2 research outputs found
Tight approximation bounds for combinatorial frugal coverage algorithms
We consider the frugal coverage problem, an interesting variation of set cover defined as follows. Instances of the problem consist of a universe of elements and a collection of sets over these elements; the objective is to compute a subcollection of sets so that the number of elements it covers plus the number of sets not chosen is maximized. The problem was introduced and studied by Huang and Svitkina (Proceedings of the 29th IARCS annual conference on foundations of software technology and theoretical computer science (FSTTCS), pp. 227β238, 2009) due to its connections to the donation center location problem. We prove that the greedy algorithm has approximation ratio at least 0.782, improving a previous bound of 0.731 in Huang and Svitkina (Proceedings of the 29th IARCS annual conference on foundations of software technology and theoretical computer science (FSTTCS), pp. 227β238, 2009). We also present a further improvement that is obtained by adding a simple corrective phase at the end of the execution of the greedy algorithm. The approximation ratio achieved in this way is at least 0.806. Finally, we consider a packing based algorithm that uses semi-local optimization, and show that its approximation ratio is not less than 0.872. Our analysis is based on the use of linear programs which capture the behavior of the algorithms in worst-case examples. The obtained bounds are proved to be tight
Near-optimal asymmetric binary matrix partitions
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 binary matrix and a
probability distribution over its columns. A partition scheme
consists of a partition for each row of . The partition acts
as a smoothing operator on row that distributes the expected value of each
partition subset proportionally to all its entries. Given a scheme that
induces a smooth matrix , the partition value is the expected maximum
column entry of . The objective is to find a partition scheme such that
the resulting partition value is maximized. We present a -approximation
algorithm for the case where the probability distribution is uniform and a
-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.Comment: 17 page