Approximation and Competitive Algorithms for Single-Minded Selling Problem

Abstract

The problem of item selling with the objective of maximizing the revenue is studied. Given a seller with k types of items and n single-minded buyers, i.e., each buyer is only interested in a particular bundle of items, to maximize the revenue, the seller must carefully assign some amount of bundles to each buyer with respect to the buyer’s accepted price. Each buyer $b_i$ is associated with a value function $v_i(\cdot )$ such that $v_i(x)$ is the accepted unit bundle price $b_i$ is willing to pay for x bundles. We show that the single-minded item selling problem is NP-hard. Moreover, we give an $O(\sqrt{k})$-approximation algorithm. For the online version, i.e., the buyers come one by one and the decision on each buyer must be made before the arrival of the next buyer, an $O(\sqrt{k}\cdot (\log h + \log k))$-competitive algorithm is achieved, where h is the highest unit item price among all buyers.