Abstract
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 n1−ε-competitive, for any ε>0. Finally, we analyze the advice complexity of various graph classes for optimality.
Highlights
IntroductionAn algorithm gets an input and produces an output. In online computation, the scenario is different
In classic offline computation, an algorithm gets an input and produces an output
Besides giving some insight into what makes an online problem hard, advice complexity can be used as a means for proving lower bounds on the competitive ratio achievable by randomized online algorithms
Summary
An algorithm gets an input and produces an output. In online computation, the scenario is different. There is a huge gap between knowing nothing about the future and knowing everything about it To make this transition smoother, online computation with advice has been introduced as a new framework to analyze the lack of information of an online algorithm compared to an offline algorithm [8,9,18,22,26]. If the resulting graph is allowed to have isolated vertices, every online algorithm without advice (even when randomized) is forced to take every vertex into the dominating vertex that is isolated in the partial graph at the time it is presented; otherwise the algorithm risks to compute an infeasible solution.
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