Approximate envy-freeness in graphical cake cutting

We study the problem of fairly allocating a divisible resource in the form of a graph, also known as graphical cake cutting. Unlike for the canonical interval cake, a connected envy-free allocation is not guaranteed to exist for a graphical cake. We focus on the existence and computation of connected allocations with low envy. For general graphs, we show that there is always a 1/2-additive-envy-free allocation and, if the agents' valuations are identical, a (2 + epsilon)-multiplicative-envy-free allocation for any epsilon > 0. In the case of star graphs, we obtain a multiplicative factor of 3 + epsilon for arbitrary valuations and 2 for identical valuations. We also derive guarantees when each agent can receive more than one connected piece. All of our results come with efficient algorithms for computing the respective allocations. (c) 2024 The Author(s). Published by Elsevier B.V.