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Computer Science, Computer Science and Game Theory, cs.GT,Computer Science, Computational Complexity, cs.CC
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.