The graph isomorphism problems asks whether two graphs that appear dissimilar are in fact identical.

An innovative approach to solving a stubborn, but elementary, question in graph theory — the mathematical study of networks of nodes and their connections — may signal the first major theoretical advance in understanding the problem in more than 30 years.

László Babai, a mathematician and theoretical computer scientist at the University of Chicago in Illinois, outlined the advance at two talks last week (video). And now details are emerging from the mathematicians and computer scientists who attended.

“All of a sudden, an unexpected and huge step has been made,” says Stuart Kurtz, a computer scientist at the University of Chicago.

Babai's talks sketch a proof that shows that the graph isomorphism problem — determining whether two graphs are the same — can be solved much more quickly than was previously known.

A natural question

Graphs are relatively simple mathematical objects — abstract representations of networks — that arise frequently in physics, chemistry and computer science. They are defined by nodes and the links between them, and can be pictured as points connected by lines. The graph isomorphism problem simply asks whether two graphs are the same, regardless of how they are drawn or how their nodes are named.

In a sense, the connections between nodes are the essence of graphs, says Janos Simon, a theoretical computer scientist at the University of Chicago who is attending Babai’s lectures. “It shouldn’t matter what names you give them,” Simon says. He adds that it seems as if answering this fundamental question should be simple, but the best theoretical approaches have not developed much since 1983.

Computer scientists often study the complexity of an algorithm: how long it takes the algorithm to solve a problem or verify that a solution is correct. Two well-studied sets of problems are termed P — those that can be solved quickly — and NP, which have solutions that can be checked quickly, but not necessarily found quickly. Graph isomorphism is known to be in NP, which contains some problems that are believed to take a long time to solve.

A simple way to match two graphs is to check all possible ways of swapping node names on one graph, looking for a match to the second — a very inefficient procedure. But there was good reason to suspect that graph isomorphism was easier than this. For a start, it has been difficult for mathematicians to find two graphs that caused the best theoretical matching algorithm to run slowly.

“That was a unique behaviour that was very interesting,” says Luca Trevisan, a theoretical computer scientist at the University of California, Berkeley, who attended Babai’s second lecture. “Of course, now, if Professor Babai’s result is correct, these difficult inputs don’t exist.”