{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7105","name":"Trichotomy for Integer Linear Systems Based on Their Sign Patterns","abstract":"In this paper, we consider solving the integer linear systems, i.e., \r\ngiven a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b, \r\nwhere m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science. \r\n\r\nFor the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy:\r\n\r\n- ILS_=(r) is linearly solvable, if r < 1, \r\n\r\n- ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and \r\n\r\n- ILS_=(r) is strongly NP-hard, if r > 1. \r\n\r\nThis, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.","keywords":["Integer linear system","Sign pattern","Complexity index","TVPI system","Horn system"],"author":[{"@type":"Person","name":"Kimura, Kei","givenName":"Kei","familyName":"Kimura"},{"@type":"Person","name":"Makino, Kazuhisa","givenName":"Kazuhisa","familyName":"Makino"}],"position":55,"pageStart":613,"pageEnd":623,"dateCreated":"2012-02-24","datePublished":"2012-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kimura, Kei","givenName":"Kei","familyName":"Kimura"},{"@type":"Person","name":"Makino, Kazuhisa","givenName":"Kazuhisa","familyName":"Makino"}],"copyrightYear":"2012","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2012.613","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6217","volumeNumber":14,"name":"29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)","dateCreated":"2012-02-24","datePublished":"2012-02-24","editor":[{"@type":"Person","name":"D\u00fcrr, Christoph","givenName":"Christoph","familyName":"D\u00fcrr"},{"@type":"Person","name":"Wilke, Thomas","givenName":"Thomas","familyName":"Wilke"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7105","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":"#volume6217"}}}