Pebble games with algebraic rules

TL;DR

This paper introduces a general framework of partition pebble games over finite structures, including the invertible-map game, which provides a stronger polynomial-time approximation of graph isomorphism than Weisfeiler-Lehman.

Contribution

It presents a unified framework for various pebble games, introduces the invertible-map game, and demonstrates its superior approximation power over existing methods.

Findings

Invertible-map game yields stronger graph isomorphism approximations.

Framework includes pebble games for finite-variable logics with/without counting.

Introduces a matrix-equivalence game characterising matrix-rank logic.

Abstract

We define a general framework of partition games for formulating two-player pebble games over finite structures. We show that one particular such game, which we call the invertible-map game, yields a family of polynomial-time approximations of graph isomorphism that is strictly stronger than the well-known Weisfeiler-Lehman method. The general framework we introduce includes as special cases the pebble games for finite-variable logics with and without counting. It also includes a matrix-equivalence game, introduced here, which characterises equivalence in the finite-variable fragments of matrix-rank logic. We show that the equivalence defined by the invertible-map game is a refinement of the equivalence defined by each of these games for finite-variable logics.