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The value of a piece to a partner is the integral of his value-density over that piece (this means that the value is a nonatomic measure over the cake). The envy-free cake-cutting problem is to partition the cake to n disjoint pieces, one piece per agent, such for each agent, the value of his piece is weakly larger than the values of all other ...
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.
Level 3: a node for each task. There is an arc from each agent i to each task j, with the corresponding cost, and capacity 1. Level 4: One sink-node t. There is an arc from each task to t, with cost 0 and capacity d j. An integral maximum flow of minimum cost can be found in polynomial time; see network flow problem.
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In case the "cake" is a 1-dimensional interval, this translates to the requirement that each piece is also an interval. In case the cake is a 1-dimensional circle ("pie"), this translates to the requirement that each piece be an arc; see fair pie-cutting. Another constraint is adjacency. This constraint applies to the case when the "cake" is a ...
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