{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10309","name":"Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable","abstract":"Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP\u00a010, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X_1=(V,E_1) and X_2=(V,E_2) and a subset R\\subseteq V\\times V of forbidden pairs whether there is an isomorphism \\pi from X_1 to X_2 such that i^\\pi\\ne j for all (i,j)\\in R. We prove that this problem and several of its generalizations are in fact in \\FPT:\r\n\r\n- The problem of deciding whether there is an isomorphism between two graphs that moves k\u00a0vertices and satisfies Lubiw-style constraints is in\u00a0FPT, with k and the size of R as parameters. The problem remains in FPT even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly\u00a0k vertices is in\u00a0FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017].\r\n\r\n- When the number of moved vertices is unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle.\r\n\r\n- Checking if there is an isomorphism\u00a0\u03c0 between two graphs with complexity\u00a0t is also in\u00a0FPT with\u00a0t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number\u00a0t such that \\pi\u00a0can be expressed as a product of t\u00a0transpositions.\r\n\r\n- We consider a more general problem in which the vertex set of a graph\u00a0X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly\u00a0k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size\u00a04 inside Red. Now, for color classes of size at most\u00a03 inside Red, we show the problem is in\u00a0FPT.\r\n\r\nIn the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most\u00a0k vertices, shown to be in FPT by Schweitzer [ESA 2011].","keywords":["parameterized algorithms","hypergraph isomorphism","mislabeled graphs"],"author":[{"@type":"Person","name":"Arvind, Vikraman","givenName":"Vikraman","familyName":"Arvind"},{"@type":"Person","name":"K\u00f6bler, Johannes","givenName":"Johannes","familyName":"K\u00f6bler"},{"@type":"Person","name":"Kuhnert, Sebastian","givenName":"Sebastian","familyName":"Kuhnert"},{"@type":"Person","name":"Tor\u00e1n, Jacobo","givenName":"Jacobo","familyName":"Tor\u00e1n"}],"position":2,"pageStart":"2:1","pageEnd":"2:13","dateCreated":"2018-03-02","datePublished":"2018-03-02","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arvind, Vikraman","givenName":"Vikraman","familyName":"Arvind"},{"@type":"Person","name":"K\u00f6bler, Johannes","givenName":"Johannes","familyName":"K\u00f6bler"},{"@type":"Person","name":"Kuhnert, Sebastian","givenName":"Sebastian","familyName":"Kuhnert"},{"@type":"Person","name":"Tor\u00e1n, Jacobo","givenName":"Jacobo","familyName":"Tor\u00e1n"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2017.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.4230\/LIPIcs.STACS.2017.7","http:\/\/dx.doi.org\/10.1145\/2897518.2897542","http:\/\/dx.doi.org\/10.1145\/800070.802206","http:\/\/dx.doi.org\/10.1007\/978-1-4612-0731-3","http:\/\/dx.doi.org\/10.1007\/978-1-4612-0515-9","http:\/\/dx.doi.org\/10.1145\/828.1884","http:\/\/dx.doi.org\/10.1109\/SFCS.1980.34","http:\/\/dx.doi.org\/10.1007\/978-1-4612-4478-3_5","http:\/\/dx.doi.org\/10.1016\/S0022-0000(03)00042-4","https:\/\/arxiv.org\/abs\/1607.03918","http:\/\/dx.doi.org\/10.1137\/140999980","http:\/\/dx.doi.org\/10.1137\/0210002","http:\/\/dx.doi.org\/10.1007\/978-3-642-23719-5_32"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6292","volumeNumber":89,"name":"12th International Symposium on Parameterized and Exact Computation (IPEC 2017)","dateCreated":"2018-03-02","datePublished":"2018-03-02","editor":[{"@type":"Person","name":"Lokshtanov, Daniel","givenName":"Daniel","familyName":"Lokshtanov"},{"@type":"Person","name":"Nishimura, Naomi","givenName":"Naomi","familyName":"Nishimura"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10309","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6292"}}}