Logical and Structural Approaches to the Graph Isomorphism Problem

Abstract

AbstractIt is a long-standing open question whether there is a polynomial time algorithm deciding if two graphs are isomorphic. Indeed, graph isomorphism is one of the very few natural problems in NP that is neither known to be in P nor known to be NP-complete. The question is still wide open, but a number of deep partial results are known. On the complexity theoretic side, we have good reason to believe that graph isomorphism is not NP-complete: if it was NP-complete, then the polynomial hierarchy would collapse to its second level. On the algorithmic side, we know a nontrivial algorithm with a worst-case running time of \(2^{O(\sqrt{n\log n})}\) and polynomial time algorithms for many specific classes of graphs. Many of these algorithmic results have been obtained through a group theoretic approach that dominated the research on the graph isomorphism problem since the early 1980s.After an introductory survey, in my talk I will focus on approaches to the graph isomorphism problem based on structural graph theory and connections between logical definability, certain combinatorial algorithms, and mathematical programming approaches to the isomorphism problem.