13

Wikipedia's graph isomorphism problem page would seem to indicate that, no, it has not been solved. However, a friend of mine pointed out A Polynomial Time Algorithm for Graph Isomorphism . I am not sophisticated enough to follow the reasoning in the paper.

I do have my own very rough attempt at a polynomial time algorithm without anything like proof, but I'd like to know whether or not this problem has been successfully tackled before proceeding.

So, is the graph isomorphism problem solved?

Raphael
  • 73,212
  • 30
  • 182
  • 400
Tyler Spaeth
  • 171
  • 4

4 Answers4

24

No. That paper appears to be flawed. The flaw was explained in a comment by Tracy Hall on MathOverflow. A follow-up comment claims that the author later realized there is a flaw in his algorithm.

As Yuval explains, it is not uncommon to see attempts from amateurs to solve these problems; they tend to be flawed. When it comes to results on famous open problems (e.g., P vs NP, graph isomorphism, etc.), I recommend looking to published literature in reputable peer-reviewed conferences and journals -- peer review is not perfect, but peer-reviewed papers have a much higher likelihood of being correct.

Community
  • 1
D.W.
  • 167,959
  • 22
  • 232
  • 500
18

No, the graph isomorphism problem has not been solved. The paper you link to is from 2007–2008, and hasn't been accepted by the wider scientific community. (If it had been, I would have known about it.)

Graph isomorphism, like many other famous problems, attracts many attempts by amateurs. They are almost always wrong. I would advise against trying to tackle this problem without first becoming competent in research-level mathematics.

Yuval Filmus
  • 280,205
  • 27
  • 317
  • 514
8

I would be very dubious that it has (in the sense of the proof of existence of a polynomial time algorithm). While it is not impossible that the paper is correct, there are a number of warning signs:

  1. The author has not published the result in a peer reviewed venue (even after 7 years).
  2. The author does not seem to have published anything else, anywhere.
  3. The paper presents the algorithms, but the claim of correctness is an informal handwaving argument about the complexity.
  4. For a problem that has resisted the attempts of some very clever people, the maths in the paper is too simple.
  5. The author doesn't appear to be affiliated with an academic institution. The new version of the paper clarifies this.

Again, without someone identifying a flaw in the paper, these are not fool proof signs. Maybe the author had a unique flash of insight and then moved on to a completely different life, but the weight of probability is against it - extraordinary claims require extraordinary evidence.

To elaborate on (4) given recent news, László Babai recently claimed a major improvement on known graph isomorphism algorithm (no preprint yet, but a decent running commentary on his public lecture can be found here), giving a pseudo-polynomial time algorithm. Babai and his colleagues are definitely very smart people, and the mathematics used to obtain this result is difficult, deep and spans graph theory and group theory. Given the weight of probability, this is the expected level for a significant advance on a problem like this.

Luke Mathieson
  • 18,373
  • 4
  • 60
  • 87
4

Laszlo Babai has claimed to have found a quasipolynomial solution for the graph isomorphism problem as of November 11th 2015.

... and retracted the claim yesterday (4/1/2017):

Source: http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/

Luke Mathieson
  • 18,373
  • 4
  • 60
  • 87
bharv14
  • 159
  • 1