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We study the problem of fair allocation of indivisible items when agents have ternary additive valuations --- each agent values each item at some fixed integer values a, b, or c that are common to all agents. The notions of fairness we consider are max Nash welfare (MNW), when a, b, and c are non-negative, and max egalitarian welfare (MEW). We show that for any distinct non-negative a, b, and c, maximizing Nash welfare is APX-hard --- i.e., the problem does not admit a PTAS unless P = NP. We also show that for any distinct a, b, and c, maximizing egalitarian welfare is APX-hard except for a few cases when b = 0 that admit efficient algorithms. These results make significant progress towards completely characterizing the complexity of computing exact MNW allocations and MEW allocations. En route, we resolve open questions left by prior work regarding the complexity of computing MNW allocations under bivalued valuations, and MEW allocations under ternary mixed manna. 

We study the problem of fairly allocating a set of indivisible goods among agents with matroid rank valuations: every good provides a marginal value of 0 or 1 when added to a bundle and valuations are submodular.We present a simple algorithmic framework, calledGeneral Yankee Swap, that can efficiently compute allocations that maximize any justice criterion (or fairness objective) satisfying some mild assumptions. Along with maximizing a justice criterion, General Yankee Swap is guaranteed to maximize utilitarian social welfare, ensure strategyproofness and use at most a quadratic number of valuation queries. We show how General Yankee Swap can be used to compute allocations for five different well-studied justice criteria: (a) Prioritized Lorenz dominance, (b) Maximin fairness, (c) Weighted leximin, (d) Max weighted Nash welfare, and (e) Max weightedp-mean welfare.In particular, this framework provides the first polynomial time algorithms to compute weighted leximin, max weighted Nash welfare and max weightedp-mean welfare allocations for agents with matroid rank valuations. We also extend this framework to the setting of binary chores — items with marginal values -1 or 0 — and similarly show that it can be used to maximize any justice criteria satisfying some mild assumptions.