Jacobo Torán

The Graph Isomorphism Problem

This report documents the program and the outcomes of Dagstuhl Seminar 15511 “The Graph Isomorphism Problem”. The goal of the seminar was to bring together researchers working on the numerous topics closely related to the Isomorphism Problem to foster their collaboration. To this end the participants of the seminar included researchers working on the theoretical and practical aspects of isomorphism ranging from the fields of algorithmic group theory, finite model theory, combinatorial optimization to algorithmics. A highlight of the conference was the presentation of a new quasi-polynomial time algorithm for the Graph Isomorphism Problem, providing the first improvement since 1983. Seminar December 13–18, 2015 – http://www.dagstuhl.de/15511 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory

Complexity classes defined by counting quantifiers

The polynomial-time counting hierarchy, a hierarchy of complexity classes related to the notion of counting is studied. Some of their structural properties are investigated, settling many open questions dealing with oracle characterizations, closure under Boolean operations, and relations with other complexity classes. A new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes, is developed.

On counting and approximation

We introduce a new class of functions, called span functions which count the different output values that occur at the leaves of the computation tree associated with a nondeterministic polynomial time Turing machine transducer. This function class has natural complete problems; it is placed between Valiants function classes #P and #NP, and contains both Goldberg and Sipsers ranking functions for sets in NP, and Krentels optimization functions. We show that it is unlikely that the span functions coincide with any of the mentioned function classes.

On the Hardness of Graph Isomorphism

We show that the graph isomorphism problem is hard under DLOGTIME uniform AC{

Turing machines with few accepting computations and low sets for PP

^0

Classes of bounded nondeterminism

} many-one reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class {Mod}

Completeness results for graph isomorphism

_k

Graph isomorphism is low for PP

L and for the class DET of problems NC{

The complexity of algorithmic problems on succinct instances

^1

Space bounds for resolution

} reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.