{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T02:11:56Z","timestamp":1787364716544,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["34828"],"award-info":[{"award-number":["34828"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>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 of (approximately) counting independent sets in a bipartite graph. $\\#\\mathrm{BIS}$ is a complete problem in an important complexity class for approximate counting, and is believed not to have a fully polynomial randomized approximation scheme (FPRAS). If this is so, then our result shows that for every graph $H$ without trivial components, the $H$-coloring counting problem has no FPRAS. This problem was studied a decade ago by Goldberg, Kelk, and Paterson. They were able to show that approximately sampling $H$-colorings is $\\#\\mathrm{BIS}$-hard, but it was not known how to get the result for approximate counting. Our solution builds on nonconstructive ideas using the work of Lov\u00e1sz.<\/jats:p>","DOI":"10.1137\/15m1020551","type":"journal-article","created":{"date-parts":[[2016,5,19]],"date-time":"2016-05-19T21:49:04Z","timestamp":1463694544000},"page":"680-711","source":"Crossref","is-referenced-by-count":10,"title":["Approximately Counting $H$-Colorings is $\\#\\mathrm{BIS}$-Hard"],"prefix":"10.1137","volume":"45","author":[{"given":"Andreas","family":"Galanis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leslie Ann","family":"Goldberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Jerrum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,5,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-003-1073-y"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2003.09.001"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200010\/12)17:3\/4<260::AID-RSA5>3.0.CO;2-W"},{"key":"atypb4","first-page":"240","author":"Ge Q.","year":"2010","journal-title":"Germany"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1017\/S096354830600767X"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1214\/ECP.v17-1712"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702408363"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(90)90132-J"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"P. Hell and J. Ne\u0161et\u0159il,\n                      Graphs and Homomorphisms\n                      , Oxford University Press, Oxford, 2004.","DOI":"10.1093\/acprof:oso\/9780198528173.001.0001"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(86)90174-X"},{"key":"atypb11","unstructured":"S. Kelk,\n                      On the Relative Complexity of Approximately Counting H-Colourings\n                      , Ph.D. Thesis, University of Warwick, Coventry, England, 2004."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02280291"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-007-0131-9"},{"key":"atypb14","unstructured":"W. M. Schmidt,\n                      Diophantine Approximations and Diophantine Equations\n                      , Lecture Notes in Math. 1467, Springer Verlag, Berlin, 2000."}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1020551","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:08:27Z","timestamp":1787339307000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1020551"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["10.1137\/15M1020551"],"URL":"https:\/\/doi.org\/10.1137\/15m1020551","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1]]}}}