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| Content Provider | ACM Digital Library |
|---|---|
| Author | Maneva, Elitza Atserias, Albert |
| Abstract | Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity $P^{T}AP$ = 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. |
| Starting Page | 367 |
| Ending Page | 379 |
| Page Count | 13 |
| File Format | |
| ISBN | 9781450311151 |
| DOI | 10.1145/2090236.2090265 |
| Language | English |
| Publisher | Association for Computing Machinery (ACM) |
| Publisher Date | 2012-01-08 |
| Publisher Place | New York |
| Access Restriction | Subscribed |
| Subject Keyword | Weisfeiler-lehman algorithm Graph isomorphism Linear programming Pebble games Lift-and-project Counting logics Sherali-adams relaxations |
| Content Type | Text |
| Resource Type | Article |
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