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Online Matroid Intersection: Beating Half for Random Arrival

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Integer Programming and Combinatorial Optimization (IPCO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10328))

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 \(\frac{1}{2} + \delta \) 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.

Supported in part by NSF awards CCF-1319811, CCF-1536002, and CCF-1617790.

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Notes

  1. 1.

    We emphasize that our definition also works when i and j are non-integral.

  2. 2.

    We also show that for regular graphs Greedy is at least \(\left( 1- \frac{1}{e} \right) \) competitive, and that no online algorithm for OBME can be better than \(\frac{69}{84} \approx 0.821\) competitive.

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Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, Pittsburgh, USA

    Guru Prashanth Guruganesh & Sahil Singla

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  1. Guru Prashanth Guruganesh
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  2. Sahil Singla
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Correspondence to Sahil Singla .

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Editors and Affiliations

  1. Ecole Polytechnique Federale de Lausanne - EPFL, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. University of Waterloo, Waterloo, Ontario, Canada

    Jochen Koenemann

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Guruganesh, G.P., Singla, S. (2017). Online Matroid Intersection: Beating Half for Random Arrival. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_20

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  • DOI: https://doi.org/10.1007/978-3-319-59250-3_20

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