## Solution of Peter Winkler's Pizza Problem

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##### Authors

Cibulka, Josef

Kynčl, Jan

Mészáros, Viola

Stolař, Rudolf

Valtr, Pavel

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##### Abstract

##### Description

Bob cuts a pizza into slices of not necessarily equal size and shares it with
Alice by alternately taking turns. One slice is taken in each turn. The first
turn is Alice's. She may choose any of the slices. In all other turns only
those slices can be chosen that have a neighbor slice already eaten. We prove a
conjecture of Peter Winkler by showing that Alice has a strategy for obtaining
4/9 of the pizza. This is best possible, that is, there is a cutting and a
strategy for Bob to get 5/9 of the pizza. We also give a characterization of
Alice's best possible gain depending on the number of slices. For a given
cutting of the pizza, we describe a linear time algorithm that computes Alice's
strategy gaining at least 4/9 of the pizza and another algorithm that computes
the optimal strategy for both players in any possible position of the game in
quadratic time. We distinguish two types of turns, shifts and jumps. We prove
that Alice can gain 4/9, 7/16 and 1/3 of the pizza if she is allowed to make at
most two jumps, at most one jump and no jump, respectively, and the three
constants are the best possible.

Comment: 29 pages, 14 figures

Comment: 29 pages, 14 figures

##### Keywords

Computer Science - Discrete Mathematics, G.2.1