{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8974","name":"A Preliminary Investigation of Satisfiability Problems Not Harder than 1-in-3-SAT","abstract":"The parameterized satisfiability problem over a set of Boolean\r\nrelations Gamma (SAT(Gamma)) is the problem of determining\r\nwhether a conjunctive formula over Gamma has at least one\r\nmodel. Due to Schaefer's dichotomy theorem the computational\r\ncomplexity of SAT(Gamma), modulo polynomial-time reductions, has\r\nbeen completely determined: SAT(Gamma) is always either tractable\r\nor NP-complete. More recently, the problem of studying the\r\nrelationship between the complexity of the NP-complete cases of\r\nSAT(Gamma) with restricted notions of reductions has attracted\r\nattention. For example, Impagliazzo et al. studied the complexity of\r\nk-SAT and proved that the worst-case time complexity increases\r\ninfinitely often for larger values of k, unless 3-SAT is solvable in\r\nsubexponential time. In a similar line of research Jonsson et al.\r\nstudied the complexity of SAT(Gamma) with algebraic tools borrowed\r\nfrom clone theory and proved that there exists an NP-complete problem\r\nSAT(R^{neq,neq,neq,01}_{1\/3}) such that there cannot exist any NP-complete SAT(Gamma) problem with strictly lower worst-case time complexity: the easiest NP-complete SAT(Gamma) problem. In this paper\r\nwe are interested in classifying the NP-complete SAT(Gamma)\r\nproblems whose worst-case time complexity is lower than 1-in-3-SAT but\r\nhigher than the easiest problem SAT(R^{neq,neq,neq,01}_{1\/3}). Recently it was conjectured that there only exists three satisfiability problems of this form. We prove that this conjecture does not hold and that there is an infinite number of such SAT(Gamma) problems. In the process we determine several algebraic properties of 1-in-3-SAT and related problems, which could be of independent interest for constructing exponential-time algorithms.","keywords":["clone theory","universal algebra","satisfiability problems"],"author":[{"@type":"Person","name":"Lagerkvist, Victor","givenName":"Victor","familyName":"Lagerkvist"},{"@type":"Person","name":"Roy, Biman","givenName":"Biman","familyName":"Roy"}],"position":64,"pageStart":"64:1","pageEnd":"64:14","dateCreated":"2016-08-19","datePublished":"2016-08-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Lagerkvist, Victor","givenName":"Victor","familyName":"Lagerkvist"},{"@type":"Person","name":"Roy, Biman","givenName":"Biman","familyName":"Roy"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.MFCS.2016.64","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1515\/dma.1994.4.5.401","http:\/\/dx.doi.org\/10.1137\/120868177","http:\/\/knowledgecenter.siam.org\/0236-000094\/","http:\/\/edok01.tib.uni-hannover.de\/edoks\/e01dh08\/559615132.pdf"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6261","volumeNumber":58,"name":"41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)","dateCreated":"2016-08-19","datePublished":"2016-08-19","editor":[{"@type":"Person","name":"Faliszewski, Piotr","givenName":"Piotr","familyName":"Faliszewski"},{"@type":"Person","name":"Muscholl, Anca","givenName":"Anca","familyName":"Muscholl"},{"@type":"Person","name":"Niedermeier, Rolf","givenName":"Rolf","familyName":"Niedermeier"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8974","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":"#volume6261"}}}