Abstract
Given a heterogeneous cake and 5 players, we give the first bounded algorithm for computing an envy-free division of the cake, such that each person thinks he gets the largest piece. The case with 4 players was solved in a famous paper by Brams et al. in 1997. Our algorithm can be discretized to obtain an ε envy-free division in \(O({\rm polylog} \left( 1 / \epsilon \right))\) time. The algorithm is based on augmenting the irrevocable advantage graph in a new way.
We also look at the open problem of finding discrete procedures for computing envy-free division among 4 players. We present a simple algorithm that finds an envy-free division of a portion of the cake, such that each player gets at least 1/4 of the whole cake (in his valuation).
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Saberi, A., Wang, Y. (2009). Cutting a Cake for Five People. In: Goldberg, A.V., Zhou, Y. (eds) Algorithmic Aspects in Information and Management. AAIM 2009. Lecture Notes in Computer Science, vol 5564. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02158-9_25
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DOI: https://doi.org/10.1007/978-3-642-02158-9_25
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