Optimal 2-constraint satisfaction via sum–product algorithms

• Published in: Information processing letters Vol. 98; no. 1; pp. 24 - 28
• Main Author: Koivisto, Mikko
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.04.2006; Elsevier Science; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computational complexity; Computer science; control theory; systems; Exact sciences and technology; Mathematics; Matrix multiplication; Maximum satisfiability; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Studies; Sum–product; Theoretical computing; Variables; Maximum satisfiability; Algorithms; Matrix multiplication; Computational complexity; Sum–product; Computer theory; Optimal algorithm; Matrix product; Constraint satisfaction; State variable; Assignment; Sum-product; Information processing; Sum product; Algorithm analysis
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2005.11.013

We show that for a given set of m pairwise constraints over n variables, a variable assignment that satisfies maximally many m constraints (MAX-2-CSP) can be found in O ∗ ( n m d n ω / 3 ) time, where d is the maximum number of states per variable, and ω < 2.376 is the matrix product exponent over a ring; the notation O ∗ suppresses factors polylogarithmic in m and n. As a corollary, MAX-2-SAT can be solved in O ∗ ( n m 1.732 n ) time. This improves on a recent result by Williams [R. Williams, A new algorithm for optimal 2-constraint satisfaction and its implications, Theoret. Comput. Sci. 348 (2–3) (2005) 357–365] by reducing the polynomial factor from n m 3 to about nm.