A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between

ABSTRACT

We introduce "hybrid" Max 2-CSP formulas consisting of "simple clauses", namely conjunctions and disjunctions of pairs of variables, and general 2-variable clauses, which can be any integer-valued functions of pairs of boolean variables. This allows an algorithm to use both efficient reductions specific to AND and OR clauses, and other powerful reductions that require the general CSP setting.

Parametrizing an instance by the fraction p of nonsimple clauses, we give an exact (exponential-time) algorithm that is the fastest polynomial-space algorithm known for Max 2-Sat (and other p = 0 formulas, with arbitrary mixtures of AND and OR clauses); the only efficient algorithm for mixtures of AND, OR, and general integer-valued clauses; and tied for fastest for general Max 2-CSP (p = 1). Since a pure 2-Sat input instance may be transformed to a general CSP instance in the course of being solved, the algorithm's efficiency and generality go hand in hand.

Our novel analysis results in a family of running-time bounds, each optimized for a particular value of p. The algorithm uses new reductions introduced here, as well as recent reductions such as "clause-learning" and "2-reductions" adapted to our setting's mixture of simple and general clauses. Each reduction imposes constraints on various parameters, and the running-time bound is an "objective function" of these parameters and p. The optimal running-time bound is obtained by solving a convex nonlinear program, which can be done efficiently and with a certificate of optimality.

REFERENCES

• 1: Nikhil Bansal , Venkatesh Raman, Upper Bounds for MaxSat: Further Improved, Proceedings of the 10th International Symposium on Algorithms and Computation, p.247-258, December 16-18, 1999
• 2: David Eppstein, Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms, ACM Transactions on Algorithms (TALG), v.2 n.4, p.492-509, October 2006 [doi>10.1145/1198513.1198515]
• 3: F. V. Fomin, F. Grandoni, and D. Kratsch, Measure and conquer: Domination -- a case study, Proc. ICALP 2005, LNCS 3580, Springer, pp. 191--203.
• 4: Jens Gramm , Edward A. Hirsch , Rolf Niedermeier , Peter Rossmanith, Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT, Discrete Applied Mathematics, v.130 n.2, p.139-155, 15 August 2003 [doi>10.1016/S0166-218X(02)00402-X]
• 5: Edward A. Hirsch, A New Algorithm for MAX-2-SAT, Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, p.65-73, February 01, 2000
• 6: J. Kneis, D. Mölle, S. Richter, and P. Rossmanith, Algorithms based on the treewidth of sparse graphs, Proc. WG 2005, LNCS 3787, Springer, pp. 385--396.
• 7: J. Kneis and P. Rossmanith, A new satisfiability algorithm with applications to Max-Cut, Tech. Report AIB-2005-08, Department of Computer Science, RWTH Aachen, 2005.
• 8: Mikko Koivisto, Optimal 2-constraint satisfaction via sum-product algorithms, Information Processing Letters, v.98 n.1, p.24-28, 15 April 2006 [doi>10.1016/j.ipl.2005.11.013]
• 9: Arist Kojevnikov , Alexander S. Kulikov, A new approach to proving upper bounds for MAX-2-SAT, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.11-17, January 22-26, 2006, Miami, Florida [doi>10.1145/1109557.1109559]
• 10: A. S. Kulikov and K. Kutzkov, New bounds for MAX-SAT by clause learning, Proc. CSR 2007, LNCS 4649, Springer, pp. 194--204.
• 11: O. Kullmann, New methods for 3-SAT decision and worst-case analysis, Theoretical Computer Science, v.223 n.1-2, p.1-72, July 28, 1999 [doi>10.1016/S0304-3975(98)00017-6]
• 12: O. Kullmann, Worst-case analysis, 3-SAT decision and lower bounds: Approaches for improved SAT algorithms, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 35, Amer. Math. Soc., 1997, pp. 261--313.
• 13: Rolf Niedermeier , Peter Rossmanith, New upper bounds for maximum satisfiability, Journal of Algorithms, v.36 n.1, p.63-88, July 2000 [doi>10.1006/jagm.2000.1075]
• 14: D. Raible and H. Fernau, A new upper bound for Max-2-SAT: A graph-theoretic approach, Tech. Report cs:DS/0803.3531v2, arxiv.org, 2008, see http://arxiv.org/abs/cs.DM/0803.3531.
• 15: A. D. Scott and G. B. Sorkin, Faster algorithms for MAX CUT and MAX CSP, with polynomial expected time for sparse instances, Proc. RANDOM 2003, LNCS 2764, Springer, pp. 382--395.
• 16: A. D. Scott and G. B. Sorkin, A faster exponential-time algorithm for Max 2-Sat, Max Cut, and Max k-Cut, Tech. Report RC23456 (W0412-001), IBM Research Report, 2004, see http://domino.research.ibm.com/library/cyberdig.nsf.
• 17: A. D. Scott and G. B. Sorkin, Linear-programming design and analysis of fast algorithms for Max 2-CSP, Discrete Optim. 4 (2007), no. 3--4, 260--287.
• 18: Magnus Wahlström, Exact algorithms for finding minimum transversals in rank-3 hypergraphs, Journal of Algorithms, v.51 n.2, p.107-121, May 2004 [doi>10.1016/j.jalgor.2004.01.001]
• 19: Ryan Williams, A new algorithm for optimal 2-constraint satisfaction and its implications, Theoretical Computer Science, v.348 n.2, p.357-365, 8 December 2005 [doi>10.1016/j.tcs.2005.09.023]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.0 General

Additional Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Analysis of algorithms