Affiliation:
1. Department of Informatics, University of Bergen, Norway
2. Department of Computer Science, University of California, Santa Barbara, USA
3. LIRMM, University of Montpellier, CNRS, France
Abstract
We introduce the following submodular generalization of the
Shortest Cycle
problem. For a nonnegative monotone submodular cost function
f
defined on the edges (or the vertices) of an undirected graph
G
, we seek for a cycle
C
in
G
of minimum cost 𝖮𝖯𝖳 =
f(C)
. We give an algorithm that given an
n
-vertex graph
G
, parameter ɛ > 0, and the function
f
represented by an oracle, in time
n
𝒪
(log 1/ɛ) finds a cycle
C
in
G
with
f(C)
≤ (1+ɛ). 𝖮𝖯𝖳. This is in sharp contrast with the non-approximability of the closely related
Monotone Submodular Shortest (
s,t
-Path
problem, which requires exponentially many queries to the oracle for finding an
n
2/3-ɛ
-approximation Goel et al. [
7
], FOCS 2009. We complement our algorithm with a matching lower bound. We show that for every ɛ > 0, obtaining a (1+ɛ)-approximation requires at least
n
Ω (log 1/ ɛ)
queries to the oracle.
When the function
f
is integer-valued, our algorithm yields that a cycle of cost 𝖮𝖯𝖳 can be found in time
n
𝒪(log 𝖮𝖯𝖳)
. In particular, for 𝖮𝖯𝖳 =
n
𝒪(1)
this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that
n
𝒪(log
n
)
queries are required even when 𝖮𝖯𝖳= 𝒪(
n
).
We also consider special cases of monotone submodular functions, corresponding to the number of different color classes needed to cover a cycle in an edge-colored multigraph
G
. For special cases of the corresponding minimization problem, we obtain fixed-parameter tractable algorithms and polynomial-time algorithms, when restricted to certain classes of inputs.
Funder
Research Council of Norway via the project BWCA
ANR project ESIGMA
French-German Collaboration ANR/DFG Project UTMA
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)