On the advice complexity of the online dominating set problem

A dominating set S of a graph is a set of vertices such that each vertex is in S or has a neighbor in S. The goal of the dominating set problem is to find such a set of minimum cardinality. In the online setting, the graph is revealed vertex by vertex, together with edges to all previously revealed vertices. Advice complexity is a framework to measure the amount of information an online algorithm is lacking. Here, an online algorithm reads advice bits from an infinite binary tape prepared beforehand by an all-knowing oracle. The advice complexity is the total number of advice bits read during the computation. Besides giving some insight into what makes an online problem hard, advice complexity can also be used as a means for proving lower bounds on the competitive ratio achievable by randomized online algorithms. We analyze the advice complexity of the online dominating set problem. For general graphs, we show tight upper and lower bounds for optimality. Then, we use a result for c-competitiveness to prove that no randomized online algorithm can be better than n(1-epsilon)-competitive, for any epsilon > 0. Finally, we analyze the advice complexity of various graph classes for optimality. (C) 2021 The Author(s). Published by Elsevier B.V.