PPAD-Completeness of Contiguous Envy-Free Cake Cutting with Additive Valuations

We study temporal fair division with indivisible goods that arrive sequentially and must be allocated irrevocably. In contrast to the usual online model, we assume that valuations and future arrivals are known in advance, and ask how unfairness evolves during the process. We introduce cumulative maximum envy: the sum, over all rounds, of the maximum pairwise envy at that round. Equivalently, this is the area under the worst-envy curve, and it captures both the magnitude and the duration of envy. For a fixed arrival order, we show that the corresponding decision problem is strongly NP-complete and that minimizing this objective admits no constant-factor approximation unless P = NP, even under identical valuations and even under binary valuations. We complement these hardness results with a dynamic program that gives pseudopolynomial-time solvability for a constant number of agents, polynomial-time algorithms in further restricted settings, and an FPTAS for fixed $n$ under identical integer valuations. We then study a sequencing variant where the algorithm may choose the arrival order. This variant remains NP-complete even for two agents with identical valuations; however, a simple greedy algorithm achieves a $3/2$-approximation for $n=2$ agents, an $n/(n-1)$-approximation for any number of agents, and an additive guarantee depending on the maximum value of any good.

We study proportional cake cutting under an equal-length constraint, where each of $n$ agents must receive a bundle of length exactly $1/n$ that is worth at least $1/n$ to that agent. Our main result determines the exact worst-case number of cuts needed to guarantee such an allocation, namely $2n-2$ for every $n$. This improves on the previously known quadratic upper bound. The proof combines the constrained necklace-splitting theorem with a recursive assignment argument based on Hall's theorem. The same bound extends to the stronger requirement of envy-freeness when $n$ is a prime power, and an $O(n^{1.525})$ upper bound holds for arbitrary $n$. We then ask whether such allocations can be computed. We first show that no deterministic protocol in the Robertson--Webb model terminates after finitely many queries on every instance and always returns an exact equal-length proportional allocation, even for two agents with identical valuations. For every $\varepsilon>0$, however, $O(n^2/\varepsilon)$ queries suffice to produce an allocation in which every agent receives a bundle of length exactly $1/n$ and value at least $1/n-\varepsilon$, again using at most $2n-2$ cuts. Finally, we augment the Robertson--Webb model with two-knife queries. We show that recursive balanced halving attains the optimal bound of $2n-2$ cuts when $n$ is a power of two. For arbitrary $n$, we give a procedure using $O(n\log n)$ cuts.

We study whether strictly positive marginal values restore the compatibility of envy-freeness up to one good (EF1) and Pareto optimality (PO) for indivisible goods. For two agents, we identify the exact threshold in the number of goods. Every instance with at most seven goods and strictly increasing valuations admits an allocation that is both EF1 and PO, without any submodularity assumption. In contrast, we construct an eight-good instance with normalized, integer-valued, strictly increasing, submodular valuations in which every EF1 allocation is strictly Pareto dominated. Thus, eight goods are necessary and sufficient for a two-agent counterexample. Finally, we strengthen the three-agent NP-hardness result of Chandramouleeswaran and Nimbhorkar (2026): deciding whether an EF1 and PO allocation exists remains NP-hard for normalized, integer-valued, monotone submodular valuations even when zero marginals are confined to eight fixed agent-good pairs, all involving a single agent.