dc.contributor |
Universitat de Barcelona |
dc.contributor.author |
Atserias, Albert |
dc.contributor.author |
Maneva, Elitza |
dc.date |
2013-02-19T11:54:16Z |
dc.date |
2013-02-19T11:54:16Z |
dc.date |
2013-01-17 |
dc.date |
2013-02-19T11:42:38Z |
dc.identifier.citation |
0097-5397 |
dc.identifier.citation |
619363 |
dc.identifier.uri |
http://hdl.handle.net/2445/33855 |
dc.format |
26 p. |
dc.format |
application/pdf |
dc.language.iso |
eng |
dc.publisher |
Society for Industrial and Applied Mathematics |
dc.relation |
Reproducció del document publicat a: http://dx.doi.org/10.1137/120867834 |
dc.relation |
SIAM Journal on Computing, 2013, vol. 42, num. 1, p. 112-137 |
dc.relation |
http://dx.doi.org/10.1137/120867834 |
dc.rights |
(c) Society for Industrial and Applied Mathematics., 2013 |
dc.rights |
info:eu-repo/semantics/openAccess |
dc.subject |
Lògica de primer ordre |
dc.subject |
Programació lineal |
dc.subject |
Teoria de grafs |
dc.subject |
First-order logic |
dc.subject |
Linear programming |
dc.subject |
Graph theory |
dc.title |
Sherali-Adams Relaxations and Indistinguishability in Counting Logics |
dc.type |
info:eu-repo/semantics/article |
dc.type |
info:eu-repo/semantics/publishedVersion |
dc.description.abstract |
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 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 in both 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 $\Omega(n)$ levels, where $n$ is the number of vertices in the graph. |