Near-optimal distributed dominating set in bounded arboricity graphs

Abstract

Abstract We 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).