3-coloring in time O (1.3289n)

  • Authors:

  • Richard Beigel;David Eppstein

  • Affiliations:

  • Department of Electrical Engineering and Computer Science, University of Illinois, Chicago, 851 South Morgan St. (m/c 154), Fl. 11, Chicago, IL 60607-7053, USA;Donald Bren School of Information and Computer Sciences, University of California, Irvine, CA 92697-3425, USA

  • Venue:
  • Journal of Algorithms
  • Year:
  • 2005

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Abstract

We consider worst case time bounds for several NP-complete problems, based on a constraint satisfaction (CSP) formulation of these problems: (a,b)-CSP instances consist of a set of variables, each with up to a possible values, and constraints disallowing certain b-tuples of variable values; a problem is solved by assigning values to all variables satisfying all constraints, or by showing that no such assignment exist. 3-SAT is equivalent to (2,3)-CSP while 3-coloring and various related problems are special cases of (3,2)-CSP; there is also a natural duality transformation from (a,b)-CSP to (b,a)-CSP. We show that n-variable (3,2)-CSP instances can be solved in time O(1.3645^n), that satisfying assignments to (d,2)-CSP instances can be found in randomized expected time O((0.4518d)^n); that 3-coloring of n-vertex graphs can be solved in time O(1.3289^n); that 3-list-coloring of n-vertex graphs can be solved in time O(1.3645^n); that 3-edge-coloring of n-vertex graphs can be solved in time O(2^n^/^2), and that 3-satisfiability of a formula with t 3-clauses can be solved in time O(n^O^(^1^)+1.3645^t).