4,892 research outputs found
Approximating the Minimum Equivalent Digraph
The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994
Short Cycles Connectivity
Short cycles connectivity is a generalization of ordinary connectivity.
Instead by a path (sequence of edges), two vertices have to be connected by a
sequence of short cycles, in which two adjacent cycles have at least one common
vertex. If all adjacent cycles in the sequence share at least one edge, we talk
about edge short cycles connectivity.
It is shown that the short cycles connectivity is an equivalence relation on
the set of vertices, while the edge short cycles connectivity components
determine an equivalence relation on the set of edges. Efficient algorithms for
determining equivalence classes are presented.
Short cycles connectivity can be extended to directed graphs (cyclic and
transitive connectivity). For further generalization we can also consider
connectivity by small cliques or other families of graphs
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