Approximating Maximin Shares with Mixed Manna

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 $\alpha$-MMS allocation. A series of works obtained efficient algorithms, improving $\alpha$ to $\frac{3}{4}$ for $n\ge 5$ agents. Computing an $\alpha$-MMS allocation for the maximum $\alpha$ for which it exists is known to be NP-hard. But the question of finding $\alpha$-MMS for the near best $\alpha$ 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 $(\alpha-\epsilon)$-MMS allocation given $\epsilon>0$ for the best possible $\alpha$. We also show that if either condition is not satisfied then finding $\alpha$-MMS for any $\alpha\in(0,1]$ is NP-hard, even when solution exists for $\alpha=1$. As a corollary, our algorithm resolves the open question of designing a PTAS for the goods only setting with constantly many agents (best known $\alpha=\frac{3}{4}$), 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.