A Discrete and Bounded Locally Envy-Free Cake Cutting Protocol on Trees

We study the classic problem of fairly allocating a divisible resource modeled as a unit interval [0, 1] and referred to as a cake. In a landmark result, Aziz and Mackenzie [4] gave the first discrete and bounded protocol for computing an envy-free cake division, but with a huge query complexity consisting of six towers of exponent in the number of agents, n. However, the best-known lower bound for the same is Ω(n²), leaving a massive gap in our understanding of the complexity of the problem. In this work, we study an important variant of the problem where agents are embedded on a graph whose edges determine agent relations. Given a graph, the goal is to find a locally envy-free allocation where every agent values her share of the cake at least as much as that of any of her neighbors’ share. We identify a non-trivial graph structure, namely a tree having depth at most 2 (Depth2Tree), that admits a query efficient protocol to find locally envy-free allocations using O(n⁴log n) queries under the standard Robertson-Webb (RW) query model. To the best of our knowledge, this is the first such non-trivial graph structure. In our second result, we develop a novel cake-division protocol that finds a locally envy-free allocation among n agents on any Tree graph using O(n^{2 n}) RW queries. Though exponential, our protocol for Tree graphs achieves a significant improvement over the best-known query complexity of six-towers-of-n for complete graphs.