Systems of linear equations over double-struck F2 and problems parameterized above average

Robert Crowston, Gregory Gutin, Mark Jones, Eun Jung Kim, Imre Z. Ruzsa

Research output: Chapter in Book/Report/Conference proceedingConference contribution

20 Citations (Scopus)

Abstract

In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over double-struck F2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least k, where k is the parameter. It is not hard to see that we may assume that no two equations in Az = b have the same left-hand side and n = rank A. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: m ≤ 2p(n) for an arbitrary fixed function p(n) = o(n). This result generalizes earlier results by Crowston et al. (arXiv:0911.5384) and Gutin et al. (Proc. IWPEC'09). We also prove that Max Lin AA is polynomial-time solvable for every fixed k and, moreover, Max Lin AA is in the parameterized complexity class W[P]. Max r-Lin AA is a special case of Max Lin AA, where each equation has at most r variables. In Max Exact r-SAT AA we are given a multiset of m clauses on n variables such that each clause has r variables and asked whether there is a truth assignment to the n variables that satisfies at least (1 - 2-r)m + k2 -r clauses. Using our maximum excess results, we prove that for each fixed r ≥ 2, Max r-Lin AA and Max Exact r-SAT AA can be solved in time 2 O(k log k) + mO(1). This improves 2O(k2) + mO(1)-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively. It is easy to see that maximization of arbitrary pseudo-boolean functions, i.e., functions f : {-1, + 1}n → ℝ, represented by their Fourier expansions is equivalent to solving Max Lin. Using our main maximum excess result, we obtain a tight lower bound on the maxima of pseudo-boolean functions.

Original languageEnglish
Title of host publicationAlgorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings
Pages164-175
Number of pages12
DOIs
Publication statusPublished - Jul 21 2010
Event12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010 - Bergen, Norway
Duration: Jun 21 2010Jun 23 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6139 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010
CountryNorway
CityBergen
Period6/21/106/23/10

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Crowston, R., Gutin, G., Jones, M., Kim, E. J., & Ruzsa, I. Z. (2010). Systems of linear equations over double-struck F2 and problems parameterized above average. In Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings (pp. 164-175). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6139 LNCS). https://doi.org/10.1007/978-3-642-13731-0_17