Cake-Cutting Is Not a Piece of Cake

Magdon‐Ismail, Malik; Busch, Costas; Krishnamoorthy, Mukkai S.

doi:10.1007/3-540-36494-3_52

“…A recent paper [4] by Magdon-Ismail, Busch and Krishnamoorthy proves an Ω (n log n) lower bound for a certain non-standard cake cutting model: The lower bound does not hold for the number of cuts performed or evaluation queries, but for the number of comparisons needed to administer these cuts. For more information on this fair cake cutting problem and on many of its variants, we refer the reader to the books by Brams and Taylor [1] and by Robertson and Webb [7].…”

Section: Previous Resultsmentioning

confidence: 95%

On the complexity of cake cutting

Woeginger

¹

,

Sgall

²

2007

Discrete Optimization

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In the cake cutting problem, n ≥ 2 players want to cut a cake into n pieces so that every player gets a 'fair' share of the cake by his/her own measure.We prove that in a certain, natural cake cutting model, every fair cake division protocol for n players must use Ω (n log n) cuts and evaluation queries in the worst case. Up to a small constant factor, this lower bound matches a corresponding upper bound in the same model obtained by Even and Paz, from 1984.

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“…The most obvious way to prove such a lower bound is to try to reduce sorting (or more precisely, learning an unknown permutation) to cake cutting. A first step in this direction was taken by Magdon-Ismail, Busch, and Krishnamoothy [4], who were able to show that any protocol must make Ω(n log n) comparisons to compute the assignment. So this result did not really lower bound the number of queries.…”

Section: Previous Resultsmentioning

confidence: 99%

Cake cutting really is not a piece of cake

Edmonds

¹

,

Pruhs

²

2011

ACM Trans. Algorithms

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We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard RobertsonWebb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. Improving on previous lower bounds, we provide an Ω(n log n) lower bound on the complexity of any deterministic protocol, even if the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols.

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“…The lower bound of Sgall and Woeginger [7] can be seen to hold against randomized protocols. However, note that neither of these lower bounds [4,7] hold if the protocol is only required to achieve approximate fairness.…”

Section: Previous Resultsmentioning

confidence: 99%

Cake cutting really is not a piece of cake

Edmonds

¹

,

Pruhs

²

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard RobertsonWebb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. Improving on previous lower bounds, we provide an Ω(n log n) lower bound on the complexity of any deterministic protocol, even if the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols.

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Cake-Cutting Is Not a Piece of Cake

Cited by 21 publications

References 15 publications

On the complexity of cake cutting

On the complexity of cake cutting

Cake cutting really is not a piece of cake

Cake cutting really is not a piece of cake

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