Why the graph isomorphism problem is easier than propositional satisfiability: a possible qualitative explanation

Abstract

A recent result has shown that the graph isomorphism problem can be solved in quasi-polynomial time, while the general belief is that only exponential time algorithms are possible for propositional satisfiability. This is somewhat counter-intuitive, since for propositional satisfiability, we need to look for one of 2 options, while in graph isomorphism, we need to look for one of n! options, and n! ≫ 2. Our qualitative explanation for this counter-intuitive fact comes from the fact that, in general, a graph isomorphism problem has a unique solution – in contrast to propositional satisfiability which, in general, has many solutions – and it is known that problems with unique solutions are often easier to solve. 1 Complexity of Different NP-Problems and a Recent Result on Graph Isomorphism To explain the result about graph isomorphisms, it is necessary to recall the corresponding definitions: what is a feasible algorithm, what is an NP-problem, etc. For detailed description of these notions, see, e.g., [7]. Feasible algorithms: reminder. Some theoretical algorithms require so much computation time that they are not practically useful. For example, if on inputs of bit size n, the algorithm requires exponentially many 2 steps, then even for a moderate-size input, with n = 1000, this algorithm needs time which is larger than the lifetime of the Universe. It is therefore desirable to find out which algorithms are feasible. Usually: