2012.06.039Google Scholar
Digital Library
- Frank Harary and Allen Schwenk. 1971. Trees with Hamiltonian square. Mathematika 18 (1971), 138--140. DOI:http://dx.doi.org/10.1112/S0025579300008494Google Scholar
Cross Ref
- Pavol Hell. 1973. Retractions des graphes. ProQuest LLC, Ann Arbor, MI. Retrieved from http://ezproxy-prd.bodleian.ox.ac.uk:2175/openurl?url_ver=Z39.88-2004&rft_val_fmt==info:ofi/fmt:kev:mtx:dissertation&res_dat==xri:pqdiss&rft_dat==xri:pqdiss:0289365. Ph.D. thesis, Universite de Montreal.Google Scholar
- Pavol Hell. 1974. Absolute retracts in graphs. Lecture Notes in Math. Vol. 406, pp. 291-301.Google Scholar
Cross Ref
- Pavol Hell and Jaroslav Nešetřil. 1990. On the complexity of H-coloring. J. Combin. Theory Ser. B 48, 1 (1990), 92--110. DOI:http://dx.doi.org/10.1016/0095-8956(90)90132-JGoogle Scholar
Digital Library
- Pavol Hell and Jaroslav Nešetřil. 2004a. Counting list homomorphisms for graphs with bounded degrees. In Graphs, Morphisms and Statistical Physics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 63. Amer. Math. Soc., Providence, RI, 105--112. DOI:http://dx.doi.org/10.1093/acprof:oso/9780198528173.001.0001Google Scholar
Cross Ref
- Pavol Hell and Jaroslav Nešetřil. 2004b. Graphs and Homomorphisms. Oxford Lecture Series in Mathematics and its Applications, Vol. 28. Oxford University Press, Oxford. DOI:http://dx.doi.org/10.1093/acprof:oso/9780198528173.001.0001Google Scholar
- Pavol Hell and Jaroslav Nešetřil. 2008. Colouring, constraint satisfaction, and complexity. Comput. Sci. Rev. 2, 3 (2008), 143--163. DOI:http://dx.doi.org/10.1016/j.cosrev.2008.10.003Google Scholar
Digital Library
- Pavol Hell and Ivan Rival. 1987. Absolute retracts and varieties of reflexive graphs. Canad. J. Math. 39, 3 (1987), 544--567. DOI:http://dx.doi.org/10.4153/CJM-1987-025-1Google Scholar
Cross Ref
- Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. 1986. Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43, 2--3 (1986), 169--188. DOI:http://dx.doi.org/10.1016/0304-3975(86)90174-XGoogle Scholar
Digital 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
- Jan Kratochvíl and András Sebő. 1997. Coloring precolored perfect graphs. J. Graph Theory 25, 3 (1997), 207--215. DOI:http://dx.doi.org/10.1002/(SICI)1097-0118(199707)25:3<207::AID-JGT4>3.0.CO;2-PGoogle Scholar
Digital Library
- Barnaby Martin and Daniël Paulusma. 2015. The computational complexity of disconnected cut and 2K2-partition. J. Combin. Theory Ser. B 111 (2015), 17--37. DOI:http://dx.doi.org/10.1016/j.jctb.2014.09.002Google Scholar
Digital Library
- Dániel Marx. 2006. Parameterized coloring problems on chordal graphs. Theoret. Comput. Sci. 351, 3 (2006), 407--424. DOI:http://dx.doi.org/10.1016/j.tcs.2005.10.008Google Scholar
Digital Library
- Michael Mitzenmacher and Eli Upfal. 2017. Probability and Computing (2nd ed.). Cambridge University Press, Cambridge.Google Scholar
- Erwin Pesch. 1988. Retracts of Graphs. Mathematical Systems in Economics, Vol. 110. Athenäum Verlag GmbH, Frankfurt am Main.Google Scholar
- R. B. Potts. 1952. Some generalized order-disorder transformations. Proc. Cambridge Philos. Soc. 48 (1952), 106--109.Google Scholar
Cross Ref
- Wolfgang M. Schmidt. 1991. Diophantine Approximations and Diophantine Equations. Lecture Notes in Mathematics, Vol. 1467. Springer-Verlag, Berlin. viii+217 pages. DOI:http://dx.doi.org/10.1007/BFb0098246Google Scholar
- Claus-Peter Schnorr. 1976. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages and Programming, S. Michaelson and Robin Milner (Eds.). Edinburgh University Press, 322--337.Google Scholar
- Zsolt Tuza. 1997. Graph colorings with local constraints—A survey. Discuss. Math. Graph Theory 17, 2 (1997), 161--228. DOI:http://dx.doi.org/10.7151/dmgt.1049Google Scholar
Cross Ref
- Narayan Vikas. 2002. Computational complexity of compaction to reflexive cycles. SIAM J. Comput. 32, 1 (2002/03), 253--280. DOI:http://dx.doi.org/10.1137/S0097539701383522Google Scholar
Digital Library
- Narayan Vikas. 2004a. Compaction, retraction, and constraint satisfaction. SIAM J. Comput. 33, 4 (2004), 761--782. DOI:http://dx.doi.org/10.1137/S0097539701397801Google Scholar
Digital Library
- Narayan Vikas. 2004b. Computational complexity of compaction to irreflexive cycles. J. Comput. Syst. Sci. 68, 3 (2004), 473--496. DOI:http://dx.doi.org/10.1016/S0022-0000(03)00034-5Google Scholar
Digital Library
- Narayan Vikas. 2005. A complete and equal computational complexity classification of compaction and retraction to all graphs with at most four vertices and some general results. J. Comput. Syst. Sci. 71, 4 (2005), 406--439. DOI:http://dx.doi.org/10.1016/j.jcss.2004.07.003Google Scholar
Digital Library
- Narayan Vikas. 2013. Algorithms for partition of some class of graphs under compaction and vertex-compaction. Algorithmica 67, 2 (2013), 180--206. DOI:http://dx.doi.org/10.1007/s00453-012-9720-9Google Scholar
Cross Ref
- Narayan Vikas. 2017. Computational complexity of graph partition under vertex-compaction to an irreflexive hexagon. In Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science. LIPIcs. Leibniz Int. Proc. Inform., Vol. 83. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, Art. No. 69, 14.Google Scholar
- Narayan Vikas. 2018. Computational complexity relationship between compaction, vertex-compaction, and retraction. In Combinatorial Algorithms. Lecture Notes in Computer Science, Vol. 10765. Springer, Cham, 154--166.Google Scholar
- Benjamin Widom and John S. Rowlinson. 1970. New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 4 (1970), 1670--1684. DOI:http://dx.doi.org/10.1063/1.167\3203Google Scholar
Cross Ref
- Dmitriy Zhuk. 2017. A proof of CSP dichotomy conjecture. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS’17). IEEE Computer Society, Los Alamitos, CA, 331--342.Google Scholar
Cross Ref
Index Terms
The Complexity of Approximately Counting Retractions
Recommendations
The Complexity of Counting Surjective Homomorphisms and Compactions
A homomorphism from a graph $G$ to a graph $H$ is a function from the vertices of $G$ to the vertices of $H$ that preserves edges. A homomorphism is surjective if it uses all of the vertices of $H$, and it is a compaction if it uses all of the vertices of ...
The Complexity of Approximately Counting Retractions to Square-free Graphs
A retraction is a homomorphism from a graph G to an induced subgraph H of G that is the identity on H. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting ...
A Complexity Trichotomy for Approximately Counting List H-Colorings
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 ...






Comments