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TR11-077 | 8th May 2011 10:15

Graph Isomorphism, Sherali-Adams Relaxations and Expressibility in Counting Logics



Two graphs with adjacency matrices $\mathbf{A}$ and $\mathbf{B}$ are isomorphic if there exists a permutation matrix $\mathbf{P}$ for which the identity $\mathbf{P}^{\mathrm{T}} \mathbf{A} \mathbf{P} = \mathbf{B}$ holds. Multiplying through by $\mathbf{P}$ and relaxing the permutation matrix to a doubly stochastic matrix leads to the notion of fractional isomorphism. We show that the levels of the Sherali-Adams 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. 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 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, F├╝rer 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 $\Omega(n)$ levels, where $n$ is the number of vertices in the graph.

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