Approximate Envy-Freeness in Graphical Cake Cutting
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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.
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