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 + ϵ) -multiplicative-envy-free allocation for any ϵ > 0. In the case of star graphs, we obtain a multiplicative factor of 3 + ϵ 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. [ABSTRACT FROM AUTHOR]