Maximizing supermodular functions on product lattices, with application to maximum constraint satisfaction

SIAM Journal on Discrete Mathematics

Volumn 22, Issue 1, 2008, Pages 312-328

  Krokhin, Andrei ^a   Larose, Benoit ^b  

^a DURHAM UNIVERSITY   (United Kingdom)

^b Gina Cody School of Engineering and Computer Science   (Canada)

Author keywords

Constraint satisfaction; Lattices; Optimization; Supermodular function; Tractability

Indexed keywords

OPTIMIZATION; POLYNOMIAL APPROXIMATION; STEELMAKING;

CONSTRAINT SATISFACTION; FINITE LATTICES; LARGE CLASS; LATTICES; MAXIMUM CONSTRAINT SATISFACTION PROBLEMS; MAXIMUM CONSTRAINTS; OPTIMIZATION PROBLEMS; POLYNOMIAL-TIME; PRODUCT LATTICES; STRONG LINKS; SUPERMODULAR FUNCTION; SUPERMODULARITY; TRACTABILITY;

CONSTRAINED OPTIMIZATION;

EID: 52249122574     PISSN: 08954801     EISSN: None     Source Type: Journal    
DOI: 10.1137/060669565     Document Type: Article

References (28)

1. R. E. BURKARD, B. KLINZ, AND R. RUDOLF, Perspectives of Monge properties in optimization, Discrete Appl. Math., 70 (1996), pp. 95-161.
2. D. COHEN, M. COOPER, AND P. JEAVONS, An algebraic characterization of complexity for valued constraints, Lecture Notes in Comput. Sci. 4204, Springer-Verlag, Berlin, 2006, pp. 107-121.
3. D. COHEN, M. COOPER, P. JEAVONS, AND A. KROKHIN, Supermodular functions and the complexity of MAX CSP, Discrete Appl. Math., 149 (2005), pp. 53-72.
4. D. COHEN, M. COOPER, P. JEAVONS, AND A. KROKHIN, The complexity of soft constraint satisfaction, Artificial Intelligence, 170 (2006), pp. 983-1016.
5. N. CREIGNOU, S. KHANNA, AND M. SUDAN, Complexity Classifications of Boolean Constraint Satisfaction Problems, SIAM Monogr. Discrete Math. Appl. 7, SIAM, Philadelphia, 2001.
6. R. DECHTER, Constraint Processing, Morgan Kaufmann, San Francisco, CA, 2003.
7. V. DEINEKO, P. JONSSON, M. KLASSON, AND A. KROKHIN, Supermodularity on chains and complexity of maximum constraint satisfaction, in Proceedings of Eurocomb'05, Vol. AE, DMTCS, Berlin, Germany, 2005, pp. 51-56.
8. V. DEINEKO, P. JONSSON, M. KLASSON, AND A. KROKHIN, The Approximability of MAX CSP with Fixed- Value Constraints, Technical report CoRR cs.CC/0602075, 2006.
9. A. FRANK, Applications of submodular functions, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 187, K. Walker, ed., Cambridge University Press, Cambridge, UK, 1993, pp. 85-136.
10. A. FRANK, Matroids and submodular functions, in Annotated Bibliographies in Combinatorial Optimization, M. Dell'Amico, F. Maffoli, and S. Martello, eds., John Wiley, Chichester, UK, 1997, pp. 65-80.
11. S. FUJISHIGE, Submodular Functions and Optimization, Ann. Discrete Math. 58, 2nd ed., Elsevier, Amsterdam, 2005.
12. A. GOLDBERG AND R. E. TARJAN, A new approach to the maximum-flow problem, J. Assoc. Comput. Mach., 35 (1988), pp. 921-940.
13. G. GRÄTZER, General Lattice Theory, 2nd ed., Birkhäuser-Verlag, Basel, Switzerland, 2002.
14. G. GRÄTZER AND D. KELLY, Products of lattice varieties, Algebra Universalis, 21 (1985), pp. 33-45.
15. D. HOBBY AND R. N. MCKENZIE, The Structure of Finite Algebras, Contemp. Math. 76, AMS, Providence, RI, 1988.
16. S. IWATA, A fully combinatorial algorithm for submodular function minimization, J. Combin. Theory Ser. B, 84 (2002), pp. 203-212.
17. S. IWATA, L. FLEISCHER, AND S. FUJISHIGE, A combinatorial strongly polynomial algorithm for minimizing submodular functions, J. ACM, 48 (2001), pp. 761-777.
18. P. JIPSEN AND H. ROSE, Varieties of Lattices, Lecture Notes in Math. 1533, Springer-Verlag, Berlin, 1992.
19. P. JONSSON, M. KLASSON, AND A. KROKHIN, The approximability of three-valued MAX CSP, SIAM J. Comput., 35 (2006), pp. 1329-1349.
20. A. KROKHIN AND B. LAROSE, Maximum Constraint Satisfaction on Diamonds, Technical report CS-RR-408, Department of Computer Science, University of Warwick, Warwick, UK, 2004.
21. A. KROKHIN AND B. LAROSE, Supermodular Sublattices in Products of Relatively Complemented Lattices, manuscript, 2006.
22. V. B. LENDER, The groupoid of prevarieties of lattices, Siberian Math. J., 16 (1975), pp. 1214-1223 (in Russian).
23. A. MÜLLER AND D. STOYAN, Comparison Methods for Stochastic Models and Risks, Wiley Ser. Prob. Stat., Wiley & Sons, Chichester, UK, 2002.
24. H. NARAYANAN, Submodular Functions and Electrical Networks, Ann. Discrete Math. 54, North-Holland, Amsterdam, 1997.
25. S. D. PROMISLOW AND V. R. YOUNG, Supermodular functions on finite lattices, Order, 22 (2005), pp. 389-413.
26. A. SCHRIJVER, A combinatorial algorithm minimizing submodular functions in polynomial time, J. Combin. Theory Ser. B, 80 (2000), pp. 346-355.
27. A. SCHRIJVER, Combinatorial Optimization: Polyhedra and Efficiency, Vol. B, Matroids, Trees, Stable Sets, Springer-Verlag, Berlin, 2003.
28. D. TOPKIS, Supermodularity and Complementarity, Frontiers Econom. Res., Princeton University Press, Princeton, NJ, 1998.