Author:
Dory Michal,Ghaffari Mohsen,Ilchi Saeed
Abstract
AbstractWe describe a simple deterministic $$O( \varepsilon ^{-1} \log \Delta )$$
O
(
ε
-
1
log
Δ
)
round distributed algorithm for $$(2\alpha +1)(1 + \varepsilon )$$
(
2
α
+
1
)
(
1
+
ε
)
approximation of minimum weighted dominating set on graphs with arboricity at most $$\alpha $$
α
. Here $$\Delta $$
Δ
denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $$O(\alpha ^2)$$
O
(
α
2
)
approximation in $$O(\log n)$$
O
(
log
n
)
rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic $$O(\alpha \log \Delta )$$
O
(
α
log
Δ
)
approximation in $$O(\log \Delta )$$
O
(
log
Δ
)
rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic $$O(\alpha )$$
O
(
α
)
approximation in $$O(\log ^2 \Delta )$$
O
(
log
2
Δ
)
rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized $$O(\alpha )$$
O
(
α
)
approximation in $$O(\alpha \log n)$$
O
(
α
log
n
)
rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized $$O(\alpha \log \Delta )$$
O
(
α
log
Δ
)
round distributed algorithm that sharpens the approximation factor to $$\alpha (1+o(1))$$
α
(
1
+
o
(
1
)
)
. If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $$\alpha - 1 - \varepsilon $$
α
-
1
-
ε
approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).
Funder
HORIZON EUROPE European Research Council,European Union
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Swiss Federal Institute of Technology Zurich
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Computer Networks and Communications,Hardware and Architecture,Theoretical Computer Science