Years and Authors of Summarized Original Work
2010; Crowston, Gutin, Jones, Kim, Ruzsa
2011; Gutin, Kim, Szeider, Yeo
2014; Crowston, Fellows, Gutin, Jones, Kim, Rosamond, Ruzsa, Thomassé, Yeo
Problem Definition
The problem MaxLin2 can be stated as follows. We are given a system of m equations in variables x1, …, x n where each equation is \(\prod _{i\in I_{j}}x_{i} = b_{j}\), for some \(I_{j} \subseteq \{ 1,2,\ldots,n\}\) and x i , b j ∈ {−1, 1} and j = 1, …, m. Each equation is assigned a positive integral weight w j . We are required to find an assignment of values to the variables in order to maximize the total weight of the satisfied equations. MaxLin2 is a well-studied problem, which according to Håstad [8] “is as basic as satisfiability.”
Note that one can think of MaxLin2 as containing equations, \(\sum _{i\in I_{j}}y_{i} = a_{j}\) over \(\mathbb{F}_{2}\). This is equivalent to the previous definition by letting y i = 0 if and only if x i = 1 and letting y i = 1 if and only...
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Crowston R, Fellows M, Gutin G, Jones M, Kim EJ, Rosamond F, Ruzsa IZ, Thomassé S, Yeo A (2014) Satisfying more than half of a system of linear equations over GF(2): a multivariate approach. J Comput Syst Sci 80(4):687–696
Crowston R, Gutin G, Jones M (2010) Note on Max Lin-2 above average. Inform Proc Lett 110:451–454
Crowston R, Gutin G, Jones M, Kim EJ, Ruzsa I (2010) Systems of linear equations over \(\mathbb{F}_{2}\) and problems parameterized above average. In: SWAT 2010, Bergen. Lecture notes in computer science, vol 6139, pp 164–175
Downey RG, Fellows MR (2013) Fundamentals of parameterized complexity. Springer, London/Heidelberg/New York
Flum J, Grohe M (2006) Parameterized complexity theory. Springer, Berlin
Gutin G, Kim EJ, Szeider S, Yeo A (2011) A probabilistic approach to problems parameterized above or below tight bounds. J Comput Syst Sci 77:422–429
Gutin G, Yeo A (2012) Constraint satisfaction problems parameterized above or below tight bounds: a survey. Lect Notes Comput Sci 7370:257–286
Håstad J (2001) Some optimal inapproximability results. J ACM 48:798–859
Håstad J, Venkatesh S (2004) On the advantage over a random assignment. Random Struct Algorithms 25(2):117–149
Kim EJ, Williams R (2012) Improved parameterized algorithms for above average constraint satisfaction. In: IPEC 2011, Saarbrücken. Lecture notes in computer science, vol 7112, pp 118–131
Mahajan M, Raman V (1999) Parameterizing above guaranteed values: MaxSat and MaxCut. J Algorithms 31(2):335–354. Preliminary version in Electr. Colloq. Comput. Complex. (ECCC), TR-97-033, 1997
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Yeo, A. (2016). Kernelization, MaxLin Above Average. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_532
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