Affiliation:
1. University of Birmingham, Edgbaston, Birmingham, UK
2. Charles University, Praha, Czechia
3. University of California, Berkeley, CA, USA
Abstract
The D
irected
S
teiner
N
etwork
(DSN) problem takes as input a directed graph
G
=(
V
,
E
) with non-negative edge-weights and a set
D
⊆
V
×
V
of
k
demand pairs. The aim is to compute the cheapest network
N⊆ G
for which there is an
s\rightarrow t
path for each (
s
,
t
)∈ D. It is known that this problem is notoriously hard, as there is no
k
1/4−
o
(1)
-approximation algorithm under Gap-ETH, even when parametrizing the runtime by
k
[Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter
k
.
For the
bi
-DSNP
lanar
problem, the aim is to compute a solution
N⊆ G
whose cost is at most that of an optimum planar solution in a bidirected graph
G
, i.e., for every edge
uv
of
G
the reverse edge
vu
exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for
k
. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP
lanar
, under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of
bi
-DSNP
lanar
and obtain upper and lower bounds on obtainable runtimes parameterized by
k
.
One important special case of DSN is the S
trongly
C
onnected
S
teiner
S
ubgraph
(SCSS) problem, for which the solution network
N⊆ G
needs to strongly connect a given set of
k
terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter
k
[Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for
k
no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for
k
.
Funder
Grantová Agentura, Univerzita Karlova
European Research Council
Grantová Agentura České Republiky
Israel Science Foundation
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
2 articles.
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