Bireswar Das

Restricted space algorithms for isomorphism on bounded treewidth graphs

The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time. We give restricted space algorithms for these problems proving the following results:*Isomorphism for bounded tree distance width graphs is in L and thus complete for the class. We also show that for this kind of graphs a canon can be computed within logspace. *For bounded treewidth graphs, when both input graphs are given together with a tree decomposition, the problem of whether there is an isomorphism which respects the decompositions (i.e. when only isomorphisms are considered, mapping bags in one decomposition blockwise onto bags in the other decomposition) is in L. *For bounded treewidth graphs, when one of the input graphs is given with a tree decomposition the isomorphism problem is in LogCFL. *As a corollary the isomorphism problem for bounded treewidth graphs is in LogCFL. This improves the known TC^1 upper bound for the problem given by Grohe and Verbitsky.

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism, denoted CHI, which has running time (2bN)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 an fpt algorithm for a parameterized coset intersection problem that is used as a subroutine in our algorithm for CHI.

The space complexity of k-tree isomorphism

We show that isomorphism testing of k-trees is in the class StUSPACE(log n) (strongly unambiguous logspace). This bound follows from a deterministic logspace algorithm that accesses a strongly unambiguous logspace oracle for canonizing k-trees. Further we give a logspace canonization algorithm for k-paths.

Zero Knowledge and Circuit Minimization

We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RP MCSP .

The isomorphism problem for k-trees is complete for logspace

We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving an O(klogn) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes Lindell@?s tree canonization algorithm. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. Completeness for logspace holds even for simple structural properties of k-trees. We also show that a variant of our canonical labeling algorithm runs in time O((k+1)!n), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism.

Isomorphism and canonization of tournaments and hypertournaments

We give a polynomial-time oracle algorithm for Tournament Canonization that accesses oracles for Tournament Isomorphism and Rigid-Tournament Canonization. Extending the Babai-Luks Tournament Canonization algorithm (Babai and Luks (1983) [4]), we give an n^O^(^k^^^2^+^l^o^g^n^) algorithm for canonization and isomorphism testing of k-hypertournaments, where n is the number of vertices and k is the size of hyperedges.

On isomorphism and canonization of tournaments and hypertournaments

We give a polynomial-time oracle algorithm for Tournament Canonization that accesses oracles for Tournament Isomorphism and Rigid-Tournament Canonization. Extending the Babai-Luks Tournament Canonization algorithm, we give an nO(k+logn) algorithm for canonization and isomorphism testing of k-hypertournaments, where n is the number of vertices and k is the size of hyperedges.

Logspace and FPT Algorithms for Graph Isomorphism for Subclasses of Bounded Tree-Width Graphs

We give a deterministic logspace algorithm for the graph isomorphism problem for graphs with bounded tree-depth. We also show that the graph isomorphism problem is fixed parameter tractable for a related parameterized graph class where the graph parameter is the length of the longest cycle.

Succinct Encodings of Graph Isomorphism

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity. We introduce a new way for encoding graph problems, based on CNF or DNF formulas. We show that contrary to the other existing succinct models, there are examples of problems whose complexity does not increase when encoded in the new form, or increases to an intermediate complexity class less powerful than the exponential blow up. We also study the complexity of the succinct versions of the Graph Isomorphism problem. We show that all the versions are hard for PSPACE. Although the exact complexity of these problems is not known, we show that under most existing succinct models the different versions of the problem are equivalent. We also give an algorithm for the DNF encoded version of GI whose running time depends only on the size of the succinct representation.

Log-space Algorithms for Paths and Matchings in k-trees

Reachability and shortest path problems are \NLC\ for general graphs. They are known to be in \Log\ for graphs of tree-width