Biobjective Online Bipartite Matching

Motivated by Online Ad allocation when there are multiple conflicting objectives, we introduce and study the problem of Biobjective Online Bipartite Matching, a strict generalization of the standard setting of Karp, Vazirani and Vazirani [9], where we are allowed to have edges of two colors, and the goal is to find a matching that is both large and balanced at the same time. We study both deterministic and randomized algorithms for this problem; after showing that the single color upper bounds of 1/2 and 1− 1/e carry over to our biobjective setting as well, we show that a very natural, albeit hard to analyze, deterministic algorithm achieves a competitive ratio of 0.343. We next show how a natural randomized algorithm matches this ratio, through a simpler analysis, and how a clever – and perhaps not immediately obvious – generalization of Ranking can beat the 1/2 bound and get a competitive ratio of 0.573, coming close to matching the upper bound of 0.63.