625 research outputs found
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a logspace algorithm that constructs a canonical intersection
model for a given proper circular-arc graph, where `canonical' means that
models of isomorphic graphs are equal. This implies that the recognition and
the isomorphism problems for this class of graphs are solvable in logspace. For
a broader class of concave-round graphs, that still possess (not necessarily
proper) circular-arc models, we show that those can also be constructed
canonically in logspace. As a building block for these results, we show how to
compute canonical models of circular-arc hypergraphs in logspace, which are
also known as matrices with the circular-ones property. Finally, we consider
the search version of the Star System Problem that consists in reconstructing a
graph from its closed neighborhood hypergraph. We solve it in logspace for the
classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio
Average-case intractability vs. worst-case intractability
AbstractWe show that not all sets in NP (or other levels of the polynomial-time hierarchy) have efficient average-case algorithms unless the Arthur-Merlin classes MA and AM can be derandomized to NP and various subclasses of P/poly collapse to P. Furthermore, other complexity classes like P(PP) and PSPACE are shown to be intractable on average unless they are easy in the worst case
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
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