An Improved Approximation Algorithm for Maximin Shares
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We study the problem of fair allocation of m indivisible items among n agents with additive valuations using the popular notion of maximin share as our measure of fairness. The maximin share (MMS) of an agent is the maximum she can guarantee herself if she is allowed to choose a partition of items into n bundles (one for each agent), with the condition that all other agents get to choose a bundle before her. An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist, a series of remarkable work provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times her maximin share. Recently, in a groundbreaking work, Ghodsi et al. showed the existence of 3/4 MMS allocation and a PTAS to find a 3/4-ϵ MMS allocation. Most of the previous work utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. We present a simple algorithm that obtains a 3/4 MMS allocation and runs in Õ(n^5m) time. The idea is to start with a trivial upper bound on the MMS values based on the average and greedily assign high-value items to agents. Then, try to satisfy the remaining agents with remaining items using tentative assignment and bag-filling procedures. If the MMS upper bounds are tight enough, then these procedures are guaranteed to succeed. Otherwise, we update it for a certain agent and run the procedures again. We show that the bounds need to be updated at most O(n^3) times.
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