fair division software developers

Abstract: We study the parameterized complexity of finding an EF1 allocation of indivisible items in the group-sharing setting, which has received increasing attention in recent years. There, items are not allocated to individual agents but rather to one of two (or possibly more) groups, and agents within that group all share the assigned items. We consider both the "fixed" (EF1-Fix) and "variable" (EF1-Var) group cases, where in the former the assignments of agents to groups are prescribed. While both are NP-hard in the classical sense, we show that their algorithmic behavior differs significantly from the parameterized perspective. As our main results, we: (1) prove that both cases are NP-hard even when the number of agents is a small constant; (2) circumvent this hardness by establishing fixed-parameter tractability for EF1-Var w.r.t. the number of "subjective" items, that is, items which are viewed as chores or goods by different agents---but rule out an analogous result for EF1-Fix; and (3) establish fixed-parameter tractability for EF1-Fix w.r.t. the number of item types, but link the existence of an analogous fixed-parameter algorithm for EF1-Var to a fundamental open question in the setting of Integer Linear Programming.

Abstract: 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.