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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.

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

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The complexity classes PPA-k, k ≥ 2, have recently emerged as the main candidates for capturing the complexity of important problems in fair division, in particular Alon's Necklace-Splitting problem with k thieves. Indeed, the problem with two thieves has been shown complete for PPA = PPA-2. In this work, we present structural results which provide a solid foundation for the further study of these classes. Namely, we investigate the classes PPA-k in terms of (i) equivalent definitions, (ii) inner structure, (iii) relationship to each other and to other TFNP classes, and (iv) closure under Turing reductions.

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.