Near-Optimal Asymmetric Binary Matrix Partitions [chapter]

Fidaa Abed, Ioannis Caragiannis, Alexandros A. Voudouris
2015 Lecture Notes in Computer Science  
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 takeit-or-leave-it sales. Instances of the problem consist of an n × 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
more » ... d 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 9/10approximation 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. The objective of the asymmetric binary matrix partition problem is to find a partition scheme B such that the resulting partition value v B (A, p) is maximized. Alon et al. [2] were the first to consider the asymmetric matrix partition problem. They proved that the problem is APX-hard and provided a 0.563and a 1/13-approximation for uniform and nonuniform probability distributions, respectively. They also considered input matrices with non-negative non-binary entries and presented a 1/2and an Ω(1/ log m)-approximation algorithm for uniform and non-uniform distributions, respectively. This interesting combinatorial optimization problem has apparent relations to revenue maximization in take-it-or-leave-it sales. For example, consider the following setting. There are m items and n potential buyers. Each buyer has a value for each item. Nature selects at random (according to some probability distribution) an item for sale and, then, the seller approaches the highest valuation buyer and offers the item to her at a price equal to her valuation. Can the seller exploit the fact that she has much more accurate information about the items for sale compared to the potential buyers? In particular, information asymmetry arises since the seller knows the realization of the randomly selected item whereas the buyers do not. The approach that is discussed in [2] is to let the seller define a buyer-specific signaling scheme. That is, for each buyer, the seller can partition the set of items into disjoint subsets (bundles) and report this partition to the buyer. After nature's random choice, the seller can reveal to each buyer the bundle that contains the realization, thus enabling her to update her valuation beliefs. The relation of this problem to asymmetric matrix partition should be apparent. Interestingly, the seller can achieve revenue from items for which no buyer has any value. This scenario falls within the line of research that studies the impact of information asymmetry to the quality of markets. Akerlof [1] was the first to introduce a formal analysis of "markets of lemons", where the seller has more information than the buyers regarding the quality of the products. Crawford and Sobel [7] studied how, in such markets, the seller can exploit her advantage in order to maximize revenue. In [17] , Milgrom and Weber provided the "Linkage Principle" which states that the expected revenue is enhanced when bidders are provided with more information. This principle seems to suggest full transparency but, in [15] and [16] the authors suggest that careful bundling of the items is the best way to exploit information asymmetry. Many different frameworks that reveal information to the bidders have been proposed in the literature. More recently, Ghosh et al. [12] considered full information and proposed a clustering scheme according to which, the items are partitioned into bundles and then, for each such bundle, a separate second-price auction is performed. In this way, the potential buyers cannot bid only for the items that they actually want; they also have to compete for items that they do not care. Hence, the demand for each item is increased and the revenue generated is more. Emek et al. [10] present complexity results in similar settings and Miltersen and Sheffet [19] considered fractional bundling schemes for signaling. In this work we focus on the simplest binary case of asymmetric matrix partition which has been proved to be APX-hard. We present a 9/10-approximation algorithm for the uniform case and a (1 − 1/e)-approximation algorithm for non-uniform distributions. Both results significantly improve previous bounds of Alon et al. [2] . The analysis of our first algorithm is quite interesting because, despite its purely combinatorial nature, it exploits linear programming techniques. Similar techniques have been used in a series of papers on variants of set cover (e.g. [3, 4, 5, 6] ) by the second author; however, the application of the technique in the current context requires a quite involved reasoning about the structure of the solutions computed by the algorithm. In our second result, we exploit a nice relation of the problem to submodular welfare maximization and use well-known algorithms from the literature. First, we discuss the application of a simple greedy 1/2-approximation algorithm that has been studied by Lehmann et al. [14] and then apply Vondrák's smooth greedy algorithm [20] to achieve a (1 − 1/e)-approximation. Vondrák's algorithm is optimal in the value query model as Khot et al. [13] have proved. In a more powerful model where it is assumed that demand queries can be answered efficiently, Feige and Vondrák [11] have proved that (1 − 1/e + ǫ)approximation algorithms -where ǫ is a small positive constant -are possible. We briefly discuss the
doi:10.1007/978-3-662-48054-0_1 fatcat:pwrjxinozffvjf7sjsrbf7hdj4