An Experimental Study of Fast Greedy Algorithms for Fair Allocation Problems

Abstract

This paper is concerned with two salient allocationproblems in fair division of indivisible goods, aiming atmaximizing egalitarian and Nash product social welfare.These problems are computationally NP-hard, meaning thatachieving polynomial time algorithms is impossible, unlessP = NP. Approximation algorithms, which return near-optimalsolution with a theoretical guarantee, have been widely usedfor tackling the problems. However, most of them are often ofhigh computational complexity or not easy to implement. It istherefore of great interest to explore fast greedy methods thatcan quickly produce a good solution. This paper presents anempirical study of the performance of several such methods.Interestingly, the obtained results show that fair allocationproblems can be practically approximated by greedy algorithms.Keywords: Fair allocation, exact algorithm, greedy algorithm,mixed-integer linear program, NP-hard.I. INTRODUCTIONIn this paper, we study the fair allocation problem, whichhas shown its growing interest during last decades, with awide range of real-world applications [1]. In short, this is acombinatorial optimization problem which asks to allocate???? discrete items amongst a set of ???? agents (or players)so as to meet a certain notion of fairness. It is assumedthat every item is “indivisible” and “non-sharable”, thatis, i) it cannot be broken in pieces before allocating toagents, and ii) it cannot be shared by two or more agents.For example, laptops and cell-phones are indivisible itemswhich agents might not want to share with others. Anallocation of items to agents is simply a partition of thewhole set of items into ???? disjoint subsets. There are up to???????? such partitions, making the solution space large enoughso that an exhaustive search for an optimal solution isimpossible.It now remains to define what a fair allocation is, aconcept that is of independent interest in the field ofEconomic and Social Choice Theory [2, 3]. In general, thereare many different ways of defining fairness, depending onparticular applications. The most common way is to eitheruse a so-called Collective Utility Function (CUF), which isa function for aggregating individual agents’ utilities in afair manner, or to follow an orthogonal method relying ondetermining the fair share of agents. Since we are focusingon the first method in this paper, we refer the reader tothe paper [4] and the references therein for more details ofthe second method. Suppose that every agent evaluates thevalue of items through a utility function, which maps eachsubset of items to a numerical value representing the utilityof the agent for the subset. Then, one can define a maxmin fair allocation to be the one that maximizes the