Sherali-Adams relaxations and indistinguishability in counting logics

SIAM Journal on Computing

Volumn 42, Issue 1, 2013, Pages 112-137

  Atserias, Albert ^a   Maneva, Elitza ^b  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b UNIVERSITY OF BARCELONA   (Spain)

Author keywords

Combinatorial optimization; Counting quantifiers; First order logic; Graph isomorphism; Linear programming; Weisfeiler Lehman algorithm

Indexed keywords

COUNTING QUANTIFIERS; DOUBLY STOCHASTIC MATRIX; FINITE MODEL THEORY; FIRST ORDER LOGIC; GRAPH ISOMORPHISM; INDISTINGUISHABILITY; LINEAR PROGRAMMING RELAXATION; POLYHEDRAL COMBINATORICS;

ALGORITHMS; COMBINATORIAL OPTIMIZATION; FLOW GRAPHS; FORMAL LOGIC; LINEAR PROGRAMMING; MATRIX ALGEBRA; SET THEORY;

GRAPH THEORY;

EID: 84876244742     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/120867834     Document Type: Conference Paper

References (38)

1. S. Arora, B. Bollob'as, L. Lovász, and I. Tourlakis, Proving integrality gaps without knowing the linear program, Theory Comput., 2 (2006), pp. 19-51.
2. A. Atserias, A. A. Bulatov, and A. Dawar, Affine systems of equations and counting infinitary logic, Theoret. Comput. Sci., 410 (2009), pp. 1666-1683.
3. A. Atserias, P. G. Kolaitis, and M. Y. Vardi, Linear Programming and the Existential Pebble Game, manuscript, 2006.
4. L. Babai, P. Erdös, and S. M. Selkow, Random graph isomorphism, SIAM J. Comput., 9 (1980), pp. 628-635.
5. F. Barahona and A. Mahjoub, On the cut polytope, Math. Programming, 36 (1986), pp. 157- 173.
6. A. Blass, Y. Gurevich, and S. Shelah, On polynomial time computation over unordered structures, J. Symbolic Logic, 67 (2002), pp. 1093-1125.
7. J. Cai, M. Fürer, and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica, 12 (1992), pp. 389-410.
8. M. Charikar, K. Makarychev, and Y. Makarychev, Integrality gaps for Sherali-Adams relaxations, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), 2009, pp. 283-292.
9. H. Ebbinghaus and J. Flum, Finite Model Theory, Springer-Verlag, Berlin, 1995.
10. A. Ehrenfeucht, An application of games to the completeness problem for formalized theories, Fund. Math., 49 (1961), pp. 129-141.
11. R. Fraiss'e, Sur une nouvelle classification des systèmes de relations, Comptes Rendus, 230 (1950), pp. 1022-1024.
12. K. Georgiou, A. Magen, T. Pitassi, and I. Tourlakis, Integrality gaps of 2-o(1) for vertex cover SDPs in the Lovász-Schrijver hierarchy, in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2007, pp. 702-712.
13. M. Grohe, Fixed-point logics on planar graphs, in Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science (LICS), 1998, pp. 6-15.
14. M. Grohe, Isomorphism testing for embeddable graphs through definability, in Proceedings of the 32nd Annual ACM Symposium on the Theory of Computing (STOC), 2000, pp. 63-72.
15. M. Grohe, Fixed-point definability and polynomial time on graphs with excluded minors, in Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science (LICS), 2010, pp. 179-188.
16. M. Grohe and M. Otto, Pebble games and linear equations, in Proceedings of the 21st Annual Conference of the European Association for Computer Science Logic (CSL), LIPIcs 16, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Wadern, Germany, 2012, pp. 289- 304.
17. N. Immerman, Descriptive Complexity, Springer-Verlag, New York, 1999.
18. N. Immerman and E. Lander, Describing graphs: A first-order approach to graph canonization, in Complexity Theory Retrospective, Springer-Verlag, New York, 1990, pp. 59-81.
19. S. Khot and O. Regev, Vertex cover might be hard to approximate to within 2-epsilon, J.Comput. System Sci., 74 (2008), pp. 335-349.
20. P. G. Kolaitis and J. A. Väänänen, Generalized quantifiers and pebble games on finite structures, Ann. Pure Appl. Logic, 74 (1995), pp. 23-75.
21. P. G. Kolaitis and M. Y. Vardi, Conjunctive-query containment and constraint satisfaction, J. Comput. System Sci., 61 (2000), pp. 302-332.
22. L. Kucera, Canonical labeling of regular graphs in linear average time, in Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1987, pp. 271-279.
23. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2011), pp. 796-817.
24. M. Laurent, A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming, Math. Oper. Res., 28 (2001), pp. 470-496.
25. M. Laurent, Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope, Math. Oper. Res., 28 (2003), pp. 871-883.
26. L. Libkin, Elements of Finite Model Theory, Springer-Verlag, Berlin, 2004.
27. L. Lovász and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim., 1 (1991), pp. 166-190.
28. C. Mathieu and A. Sinclair, Sherali-Adams relaxations of the matching polytope, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), 2009, pp. 293-302.
29. M. Otto, Bounded Variable Logics and Counting: A Study in Finite Models, Lecture Notes Logic 9, Springer-Verlag, Berlin, 1997.
30. O. Pikhurko and O. Verbitsky, Logical complexity of graphs: A survey, in Model Theoretic Methods in Finite Combinatorics, Contemp. Math. 558, American Mathematical Society, Providence, RI, 2011, pp. 129-179.
31. M. V. Ramana, E. R. Scheinerman, and D. Ullman, Fractional isomorphism of graphs, Discrete Math., 132 (1994), pp. 247-265.
32. E. R. Scheinerman and D. H. Ullman, Fractional Graph Theory, Wiley-Interscience, New York, 1997.
33. G. Schoenebeck, Linear level Lasserre lower bounds for certain k-CSPs, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2008, pp. 593-602.
34. G. Schoenebeck, L. Trevisan, and M. Tulsiani, Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), 2007, pp. 302-310. (Pubitemid 47630748)
35. H. D. Sherali and W. P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. Discrete Math., 3 (1990), pp. 411-430.
36. T. Stephen and L. Tunçel, On a representation of the matching polytope via semi-definite liftings, Math. Oper. Res., 24 (1999), pp. 1-7.
37. I. Tourlakis, New lower bounds for vertex cover in the Lovasz-Schrijver hierarchy, in Proceedings of the 21st Annual IEEE Conference on Computational Complexity (CCC), 2006, pp. 170-182.
38. M. Tulsiani, CSP gaps and reductions in the Lasserre hierarchy, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), 2009, pp. 303-312.