{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8372","name":"Quantum Query Complexity of Subgraph Isomorphism and Homomorphism","abstract":"Let H be a (non-empty) graph on n vertices, possibly containing isolated vertices. Let f_H(G) = 1 iff the input graph G on n vertices contains H as a (not necessarily induced) subgraph. Let alpha_H denote the cardinality of a maximum independent set of H. In this paper we show: Q(f_H) = Omega( sqrt{alpha_H * n}), where Q(f_H) denotes the quantum query complexity of f_H.\r\n\r\nAs a consequence we obtain lower bounds for Q(f_H) in terms of several other parameters of H such as the average degree, minimum vertex cover, chromatic number, and the critical probability.\r\n\r\nWe also use the above bound to show that Q(f_H) = Omega(n^{3\/4}) for any H, improving on the previously best known bound of Omega(n^{2\/3}) [M. Santha\/A. Chi-Chih Yao, unpublished manuscript]. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our Omega(n^{3\/4}) bound for Q(f_H) matches the square root of the current best known bound for the randomized query complexity of f_H, which is Omega(n^{3\/2}) due to Groger. Interestingly, the randomized bound of Omega(alpha_H * n) for f_H still remains open.\r\n\r\nWe also study the Subgraph Homomorphism Problem, denoted by f_{[H]}, and show that Q(f_{[H]}) = Omega(n).\r\n\r\nFinally we extend our results to the 3-uniform hypergraphs. In particular, we show an Omega(n^{4\/5}) bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known Omega(n^{3\/4}) bound. For the Subgraph Homomorphism, we obtain an Omega(n^{3\/2}) bound for the same.","keywords":["quantum query complexity","subgraph isomorphism","monotone graph properties"],"author":[{"@type":"Person","name":"Kulkarni, Raghav","givenName":"Raghav","familyName":"Kulkarni"},{"@type":"Person","name":"Podder, Supartha","givenName":"Supartha","familyName":"Podder"}],"position":48,"pageStart":"48:1","pageEnd":"48:13","dateCreated":"2016-02-16","datePublished":"2016-02-16","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kulkarni, Raghav","givenName":"Raghav","familyName":"Kulkarni"},{"@type":"Person","name":"Podder, Supartha","givenName":"Supartha","familyName":"Podder"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2016.48","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6250","volumeNumber":47,"name":"33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)","dateCreated":"2016-02-16","datePublished":"2016-02-16","editor":[{"@type":"Person","name":"Ollinger, Nicolas","givenName":"Nicolas","familyName":"Ollinger"},{"@type":"Person","name":"Vollmer, Heribert","givenName":"Heribert","familyName":"Vollmer"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8372","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":"#volume6250"}}}