Envy-free cake-cutting

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An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.

When there are only two partners, the problem is easy and was solved in antiquity by the divide and choose protocol. When there are three or more partners, the problem becomes much more challenging.

Two major variants of the problem have been studied:

• Connected pieces, e.g. if the cake is a 1-dimensional interval then each partner must receive a single sub-interval. If there are ${\displaystyle n}$ partners, only ${\displaystyle n-1}$ cuts are needed.
• General pieces, e.g. if the cake is a 1-dimensional interval then each partner can receive a union of disjoint sub-intervals.