Journal of the ACM (JACM)
Mathematical Programming: Series A and B
A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Fractional isomorphism of graphs
Discrete Mathematics
On a Representation of the Matching Polytope Via Semidefinite Liftings
Mathematics of Operations Research
Isomorphism testing for embeddable graphs through definability
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Conjunctive-query containment and constraint satisfaction
Journal of Computer and System Sciences - Special issue on the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on principles of database systems
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Fixed-Point Logics on Planar Graphs
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
Lower Bound for the Number of Iterations in Semidefinite Hierarchies for the Cut Polytope
Mathematics of Operations Research
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
New Lower Bounds for Vertex Cover in the Lovasz-Schrijver Hierarchy
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Canonical labeling of regular graphs in linear average time
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Linear Level Lasserre Lower Bounds for Certain k-CSPs
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Affine systems of equations and counting infinitary logic
Theoretical Computer Science
Integrality gaps for Sherali-Adams relaxations
Proceedings of the forty-first annual ACM symposium on Theory of computing
Sherali-adams relaxations of the matching polytope
Proceedings of the forty-first annual ACM symposium on Theory of computing
CSP gaps and reductions in the lasserre hierarchy
Proceedings of the forty-first annual ACM symposium on Theory of computing
Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors
LICS '10 Proceedings of the 2010 25th Annual IEEE Symposium on Logic in Computer Science
Affine systems of equations and counting infinitary logic
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Maximum Matching and Linear Programming in Fixed-Point Logic with Counting
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity PTAP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali-Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known color-refinement heuristic for graph isomorphism called the Weisfeiler-Lehman algorithm, or equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers, that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications both in finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow-circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to Ω(n) levels, where n is the number of vertices in the graph.