Biobjective online bipartite matching

Abstract

Online Matching has been a problem of considerable interest recently, particularly due to its applicability in Online Ad Allocation. In practice, there are usually multiple objectives which need to be simultaneously optimized, e.g., revenue and quality. We capture this motivation by introducing the problem of Biobjective Online Bipartite Matching. This is a strict generalization of the standard setting. In our problem, the graph has edges of two colors, Red and Blue. The goal is to find a single matching that contains a large number of edges of each color. We first show how this problem is a departure from previous settings: In all previous problems, the Greedy algorithm gives a non-trivial ratio, typically 1/2. In the biobjective problem, we show that the competitive ratio of Greedy is 0, and in fact, any reasonable algorithm would have to skip vertices, i.e., not match some incoming vertices even though they have an edge available. As our main result, we introduce an algorithm which randomly discards some edges of the graph in a particular manner – thus enabling the necessary skipping of vertices – and simultaneously runs the coloroblivious algorithm Ranking. We prove that this algorithm achieves a competitive ratio of 3 - 4/√e ≃ 0.573 for graphs which have a perfect matching of each color. This beats the upper bound of 1/2 for deterministic algorithms, and comes close to the upper bound of 1-1/√e ≃ 0.63 for randomized algorithms, both of which we prove carry over to the bicriteria setting, even with the perfect matching restriction. The technical difficulty lies in analyzing the expected minimum number of blue and red edges in the matching (rather than the minimum of the two expectations). To achieve this, we introduce a charging technique which has a new locality property, i.e., misses are charged to nearby hits, according to a certain metric. Along the way we develop and analyze simpler algorithms for the problem: a deterministic algorithm which achieves a ratio of 0.343, and a simpler randomized algorithm, which achieves, intriguingly, precisely the same ratio.