Approximation Algorithms for Computing Maximin Share Allocations

Abstract

We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, hence, we resort to approximation algorithms. Our main result is a 2/3-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang [14], which also produces a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in [14], exploiting the construction of certain, appropriately selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances there exists a maximin share allocation with high probability. This can be seen as a justification of the experimental evidence reported in [5,14], that maximin share allocations exist almost always.Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of 3 agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of 2 agents). We provide a 7/8-approximation algorithm, improving the previously known result of 3/4 [14]. The second case is when all item values belong to {0, 1, 2}, extending the {0, 1} setting studied in [5]. We obtain an exact algorithm for any number of agents in this case.

Maximin Share Allocations on Cycles

“…This representation consists of nm + 1 rational numbers (one of them, m, an integer). 1 In the case when agents can be grouped into t types, where t is fixed and not part of input, the case we are now considering, input instances can be represented more concisely by a non-negative integer m, t sequences of m non-negative rational numbers (t utility functions) and t non-negative integers s 1 , . .…”

“…Both the result and the algorithm are based on deep combinatorial insights. Building on that work, Amanatidis, Markakis, Nikzad and Saberi [1] proved that for every constant ǫ, 0 < ǫ < 2/3, there is a polynomial-time algorithm finding an allocation guaranteeing to each agent at least (2/3 −ǫ) of A PRELIMINARY VERSION OF THE PAPER APPEARED IN THE PROCEEDINGS OF IJCAI 2018. 1 their maximin share (with the number of agents a part of the input).…”

“…Building on that work, Amanatidis, Markakis, Nikzad and Saberi [1] proved that for every constant ǫ, 0 < ǫ < 2/3, there is a polynomial-time algorithm finding an allocation guaranteeing to each agent at least (2/3 −ǫ) of A PRELIMINARY VERSION OF THE PAPER APPEARED IN THE PROCEEDINGS OF IJCAI 2018. 1 their maximin share (with the number of agents a part of the input). Ghodsi, Hajiaghayi, Seddighin, Seddighin and Yami [8] proved that in the results by Procaccia and Wang and by Amanatidis et al, the constant 2/3 can be replaced with 3/4.…”

“…Our main contributions are as follows. 1. We show that for goods on a cycle the maximin share value for an agent can be computed in polynomial time.…”

Strategyproof and Approximately Maxmin Fair Share Allocation of Chores

“…This work is partially supported by NSF CAREER Award No. 1553385. 2014; Amanatidis et al, 2015;Barman and Murthy, 2017;Ghodsi et al, 2018;Aziz et al, 2017]. None of these works took a mechanism design perspective to the problem of computing approximately MMS allocation.…”