232 research outputs found
Testing Cluster Structure of Graphs
We study the problem of recognizing the cluster structure of a graph in the
framework of property testing in the bounded degree model. Given a parameter
, a -bounded degree graph is defined to be -clusterable, if it can be partitioned into no more than parts, such
that the (inner) conductance of the induced subgraph on each part is at least
and the (outer) conductance of each part is at most
, where depends only on . Our main
result is a sublinear algorithm with the running time
that takes as
input a graph with maximum degree bounded by , parameters , ,
, and with probability at least , accepts the graph if it
is -clusterable and rejects the graph if it is -far from
-clusterable for , where depends only on . By the lower
bound of on the number of queries needed for testing graph
expansion, which corresponds to in our problem, our algorithm is
asymptotically optimal up to polylogarithmic factors.Comment: Full version of STOC 201
Helly-Type Theorems in Property Testing
Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If is a set of
points in , we say that is -clusterable if it can be
partitioned into clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object . In this paper, as an
application of Helly's theorem, by taking a constant size sample from , we
present a testing algorithm for -clustering, i.e., to distinguish
between two cases: when is -clusterable, and when it is
-far from being -clusterable. A set is -far
from being -clusterable if at least
points need to be removed from to make it -clusterable. We solve
this problem for and when is a symmetric convex object. For , we
solve a weaker version of this problem. Finally, as an application of our
testing result, in clustering with outliers, we show that one can find the
approximate clusters by querying a constant size sample, with high probability
Six signed Petersen graphs, and their automorphisms
Up to switching isomorphism there are six ways to put signs on the edges of
the Petersen graph. We prove this by computing switching invariants, especially
frustration indices and frustration numbers, switching automorphism groups,
chromatic numbers, and numbers of proper 1-colorations, thereby illustrating
some of the ideas and methods of signed graph theory. We also calculate
automorphism groups and clusterability indices, which are not invariant under
switching. In the process we develop new properties of signed graphs,
especially of their switching automorphism groups.Comment: 39 pp., 7 fi
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