Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

TL;DR

This paper proves that various constrained graph isomorphism problems, including those with weight, complexity, and additional restrictions, are fixed-parameter tractable, solving open questions and extending the scope of efficient algorithms.

Contribution

The paper establishes FPT algorithms for constrained graph isomorphism problems with additional restrictions, including exact weight automorphisms and automorphisms with bounded complexity, addressing open problems.

Findings

Deciding isomorphisms that move exactly k vertices is in FPT.

Finding isomorphisms with CNF constraints is in FPT with a GI oracle.

Automorphisms with bounded Cayley measure complexity are in FPT.

Abstract

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs $X_1=(V,E_1)$ and $X_2=(V,E_2)$ and a subset $R\subseteq V\times V$ of forbidden pairs whether there is an isomorphism $\pi$ from $X_1$ to $X_2$ such that $\pi(i)\neq j$ for all $(i,j)\in R$. We prove that this problem and several of its generalizations are in fact in FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of $R$ as parameters. The problem remains in FPT if a CNF of such constraints is allowed. It follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the weight and complexity are unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism $\pi$ between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that $\pi$ can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT.