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We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations.

In the ε-Consensus-Halving problem, a fundamental problem in fair division, there are n agents with valuations over the interval [0, 1], and the goal is to divide the interval into pieces and assign a label "+" or "−" to each piece, such that every agent values the total amount of "+" and the total amount of "−" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [18,19] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question.In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [19], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [19]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving ε-Consensus-Halving for any ε, as well as a polynomial-time algorithm for ε = 1/2; these are the first algorithmic results for the problem. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-1/k-Division problem [33]. In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard.

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A. Hollender is supported by an EPSRC doctoral studentship (Reference 1892947).