30 research outputs found
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
[Bod90],[YBFT99]. 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. considering only isomorphisms 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 TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Log-space Algorithms for Paths and Matchings in k-trees
Reachability and shortest path problems are NL-complete for general graphs.
They are known to be in L for graphs of tree-width 2 [JT07]. However, for
graphs of tree-width larger than 2, no bound better than NL is known. In this
paper, we improve these bounds for k-trees, where k is a constant. In
particular, the main results of our paper are log-space algorithms for
reachability in directed k-trees, and for computation of shortest and longest
paths in directed acyclic k-trees.
Besides the path problems mentioned above, we also consider the problem of
deciding whether a k-tree has a perfect macthing (decision version), and if so,
finding a perfect match- ing (search version), and prove that these two
problems are L-complete. These problems are known to be in P and in RNC for
general graphs, and in SPL for planar bipartite graphs [DKR08].
Our results settle the complexity of these problems for the class of k-trees.
The results are also applicable for bounded tree-width graphs, when a
tree-decomposition is given as input. The technique central to our algorithms
is a careful implementation of divide-and-conquer approach in log-space, along
with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201
The Isomorphism Problem of Power Graphs and a Question of Cameron
The isomorphism problem for graphs (GI) and the isomorphism problem for
groups (GrISO) have been studied extensively by researchers. The current best
algorithms for both these problems run in quasipolynomial time. In this paper,
we study the isomorphism problem of graphs that are defined in terms of groups,
namely power graphs, directed power graphs, and enhanced power graphs. It is
not enough to check the isomorphism of the underlying groups to solve the
isomorphism problem of such graphs as the power graphs (or the directed power
graphs or the enhanced power graphs) of two nonisomorphic groups can be
isomorphic. Nevertheless, it is interesting to ask if the underlying group
structure can be exploited to design better isomorphism algorithms for these
graphs. We design polynomial time algorithms for the isomorphism problems for
the power graphs, the directed power graphs and the enhanced power graphs
arising from finite nilpotent groups. In contrast, no polynomial time algorithm
is known for the group isomorphism problem, even for nilpotent groups of class
2.
We note that our algorithm does not require the underlying groups of the
input graphs to be given. The isomorphism problems of power graphs and enhanced
power graphs are solved by first computing the directed power graphs from the
input graphs. The problem of efficiently computing the directed power graph
from the power graph or the enhanced power graph is due to Cameron [IJGT'22].
Therefore, we give a solution to Cameron's question.Comment: 23 page
The Entropy Influence Conjecture Revisited
In this paper, we prove that most of the boolean functions, satisfy the Fourier Entropy Influence (FEI) Conjecture
due to Friedgut and Kalai (Proc. AMS'96). The conjecture says that the Entropy
of a boolean function is at most a constant times the Influence of the
function. The conjecture has been proven for families of functions of smaller
sizes. O'donnell, Wright and Zhou (ICALP'11) verified the conjecture for the
family of symmetric functions, whose size is . They are in fact able
to prove the conjecture for the family of -part symmetric functions for
constant , the size of whose is . Also it is known that the
conjecture is true for a large fraction of polynomial sized DNFs (COLT'10).
Using elementary methods we prove that a random function with high probability
satisfies the conjecture with the constant as , for any constant
.Comment: We thank Kunal Dutta and Justin Salez for pointing out that our
result can be extended to a high probability statemen
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.by Bireswar Da
Algorithms for the Minimum Generating Set Problem
For a finite group , the size of a minimum generating set of is
denoted by . Given a finite group and an integer , deciding if
is known as the minimum generating set (MIN-GEN) problem.
A group of order has generating set of size
where is the smallest prime dividing . This fact is used to design
an -time algorithm for the group isomorphism problem of
groups specified by their Cayley tables (attributed to Tarjan by Miller, 1978).
The same fact can be used to give an -time algorithm for the
MIN-GEN problem. We show that the MIN-GEN problem can be solved in time
for general groups given by their Cayley tables. This
runtime incidentally matches with the runtime of the best known algorithm for
the group isomorphism problem.
We show that if a group , given by its Cayley table, is the product of
simple groups then a minimum generating set of can be computed in time
polynomial in .
Given groups along with for the problem of
computing is nontrivial. As a consequence of our result
for products of simple groups we show that this problem also can be solved in
polynomial time for Cayley table representation.
For the MIN-GEN problem for permutation groups, to the best of our knowledge,
no significantly better algorithm than the brute force algorithm is known. For
an input group , the brute force algorithm runs in time
which can be . We show that if is a primitive
permutation group then the MIN-GEN problem can be solved in time
quasi-polynomial in .
We also design a algorithm for computing a minimum
generating set of permutation groups all of whose non-abelian chief factors
have bounded orders.Comment: 20 page
SZK Proofs for Black-Box Group Problems
In this paper we classify several algorithmic problems in group theory in the classes PZK and SZK (problems with perfect/statistical zero-knowledge proofs respecticely). Prior to this, these problems were known to be in AM ∩ coAM. As PZK ⊆ SZK ⊆ AM ∩ coAM, we have a tighter upper bound for these problems. Specifically: • We show that the permutation group problems Coset Intersection, Double Coset Membership, Group Conjugacy are in PZK. Further, the complements of these problems also have perfect zero knowledge proofs (in the liberal sense). We also show that permutation group isomorphism for solvable groups is in PZK. As an ingredient of this protocol, we design a randomized algorithm for sampling short presentations of solvable permutation groups. • We show that the above problems for black-box groups are in SZK. • Finally, we also show that some of the problems have SZK protocols with efficient provers in the sense of [MV03]. Keyword Classification: Computational Complexity.