Approximation algorithms for the partial assignment problem

Abstract

In the partial assignment problem, a seller has an item set M={i1,i2,…,im}, the amount of each item is exactly one. There are n buyers N={b1,b2,...,bn}, each buyer bp has a preferred bundle Bp⊆M and a value function fp(⋅). Assume that each item should be sold integrally and thus can be only assigned to at most one buyer. In previous works, buyers are often considered to be single-minded, i.e., a buyer can be either assigned the whole preferred bundle, or nothing. In this paper, we consider a more generalized and realistic model where the buyer can be partially satisfied, i.e., buyer bp can have some positive value if the seller assigns a subset of bp's preferred bundle. However, there might be exponential number of subsets, to tackle this situation, a value oracle is implemented. We can get the value fp(Sp) for buyer bp and Sp⊆Bp by querying the value oracle. The objective is to assign items to buyers such that the total values are maximized, i.e., max⁡∑p=1nfp(Sp). We first show that in this model, maximizing the total values is NP-hard. We then propose provably efficient approximation algorithms for general and submodular value functions respectively. If the value function satisfies non-negative, monotone and normalized, an 1/m-approximation algorithm can be achieved. If the value function is submodular, the total values can be approximated within a factor of (1−1/e).