13,920 research outputs found
On the approximability of robust spanning tree problems
In this paper the minimum spanning tree problem with uncertain edge costs is
discussed. In order to model the uncertainty a discrete scenario set is
specified and a robust framework is adopted to choose a solution. The min-max,
min-max regret and 2-stage min-max versions of the problem are discussed. The
complexity and approximability of all these problems are explored. It is proved
that the min-max and min-max regret versions with nonnegative edge costs are
hard to approximate within for any unless
the problems in NP have quasi-polynomial time algorithms. Similarly, the
2-stage min-max problem cannot be approximated within unless the
problems in NP have quasi-polynomial time algorithms. In this paper randomized
LP-based approximation algorithms with performance ratio of for
min-max and 2-stage min-max problems are also proposed
Arc-Disjoint Paths and Trees in 2-Regular Digraphs
An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected
spanning subdigraph of D in which every vertex x != s has precisely one arc
entering (leaving) it and s has no arcs entering (leaving) it. We settle the
complexity of the following two problems:
1) Given a 2-regular digraph , decide if it contains two arc-disjoint
branchings B^+_u, B^-_v.
2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u
such that D remains connected after removing the arcs of B^+_u.
Both problems are NP-complete for general digraphs. We prove that the first
problem remains NP-complete for 2-regular digraphs, whereas the second problem
turns out to be polynomial when we do not prescribe the root in advance. We
also prove that, for 2-regular digraphs, the latter problem is in fact
equivalent to deciding if contains two arc-disjoint out-branchings. We
generalize this result to k-regular digraphs where we want to find a number of
pairwise arc-disjoint spanning trees and out-branchings such that there are k
in total, again without prescribing any roots.Comment: 9 pages, 7 figure
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