Search Problems in Trees with Symmetries: near optimal traversal strategies for individualization-refinement algorithms

TL;DR

This paper introduces a near-optimal randomized search algorithm for tree isomorphism problems, explaining the efficiency of certain practical graph isomorphism tools and proposing a new traversal strategy with strong theoretical guarantees.

Contribution

It presents a randomized algorithm with near-optimal performance for tree isomorphism search problems, along with lower bounds and insights into existing practical algorithms.

Findings

The randomized algorithm visits leaves in quasilinear time relative to the square root of the smaller tree.

Lower bounds show randomized algorithms are optimal up to logarithmic factors for bounded degree trees.

The new traversal strategy outperforms traditional depth-first search in practical graph isomorphism tools.

Abstract

We define a search problem on trees that closely captures the backtracking behavior of all current practical graph isomorphism algorithms. Given two trees with colored leaves, the goal is to find two leaves of matching color, one in each of the trees. The trees are subject to an invariance property which promises that for every pair of leaves of equal color there must be a symmetry (or an isomorphism) that maps one leaf to the other. We describe a randomized algorithm with errors for which the number of visited leaves is quasilinear in the square root of the size of the smaller of the two trees. For inputs of bounded degree, we develop a Las Vegas algorithm with a similar running time. We prove that these results are optimal up to logarithmic factors. We show a lower bound for randomized algorithms on inputs of bounded degree that is the square root of the tree sizes. For inputs of unbounded degree, we show a linear lower bound for Las Vegas algorithms. For deterministic algorithms we can prove a linear bound even for inputs of bounded degree. This shows why randomized algorithms outperform deterministic ones. Our results explain why the randomized "breadth-first with intermixed experimental path" search strategy of the isomorphism tool Traces (Piperno 2008) is often superior to the depth-first search strategy of other tools such as nauty (McKay 1977) or bliss (Junttila, Kaski 2007). However, our algorithm also provides a new traversal strategy, which is theoretically near optimal with better worst case behavior than traversal strategies that have previously been used.