Online Dominating Set

TL;DR

This paper systematically studies the online dominating set problem, analyzing how the constraints of irrevocability and incremental solution maintenance affect algorithm performance across various graph classes.

Contribution

It provides the first comprehensive analysis of the impact of incremental and irrevocable decision-making on online dominating set algorithms across multiple graph classes.

Findings

Tight bounds for trees, bipartite, planar, and general graphs.

Bounds for bounded-degree graphs.

Incrementality often explains the online algorithms' disadvantages.

Abstract

This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. We consider important graph classes, distinguishing between connected and not necessarily connected graphs. For the classic graph classes of trees, bipartite, planar, and general graphs, we obtain tight results in almost all cases. We also derive upper and lower bounds for the class of bounded-degree graphs. From these analyses, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm's disadvantage.