Approximating Maximin Shares with Mixed Manna
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We initiate the study of fair allocations of a mixed manna under the popular fairness notion of maximin share (MMS). A mixed manna allows an item to be a good for some agents and chore for others, hence strictly generalizes the well-studied goods (chores) only manna. For the good manna, Procaccia and Wang [PW14] showed non-existence of MMS allocation. This prompted works on finding an α-MMS allocation. A series of works obtained efficient algorithms, improving α to 3/4 for n≥ 5 agents. Computing an α-MMS allocation for the maximum α for which it exists is known to be NP-hard. But the question of finding α-MMS for the near best α remains unresolved. We make significant progress towards this question for mixed manna by showing a striking dichotomy: We derive two conditions and show that the problem is tractable under these, while dropping either renders the problem intractable. The conditions are: (i) number of agents is constant, and (ii) for every agent, her total value for goods differs significantly from that for chores. For instances satisfying (i) and (ii) we design a PTAS - an efficient algorithm to find (α-ϵ)-MMS allocation given ϵ>0 for the best possible α. We also show that if either condition is not satisfied then finding α-MMS for any α∈(0,1] is NP-hard, even when solution exists for α=1. As a corollary, our algorithm resolves the open question of designing a PTAS for the goods only setting with constantly many agents (best known α=3/4), and similarly also for chores only setting. In terms of techniques, we use market equilibrium as a tool to solve an MMS problem, which may be of independent interest.
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