2 research outputs found
Simple and Fast Rounding Algorithms for Directed and Node-weighted Multiway Cut
In Directed Multiway Cut(Dir-MC) the input is an edge-weighted directed graph
and a set of terminal nodes ;
the goal is to find a min-weight subset of edges whose removal ensures that
there is no path from to for any . In Node-weighted
Multiway Cut(Node-MC) the input is a node-weighted undirected graph and a
set of terminal nodes ; the goal is to
remove a min-weight subset of nodes to disconnect each pair of terminals.
Dir-MC admits a -approximation [Naor, Zosin '97] and Node-MC admits a
-approximation [Garg, Vazirani, Yannakakis '94], both via
rounding of LP relaxations. Previous rounding algorithms for these problems,
from nearly twenty years ago, are based on careful rounding of an "optimum"
solution to an LP relaxation. This is particularly true for Dir-MC for which
the rounding relies on a custom LP formulation instead of the natural distance
based LP relaxation [Naor, Zosin '97].
In this paper we describe extremely simple and near linear-time rounding
algorithms for Dir-MC and Node-MC via a natural distance based LP relaxation.
The dual of this relaxation is a special case of the maximum multicommodity
flow problem. Our algorithms achieve the same bounds as before but have the
significant advantage in that they can work with "any feasible" solution to the
relaxation. Consequently, in addition to obtaining "book" proofs of LP rounding
for these two basic problems, we also obtain significantly faster approximation
algorithms by taking advantage of known algorithms for computing near-optimal
solutions for maximum multicommodity flow problems. We also investigate lower
bounds for Dir-MC when and in particular prove that the integrality gap
of the LP relaxation is even in directed planar graphs
Approximating Multicut and the Demand Graph
In the minimum Multicut problem, the input is an edge-weighted supply graph
and a simple demand graph . Either and are directed
(DMulC) or both are undirected (UMulC). The goal is to remove a minimum weight
set of edges in such that there is no path from to in the remaining
graph for any . UMulC admits an -approximation where
is the vertex cover size of while the best known approximation for
DMulC is . These approximations are obtained
by proving corresponding results on the multicommodity flow-cut gap. In
contrast to these results some special cases of Multicut, such as the
well-studied Multiway Cut problem, admit a constant factor approximation in
both undirected and directed graphs. Motivated by both concrete instances from
applications and abstract considerations, we consider the role that the
structure of the demand graph plays in determining the approximability of
Multicut.
In undirected graphs our main result is a -approximation in
time when the demand graph excludes an induced matching of size . This
gives a constant factor approximation for a specific demand graph that
motivated this work.
In contrast to undirected graphs, we prove that in directed graphs such
approximation algorithms can not exist. Assuming the Unique Games Conjecture
(UGC), for a large class of fixed demand graphs DMulC cannot be approximated to
a factor better than worst-case flow-cut gap. As a consequence we prove that
for any fixed , assuming UGC, DMulC with demand pairs is hard to
approximate to within a factor better than . On the positive side, we prove
an approximation of when the demand graph excludes certain graphs as an
induced subgraph. This generalizes the Multiway Cut result to a much larger
class of demand graphs