Affiliation:
1. Toyota Technological Institute at Chicago, Chicago, IL
2. Princeton University
3. Carnegie Mellon University
Abstract
We study the
k
-route cut problem: given an undirected edge-weighted graph
G
= (
V
,
E
), a collection {(
s
1
,
t
1
), (
s
2
,
t
2
), …, (
s
r
,
t
r
)} of source-sink pairs, and an integer connectivity requirement
k
, the goal is to find a minimum-weight subset
E
′ of edges to remove, such that the connectivity of every pair (
s
i
,
t
i
) falls below
k
. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (
k
− 1) edge-disjoint paths connecting
s
i
to
t
i
in
G
∖
E
′, while in the vertex-connectivity version, VC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where
k
⩽ 3, but no non-trivial approximation algorithms were known for any value
k
> 3, except in the single-source setting. We show an
O
(
k
log
3/2
r
)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and VC-kRC, where the connectivity requirement
k
is violated by a constant factor. We complement these upper bounds by proving that VC-kRC is hard to approximate to within a factor of
k
ϵ
for some fixed ϵ > 0. We then turn to study a simpler version of VC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of VC-kRC has no constant-factor approximation, assuming Feige’s Random κ-AND assumption.
Funder
Sloan Research Fellowship
National Science Foundation
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
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