Online Matroid Intersection: Beating Half for Random Arrival

We study the online matroid intersection problem, which is related to the well-studied online bipartite matching problem in the vertex arrival model. For two matroids M1 and M2 defined on the same ground set E, the problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniformly random order. At each step, the algorithm must irrevocably decide whether to pick the element, while always maintaining a common independent set. Since the greedy algorithm — pick the element whenever possible — has a competitive ratio of half, the natural question is whether we can beat half. This problem generalizes online bipartite matching in the edge arrival model where a random edge is presented at each step; nothing better than half competitiveness was previously known. In this paper, we present a simple randomized algorithm that has a 1 2 + δ competitive ratio in expectation, for a constant δ > 0. The expectation is over the randomness of the input order and the coin tosses of the algorithm. We also extend our result to intersection of k matroids and to general graphs, cases not captured by intersection of two matroids. Supported in part by NSF awards CCF-1319811 and CCF-1536002 Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA, e-mail: [email protected] Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA, e-mail: [email protected]