1 research outputs found
Minimum Certificate Dispersal with Tree Structures
Given an n-vertex graph G=(V,E) and a set R \subseteq {{x,y} | x,y \in V} of
requests, we consider to assign a set of edges to each vertex in G so that for
every request {u, v} in R the union of the edge sets assigned to u and v
contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is
defined as one to find an assignment that minimizes the sum of the cardinality
of the edge set assigned to each vertex. This problem has been shown to be
LOGAPX-complete for the most general setting, and APX-hard and 2-approximable
in polynomial time for dense request sets, where R forms a clique. In this
paper, we investigate the complexity of MCD with sparse (tree) structures. We
first show that MCD is APX-hard when R is a tree, even a star. We then explore
the problem from the viewpoint of the maximum degree \Delta of the tree: MCD
for tree request set with constant \Delta is solvable in polynomial time, while
that with \Delta=\Omega(n) is 2.56-approximable in polynomial time but hard to
approximate within 1.01 unless P=NP. As for the structure of G itself, we show
that the problem can be solved in polynomial time if G is a tree