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arXiv · 1512.06271

Online Matroid Intersection: Beating Half for Random Arrival

Guru Guruganesh·Sahil Singla
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Abstract

For two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ defined on the same ground set $E$, the online matroid intersection 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. While the natural greedy algorithm---pick an element whenever possible---is half competitive, nothing better was previously known; even for the special case of online bipartite matching in the edge arrival model. We present the first randomized online algorithm that has a $\frac12 + \delta$ competitive ratio in expectation, where $\delta >0$ is a constant. The expectation is over the random order and the coin tosses of the algorithm. As a corollary, we also obtain the first linear time algorithm that beats half competitiveness for offline matroid intersection.

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Guru Guruganesh, Sahil Singla. 2015-12-19. Online Matroid Intersection: Beating Half for Random Arrival. https://arxiv.org/abs/1512.06271

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