The Graph Clustering Problem is parameterized by a sequence
of positive integers, $m_1,...,m_t$.
The input is a sequence of $\sum_{i=1}^{t}m_i$ graphs,
and the question is whether the equivalence classes
under the graph isomorphism relation have sizes which match
the sequence of parameters.
In this note
we show that ... more >>>

The input to the {\em Graph Clustering Problem}\/
consists of a sequence of integers $m_1,...,m_t$
and a sequence of $\sum_{i=1}^{t}m_i$ graphs.
The question is whether the equivalence classes,
under the graph isomorphism relation,
of the input graphs have sizes which match the input sequence of integers.
In this note we ... more >>>

We show that Graph Isomorphism is in the complexity class
SPP, and hence it is in $\ParityP$ (in fact, it is in $\ModkP$ for
each $k\geq 2$). We derive this result as a corollary of a more
general result: we show that a {\em generic problem} $\FINDGROUP$ has
an $\FP^{\SPP}$ ... more >>>

In this paper we study the complexity of Bounded Color
Multiplicity Graph Isomorphism (BCGI): the input is a pair of
vertex-colored graphs such that the number of vertices of a given
color in an input graph is bounded by $b$. We show that BCGI is in the
#L hierarchy (more ... more >>>

We provide <i>unconditional</i> constructions of <i>concurrent</i>
statistical zero-knowledge proofs for a variety of non-trivial
problems (not known to have probabilistic polynomial-time
algorithms). The problems include Graph Isomorphism, Graph
Nonisomorphism, Quadratic Residuosity, Quadratic Nonresiduosity, a
restricted version of Statistical Difference, and approximate
versions of the (<b>coNP</b> forms of the) Shortest Vector ... more >>>

The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3-connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC^1.

In this paper we improve the upper bound for planar 3-connected graphs to unambiguous logspace, in fact to ... more >>>

We show that several reducibility notions coincide when applied to the
Graph Isomorphism (GI) problem. In particular we show that if a set is
many-one logspace reducible to GI, then it is in fact many-one AC^0
reducible to GI. For the case of Turing reducibilities we show that ... more >>>

We show that k-tree isomorphism can be decided in logarithmic
space by giving a logspace canonical labeling algorithm. This improves
over the previous StUL upper bound and matches the lower bound. As a
consequence, the isomorphism, the automorphism, as well as the
canonization problem for k-trees ... more >>>

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism} which has running time $b!2^{O(b)}N^{O(1)}$, where the parameter $b$ is the maximum size of the color classes of the given hypergraphs and $N$ is the input size. We also describe fpt algorithms for certain permutation group problems that ... more >>>

We present a logspace algorithm for computing a canonical interval representation and a canonical labeling of interval graphs. As a consequence, the isomorphism and automorphism problems for interval graphs are solvable in logspace.

The group isomorphism problem consists in deciding whether two groups $G$ and $H$
given by their multiplication tables are isomorphic.
An algorithm for group isomorphism attributed to Tarjan runs in time $n^{\log n + O(1)}$, c.f. [Mil78].

Miller and Monk showed in [Mil79] that group isomorphism can be many-one ... more >>>

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set $\mathcal{L}$ of matrices such that $A,B \in \mathcal{L}$ implies$AB - BA \in ... more >>>