On the Weisfeiler-Leman Dimension of Fractional Packing

TL;DR

This paper explores how the Weisfeiler-Leman procedure relates to fractional packing numbers in graphs, analyzing its ability to preserve certain parameters and estimate integrality gaps in graph isomorphism problems.

Contribution

It investigates the preservation of fractional packing numbers under $k$-WL-equivalence and discusses the potential of $k$-WL to estimate integrality gaps in LP relaxations.

Findings

$k$-WL-equivalence preserves certain fractional packing parameters.

Analysis of $k$-WL's ability to estimate integrality gaps.

Insights into the relation between Weisfeiler-Leman procedure and graph packing numbers.

Abstract

The $k$-dimensional Weisfeiler-Leman procedure ($k$-WL), which colors $k$-tuples of vertices in rounds based on the neighborhood structure in the graph, has proven to be immensely fruitful in the algorithmic study of Graph Isomorphism. More generally, it is of fundamental importance in understanding and exploiting symmetries in graphs in various settings. Two graphs are $k$-WL-equivalent if the $k$-dimensional Weisfeiler-Leman procedure produces the same final coloring on both graphs. 1-WL-equivalence is known as fractional isomorphism of graphs, and the $k$-WL-equivalence relation becomes finer as $k$ increases. We investigate to what extent standard graph parameters are preserved by $k$-WL-equivalence, focusing on fractional graph packing numbers. The integral packing numbers are typically NP-hard to compute, and we discuss applicability of $k$-WL-invariance for estimating the integrality gap of the LP relaxation provided by their fractional counterparts.