Near-optimal asymmetric binary matrix partitions

TL;DR

This paper introduces approximation algorithms for the asymmetric binary matrix partition problem, improving previous results by providing near-optimal solutions for both uniform and non-uniform distributions using combinatorial and submodular welfare maximization techniques.

Contribution

It presents a simple combinatorial 9/10-approximation and a (1-1/e)-approximation for non-uniform distributions, advancing the understanding of the problem's approximability.

Findings

Achieved a 9/10-approximation for uniform distributions.

Developed a (1-1/e)-approximation for non-uniform distributions.

Connected the problem to submodular welfare maximization.

Abstract

We study the asymmetric binary matrix partition problem that was recently introduced by Alon et al. (WINE 2013) to model the impact of asymmetric information on the revenue of the seller in take-it-or-leave-it sales. Instances of the problem consist of an $n \times m$ binary matrix $A$ and a probability distribution over its columns. A partition scheme $B=(B_1,...,B_n)$ 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.