Abstract
We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm. More precisely, the runtime bound has a constant factor improvement in the base of the exponent: the algorithm can count the number of optima in MAX-2-SAT and MAX-CUT instances in O(m 3 2ωn/3) time, where ω < 2.376 is the matrix product exponent over a ring. When constraints have arbitrary weights, there is a (1+ε)-approximation with roughly the same runtime, modulo polynomial factors. Our construction shows that improvement in the runtime exponent of either k-clique solution (even when k = 3) or matrix multiplication over GF(2) would improve the runtime exponent for solving 2-CSP optimization. The overall approach also yields connections between the complexity of some (polynomial time) high dimensional search problems and some NP-hard problems. For example, if there are sufficiently faster algorithms for computing the diameter of n points in ℓ1, then there is an (2–ε)n algorithm for MAX-LIN. These results may be construed as either lower bounds on the high-dimensional problems, or hope that better algorithms exist for the corresponding hard problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alber, J., Gramm, J., Niedermeier, R.: Faster exact algorithms for hard problems: a parameterized point of view. Discrete Mathematics 229, 3–27 (2001)
Alon, N., Naor, M.: Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions. Algorithmica 16, 434–449 (1996)
Bansal, N., Raman, V.: Upper bounds for Max-Sat: Further Improved. In: Aggarwal, A.K., Pandu Rangan, C. (eds.) ISAAC 1999. LNCS, vol. 1741, pp. 247–258. Springer, Heidelberg (1999)
Charikar, M., Indyk, P., Panigrahy, R.: New Algorithms for Subset Query, Partial Match, Orthogonal Range Searching, and Related Problems. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 451–462. Springer, Heidelberg (2002)
Chen, J., Kanj, I.: Improved Exact Algorithms for MAX-SAT. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 341–355. Springer, Heidelberg (2002)
Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. JSC 9(3), 251–280 (1990)
Gramm, J., Niedermeier, R.: Faster exact solutions for Max2Sat. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 174–186. Springer, Heidelberg (2000)
Gramm, J., Hirsch, E.A., Niedermeier, R., Rossmanith, P.: Worst-case upper bounds for MAX-2-SAT with application to MAX-CUT. Discrete Applied Mathematics 130(2), 139–155 (2003)
Dantsin, E., Gavrilovich, M., Hirsch, E.A., Konev, B.: MAX-SAT approximation beyond the limits of polynomial-time approximation. Annals of Pure and Applied Logic 113(1-3), 81–94 (2001)
Hirsch, E.A.: Worst-case study of local search for MAX-k-SAT. Discrete Applied Mathematics 130(2), 173–184 (2003)
Hirsch, E.A.: A 2m/4-time Algorithm for MAX-2-SAT: Corrected Version. Electronic Colloquium on Computational Complexity Report TR99-036 (2000)
Hofmeister, T., Schöning, U., Schuler, R., Watanabe, O.: A probabilistic 3-SAT algorithm further improved. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 192–202. Springer, Heidelberg (2002)
Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. JACM 21, 277–292 (1974)
Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM J. Computing 7(4), 413–423 (1978)
Kulikov, A.S., Fedin, S.S.: A 2|E|/4-time Algorithm for MAX-CUT. Zapiski nauchnyh seminarov POMI No.293, 129–138 (2002)
Mahajan, M., Raman, V.: Parameterizing above Guaranteed Values: MAXSAT and MAXCUT. J. Algorithms 31(2), 335–354 (1999)
Nesetril, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae 26(2), 415–419 (1985)
Niedermeier, R., Rossmanith, P.: New upper bounds for maximum satisfiability. J. Algorithms 26, 63–88 (2000)
Paturi, R., Pudlak, P., Saks, M.E., Zane, F.: An improved exponential-time algorithm for k-SAT. In: Proceedings of the 39th IEEE FOCS, pp. 628–637 (1998)
Raman, V., Ravikumar, B., Srinivasa Rao, S.: A Simplified NP-Complete MAXSAT problem. Information Processing Letters 65, 1–6 (1998)
Rivest, R.: Partial match retrieval algorithms. SIAM J. on Computing 5, 19–50 (1976)
Robson, M.: Algorithms for maximum independent sets. J. Algorithms 7(3), 425–440 (1986)
Ruskey, F.: Simple combinatorial Gray codes constructed by reversing sublists. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 201–208. Springer, Heidelberg (1993)
Schöning, U.: A probabilistic algorithm for k-SAT and constraint satisfaction problems. In: Proceedings of the 40th IEEE FOCS, pp. 410–414 (1999)
Schroeppel, R., Shamir, A.: A T=O(2n/2), S=O(2n/4) Algorithm for Certain NP-Complete Problems. SIAM J. Comput. 10(3), 456–464 (1981)
Schuler, R.: An algorithm for the satisfiability problem of formulas in conjunctive normal form. Accepted in J. Algorithms (2003)
Scott, A., Sorkin, G.: Faster Algorithms for MAX CUT and MAX CSP, with Polynomial Expected Time for Sparse Instances. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 382–395. Springer, Heidelberg (2003)
Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. Electronic Colloquium on Computational Complexity, Report TR04-032 (2004)
Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)
Zwick, U.: All Pairs Shortest Paths using bridging sets and rectangular matrix multiplication. JACM 49(3), 289–317 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Williams, R. (2004). A New Algorithm for Optimal Constraint Satisfaction and Its Implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_101
Download citation
DOI: https://doi.org/10.1007/978-3-540-27836-8_101
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22849-3
Online ISBN: 978-3-540-27836-8
eBook Packages: Springer Book Archive