2026-04-13T02:54:55Zhttps://recercat.cat/oai/request

oai:recercat.cat:2445/338552025-12-05T09:53:20Zcom_2072_1057col_2072_478917col_2072_478920

00925njm 22002777a 4500 dc Atserias, Albert author Maneva, Elitza author 2013-02-19T11:54:16Z 2013-02-19T11:54:16Z 2013-01-17 2013-02-19T11:42:38Z 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. Lògica de primer ordre Programació lineal Teoria de grafs First-order logic Linear programming Graph theory Sherali-Adams Relaxations and Indistinguishability in Counting Logics