3,891 research outputs found
The complexity of two graph orientation problems
This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol
Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut
The multiway-cut problem is, given a weighted graph and k >= 2 terminal
nodes, to find a minimum-weight set of edges whose removal separates all the
terminals. The problem is NP-hard, and even NP-hard to approximate within
1+delta for some small delta > 0.
Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance
guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this
paper, we give improved randomized rounding schemes for their relaxation,
yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation
algorithm in general.
Our approach hinges on the observation that the problem of designing a
randomized rounding scheme for a geometric relaxation is itself a linear
programming problem. The paper explores computational solutions to this
problem, and gives a proof that for a general class of geometric relaxations,
there are always randomized rounding schemes that match the integrality gap.Comment: Conference version in ACM Symposium on Theory of Computing (1999). To
appear in Mathematics of Operations Researc
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