Online Domination: The Value of Getting to Know All your Neighbors

TL;DR

This paper investigates the online dominating set problem where the algorithm learns the entire neighborhood of each revealed node, demonstrating tight competitive ratios on various graph classes and highlighting the value of full neighborhood knowledge.

Contribution

It introduces and analyzes an online dominating set algorithm with optimal competitiveness on trees and other graph classes, emphasizing the importance of full neighborhood information.

Findings

Achieves 2-competitiveness on trees, proven optimal.

Provides algorithms with tight competitive ratios for cactus, $K_{1,t}$-free, and maximum degree graphs.

Shows certain graph classes do not admit competitive algorithms, highlighting the impact of neighborhood knowledge.

Abstract

We study the dominating set problem in an online setting. An algorithm is required to guarantee competitiveness against an adversary that reveals the input graph one node at a time. When a node is revealed, the algorithm learns about the entire neighborhood of the node (including those nodes that have not yet been revealed). Furthermore, the adversary is required to keep the revealed portion of the graph connected at all times. We present an algorithm that achieves 2-competitiveness on trees and prove that this competitive ratio cannot be improved by any other algorithm. We also present algorithms that achieve 2.5-competitiveness on cactus graphs, $(t-1)$-competitiveness on $K_{1,t}$-free graphs, and $\Theta(\sqrt{\Delta})$ for maximum degree $\Delta$ graphs. We show that all of those competitive ratios are tight. Then, we study several more general classes of graphs, such as threshold, bipartite planar, and series-parallel graphs, and show that they do not admit competitive algorithms (that is, when competitive ratio is independent of the input size). Previously, the dominating set problem was considered in a slightly different input model, where a vertex is revealed alongside its restricted neighborhood: those neighbors that are among already revealed vertices. Thus, conceptually, our results quantify the value of knowing the entire neighborhood at the time a vertex is revealed as compared to the restricted neighborhood. For instance, it was known in the restricted neighborhood model that 3-competitiveness is optimal for trees, whereas knowing the neighbors allows us to improve it to 2-competitiveness.