Allocating indivisible goods

Abstract

The problem of allocating divisible goods has enjoyed a lot of attention in both mathematics (e.g. the cake-cutting problem) and economics (e.g. market equilibria). On the other hand, the natural requirement of indivisible goods has been somewhat neglected, perhaps because of its more complicated nature. In this work we study a fairness criterion, called the Max-Min Fairness problem, for k players who want to allocate among themselves m indivisible goods. Each player has a specified valuation function on the subsets of the goods and the goal is to split the goods between the players so as to maximize the minimum valuation. Viewing the problem from a game theoretic perspective, we show that for two players and additive valuations the expected minimum of the (randomized) cut-and-choose mechanism is a 1/2-approximation of the optimum. To complement this result we show that no truthful mechanism can compute the exact optimum.We also consider the algorithmic perspective when the (true) additive valuation functions are part of the input. We present a simple 1/( m - k + 1) approximation algorithm which allocates to every player at least 1/ k fraction of the value of all but the k - 1 heaviest items. We also give an algorithm with additive error against the fractional optimum bounded by the value of the largest item. The two approximation algorithms are incomparable in the sense that there exist instances when one outperforms the other.

On Low-Envy Truthful Allocations

“…Here, the objective is to compute an allocation in which the benefit of the least happy player is maximized. The problem was studied by Bezáková and Dani [3] and Golovin [14] who obtained approximation algorithms that provably return a solution that is always a factor of O(n) within the optimal value. The problem was popularized by Bansal and Sviridenko [2] as the Santa Claus problem, where Santa Claus aims to distribute presents to the kids so as to maximize the happiness of the least happy kid.…”

A General Framework for Designing Approximation Schemes for Combinatorial Optimization Problems with Many Objectives Combined into One

“…The Santa Claus problem was first studied by Lipton et al [11]. Bezàkovà and Dani [12] proposed a linear factor approximation algorithm for this problem, which is based on a linear programming relaxation and rounding; our rounding procedure is similar to the rounding procedure used in their paper. Bansal and Sviridenko [10] obtained a tighter approximation algorithm for the restricted assignment version of the problem, where each resource can be allocated to only a subset of the agents, and each such agent has the same utility for that resource.…”

“…A closely related problem is the Santa Claus problem [10,11,12]. In this problem, each agent has a utility corresponding to each resource allocated to it, and the objective is to allocate the resources among the agents so that the minimum utility over all the agents is maximized.…”

“…The algorithm makes use of the well-known technique of parametric linear programming and rounding, which has been successfully used in obtaining approximation algorithm for scheduling problems in the past [3]. Our rounding procedure, however, differs from the one given in [3]; it is more similar to the one used by Bezàkovà and Dani [12] for the Santa Claus problem.…”

Assigning Papers to Referees

“…On the positive side, for = 2 we show that both problems can be solved in polynomial time by establishing a connection to a variant of rank-maximal matchings [14,20]. For larger values of in the case of the weighted preference order we give approximation algorithms building upon ideas of Bezáková and Dani [5], and Shmoys and Tardos [25].…”