17,889 research outputs found
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
Diffusion determines the recurrent graph
We consider diffusion on discrete measure spaces as encoded by Markovian
semigroups arising from weighted graphs. We study whether the graph is uniquely
determined if the diffusion is given up to order isomorphism. If the graph is
recurrent then the complete graph structure and the measure space are
determined (up to an overall scaling). As shown by counterexamples this result
is optimal. Without the recurrence assumption, the graph still turns out to be
determined in the case of normalized diffusion on graphs with standard weights
and in the case of arbitrary graphs over spaces in which each point has the
same mass. These investigations provide discrete counterparts to studies of
diffusion on Euclidean domains and manifolds initiated by Arendt and continued
by Arendt/Biegert/ter Elst and Arendt/ter Elst. A crucial step in our
considerations shows that order isomorphisms are actually unitary maps (up to a
scaling) in our context.Comment: 30 page
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
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