# Near-Optimal Asymmetric Binary Matrix Partitions

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## Abstract

We study the asymmetric binary matrix partition problem that was recently introduced by Alon et al. (Proceedings of the 9th Conference on Web and Internet Economics (WINE), pp 1–14, 2013). Instances of the problem consist of an $$n \times m$$ binary matrix A and a probability distribution over its columns. A partition scheme $$B=(B_1,\ldots ,B_n)$$ consists of a partition $$B_i$$ for each row i of A. The partition $$B_i$$ acts as a smoothing operator on row i that distributes the expected value of each partition subset proportionally to all its entries. Given a scheme B that induces a smooth matrix $$A^B$$, the partition value is the expected maximum column entry of $$A^B$$. The objective is to find a partition scheme such that the resulting partition value is maximized. We present a 9/10-approximation algorithm for the case where the probability distribution is uniform and a $$(1-1/e)$$-approximation algorithm for non-uniform distributions, significantly improving results of Alon et al. Although our first algorithm is combinatorial (and very simple), the analysis is based on linear programming and duality arguments. In our second result we exploit a nice relation of the problem to submodular welfare maximization.

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## Notes

1. Invoking Lemma 2.2 in order to prove property P2 is crucial here; verifying properties P3 and P4 is much easier.

2. As an example of such an extreme case, consider an instance with a $$k\times (k+1)$$ matrix that consists of the identity $$k\times k$$ matrix and an extra zero-column, and has a uniform probability distribution over the columns. The optimal partition scheme contains a full cover and all-zero bundles only, and the zero-column has no contribution to the partition value.

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## Author information

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Correspondence to Ioannis Caragiannis.

A preliminary version of the results in this paper appeared in the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS ’15). This work was partially supported by the European Social Fund and Greek national funds through the research funding program Thales on “Algorithmic Game Theory”, and by the Caratheodory research Grant E.114 from the University of Patras.

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Abed, F., Caragiannis, I. & Voudouris, A.A. Near-Optimal Asymmetric Binary Matrix Partitions. Algorithmica 80, 48–72 (2018). https://doi.org/10.1007/s00453-016-0238-4