53 research outputs found
Canonizing Graphs of Bounded Tree Width in Logspace
Graph canonization is the problem of computing a unique representative, a
canon, from the isomorphism class of a given graph. This implies that two
graphs are isomorphic exactly if their canons are equal. We show that graphs of
bounded tree width can be canonized by logarithmic-space (logspace) algorithms.
This implies that the isomorphism problem for graphs of bounded tree width can
be decided in logspace. In the light of isomorphism for trees being hard for
the complexity class logspace, this makes the ubiquitous class of graphs of
bounded tree width one of the few classes of graphs for which the complexity of
the isomorphism problem has been exactly determined.Comment: 26 page
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
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
3-connected Planar Graph Isomorphism is in Log-space
We show that the isomorphism of 3-connected planar graphs can be decided in
deterministic log-space. This improves the previously known bound ULcoUL
of Thierauf and Wagner
Isomorphism testing of read-once functions and polynomials
In this paper, we study the isomorphism testing problem of formulas in
the Boolean and arithmetic settings. We show that isomorphism testing
of Boolean formulas in which a variable is read at most once (known as
read-once formulas) is complete for log-space. In contrast, we observe
that the problem becomes polynomial time equivalent to the graph
isomorphism problem, when the input formulas can be represented as OR
of two or more monotone read-once formulas. This classifies the
complexity of the problem in terms of the number of reads, as read-3
formula isomorphism problem is hard for coNP.
We address the polynomial isomorphism problem, a special case of
polynomial equivalence problem which in turn is important from a
cryptographic perspective[Patarin EUROCRYPT\u2796, and Kayal SODA\u2711]. As our main result, we propose a deterministic polynomial time
canonization scheme for polynomials computed by constant-free
read-once arithmetic formulas. In contrast, we show that when the
arithmetic formula is allowed to read a variable twice, this problem
is as hard as the graph isomorphism problem
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