Affiliation:
1. Aalto University, Finland
2. Hamburg University of Technology, Institute for Algorithms and Complexity, Germany
3. University of Sheffield, United Kingdom
4. École Normale Supérieure Paris and Université Paris Cité, CNRS, IRIF, France
Abstract
The fundamental Sparsest Cut problem takes as input a graph
G
together with edge capacities and demands and seeks a cut that minimizes the ratio between the capacities and demands across the cuts. For
n
-vertex graphs
G
of treewidth
k
, Chlamtáč, Krauthgamer, and Raghavendra (APPROX’10) presented an algorithm that yields a factor-
\(2^{2^k}\)
approximation in time
\(2^{O(k)} \cdot n^{O(1)}\)
. Later, Gupta, Talwar, and Witmer (STOC’13) showed how to obtain a 2-approximation algorithm with a blown-up runtime of
\(n^{O(k)}\)
. An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-2 approximation in time
\(2^{O(k)} \cdot n^{O(1)}\)
.
In this article, we make significant progress towards this goal via the following results:
(i)
A factor-
\(O(k^2)\)
approximation that runs in time
\(2^{O(k)} \cdot n^{O(1)}\)
, directly improving the work of Chlamtáč et al. while keeping the runtime single-exponential in
k
.
(ii)
For any
\(\varepsilon \in (0,1]\)
, a factor-
\(O(1/\varepsilon ^2)\)
approximation whose runtime is
\(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)}\)
, implying a constant-factor approximation whose runtime is nearly single-exponential in
k
and a factor-
\(O(\log ^2 k)\)
approximation in time
\(k^{O(k)} \cdot n^{O(1)}\)
.
Key to these results is a new measure of a tree decomposition that we call
combinatorial diameter
, which may be of independent interest.
Funder
European Research Council
European Union’s Horizon 2020
Academy of Finland Research Fellowship
French National Research Agency
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)