Improved Bounds for Online Dominating Sets of Trees

TL;DR

This paper improves bounds for online dominating set algorithms on trees, presenting a matching lower bound for deterministic algorithms and a new randomized algorithm with a better competitive ratio.

Contribution

It establishes a tight lower bound for deterministic algorithms and introduces a randomized algorithm with a competitive ratio of 2.5 for online dominating sets of trees.

Findings

Deterministic algorithms have a lower bound of 2 on competitive ratio.

A randomized algorithm achieves a competitive ratio of at most 2.5.

Any randomized algorithm has a lower bound of 4/3 on competitive ratio.

Abstract

The online dominating set problem is an online variant of the minimum dominating set problem, which is one of the most important NP-hard problems on graphs. This problem is defined as follows: Given an undirected graph $G = (V, E)$, in which $V$ is a set of vertices and $E$ is a set of edges. We say that a set $D \subseteq V$ of vertices is a {\em dominating set} of $G$ if for each $v \in V \setminus D$, there exists a vertex $u \in D$ such that $\{ u, v \} \in E$. The vertices are revealed to an online algorithm one by one over time. When a vertex is revealed, edges between the vertex and vertices revealed in the past are also revealed. A revelaed subtree is connected at any time. Immediately after the revelation of each vertex, an online algorithm can choose vertices which were already revealed irrevocably and must maintain a dominating set of a graph revealed so far. The cost of an algorithm on a given tree is the number of vertices chosen by it, and its objective is to minimize the cost. Eidenbenz (Technical report, Institute of Theoretical Computer Science, ETH Z\"{u}rich, 2002) and Boyar et al.\ (SWAT 2016) studied the case in which given graphs are trees. They designed a deterministic online algorithm whose competitive ratio is at most three, and proved that a lower bound on the competitive ratio of any deterministic algorithm is two. In this paper, we also focus on trees. We establish a matching lower bound for any deterministic algorithm. Moreover, we design a randomized online algorithm whose competitive ratio is at most $5/2 = 2.5$, and show that the competitive ratio of any randomized algorithm is at least $4/3 \approx 1.333$.