{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article15937","name":"CNF Satisfiability in a Subspace and Related Problems","abstract":"We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over \ud835\udd3d\u2082 each of which is a product of affine forms.\r\n\r\nWe focus on the case of k-CNF formulas (the k-Sub-Sat problem). Clearly, k-Sub-Sat is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that 2-Sub-Sat is NP-hard, and W[1]-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-2-Sub-Sat is NP-hard to approximate better than the trivial 3\/4 ratio even on satisfiable instances.\r\n\r\nOn the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with running time (1.5)^r for 2-Sub-Sat, where r is the subspace dimension, as well as an O^*(1.4312)\u207f time algorithm where n is the number of variables.\r\n\r\nTurning to k-Sub-Sat for k \u2a7e 3, while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time \u2248 2^{r(1-1\/2k)}, we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time \u2248 {n choose {\u2a7dt}} 2^{n-n\/k} where n is the number of variables and t is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for k-Sub-Sat with running time \u2248 2^{n-n\/2k}. This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over \ud835\udd3d\u2082, we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick.","keywords":["CNF Satisfiability","Exact exponential algorithms","Hardness results"],"author":[{"@type":"Person","name":"Arvind, Vikraman","givenName":"Vikraman","familyName":"Arvind","email":"mailto:arvind@imsc.res.in","affiliation":"The Institute of Mathematical Sciences (HBNI), Chennai, India"},{"@type":"Person","name":"Guruswami, Venkatesan","givenName":"Venkatesan","familyName":"Guruswami","email":"mailto:venkatg@cs.cmu.edu","affiliation":"Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA","funding":"Portions of this work were done during visits to the Institute of Mathematical Sciences, Chennai. Research supported in part by the National Science Foundation grant CCF-1908125 and a Simons Investigator Award."}],"position":5,"pageStart":"5:1","pageEnd":"5:15","dateCreated":"2021-11-22","datePublished":"2021-11-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arvind, Vikraman","givenName":"Vikraman","familyName":"Arvind","email":"mailto:arvind@imsc.res.in","affiliation":"The Institute of Mathematical Sciences (HBNI), Chennai, India"},{"@type":"Person","name":"Guruswami, Venkatesan","givenName":"Venkatesan","familyName":"Guruswami","email":"mailto:venkatg@cs.cmu.edu","affiliation":"Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA","funding":"Portions of this work were done during visits to the Institute of Mathematical Sciences, Chennai. Research supported in part by the National Science Foundation grant CCF-1908125 and a Simons Investigator Award."}],"copyrightYear":"2021","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2021.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/2108.05914","http:\/\/arxiv.org\/abs\/1801.09488"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6417","volumeNumber":214,"name":"16th International Symposium on Parameterized and Exact Computation (IPEC 2021)","dateCreated":"2021-11-22","datePublished":"2021-11-22","editor":[{"@type":"Person","name":"Golovach, Petr A.","givenName":"Petr A.","familyName":"Golovach","email":"mailto:petr.golovach@ii.uib.no","sameAs":"https:\/\/orcid.org\/0000-0002-2619-2990","affiliation":"University of Bergen, Norway"},{"@type":"Person","name":"Zehavi, Meirav","givenName":"Meirav","familyName":"Zehavi","email":"mailto:meiravze@bgu.ac.il","sameAs":"https:\/\/orcid.org\/0000-0002-3636-5322","affiliation":"Ben-Gurion University of the Negev, Israel"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article15937","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":"#volume6417"}}}