Approximation Algorithms for Max-Min Share Allocations of Indivisible Chores and Goods

Abstract

We consider Max-min Share (MmS) allocations of items both in the case whereitems are goods (positive utility) and when they are chores (negative utility).We show that fair allocations of goods and chores have some fundamentalconnections but differences as well. We prove that like in the case for goods,an MmS allocation does not need to exist for chores and computing an MmSallocation - if it exists - is strongly NP-hard. In view of these non-existenceand complexity results, we present a polynomial-time 2-approximation algorithmfor MmS fairness for chores. We then introduce a new fairness concept calledoptimal MmS that represents the best possible allocation in terms of MmS thatis guaranteed to exist. For both goods and chores, we use connections toparallel machine scheduling to give (1) an exponential-time exact algorithm and(2) a polynomial-time approximation scheme for computing an optimal MmSallocation when the number of agents is fixed