Finance

Discrete mathematics

Mathematical economics

Valuation (finance)

Mathematics

Algorithm

Mathematical optimization

Piecewise

Mathematical analysis

Piecewise linear function

Economics

Fair division

Programming language

Geometry

Constant (computer programming)

Computer science

Scheme (mathematics)

Efficient algorithm

Time complexity

Division (mathematics)

Arithmetic

We consider the classic cake cutting problem where one allocates a divisible cake to n participating agents. Among all valid divisions, fairness and efficiency (a.k.a. ~social welfare) are the most critical criteria to satisfy and optimize, respectively. We study computational complexity of computing an efficiency optimal division given the conditions that the allocation satisfies proportional fairness and assigns each agent a connected piece. For linear valuation functions, we give a polynomial time approximation scheme to compute an efficiency optimal allocation. On the other hand, we show that the problem is NP-hard to approximate within a factor of Ω 1/√n for general piecewise constant functions, and is NP-hard to compute for normalized functions.

Optimal Proportional Cake Cutting with Connected Pieces