Thieves can make sandwiches

Blagojević, Pavle V. M.; Soberón, Pablo

doi:10.1112/blms.12109

Cited by 8 publications

(2 citation statements)

References 21 publications

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“…We prove the PPA-completeness of Discrete Ham Sandwich via a simple reduction from Necklace-splitting. Ours is not the first paper to develop the close relationship between the two problems: Blagojević and Soberón [9] shows a generalisation, where multiple agents may share a "sandwich", dividing it into convex pieces. Further papers to explicitly point out their computational complexity as open problems include Deng et al [23] (mentioning that both problems "show promise to be complete for PPA"), Aisenberg et al [1], and Belovs et al [7].…”

Section: Introductionmentioning

confidence: 97%

The complexity of splitting necklaces and bisecting ham sandwiches

Filos-Ratsikas

Goldberg

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We resolve the computational complexity of two problems known as Necklace-splitting and Discrete Ham Sandwich showing that they are PPA-complete. For Necklace-splitting, this result is specific to the important special case in which two thieves share the necklace. We do this via a PPA-completeness result for an approximate version of the Consensus-halving problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision. At the heart of our construction is a smooth embedding of the high-dimensional Möbius strip in the Consensus-halving problem. These results settle the status of PPA as a class that captures the complexity of "natural" problems whose definitions do not incorporate a circuit.Definition 1 An instance of the problem Leaf consists of an undirected graph G whose vertices have degree at most 2; G has 2 n vertices represented by bitstrings of length n; G is presented concisely via a circuit that takes as input a vertex and outputs its neighbour(s). We stipulate that vertex 0 n has degree 1. The challenge is to find some other vertex having degree 1.Complete problems for the class PPA seemed to be much more elusive than PPAD-complete ones, especially when one is interested in "natural" problems, where "natural" here has the very specific meaning of problems that do not explicitly contain a circuit in their definition. Besides Papadimitriou [60], other papers asking about the possible existence of natural PPA-complete problems include [36,13,19,22]. In a recent precursor [29] to the present paper we identified the first example of such a problem, namely the approximate Consensus-halving problem, dispelling the suspicion that such problems might not exist. In this paper we build on that result and settle the complexity of two natural and important problems whose complexity status were raised explicitly as open problems in Papadimitriou's paper itself, and in many other papers beginning in the 1980s. Specifically, we prove that Necklace-splitting (with two thieves, see Definition 2) and Discrete Ham Sandwich are both PPA-complete.Definition 2 (Necklace Splitting) In the k-Necklace-splitting problem there is an open necklace with ka i beads of colour i, for 1 ≤ i ≤ n. An "open necklace" means that the beads form a string, not a cycle. The task is to cut the necklace in (k − 1) · n places and partition the resulting substrings into k collections, each containing precisely a i beads of colour i, 1 ≤ i ≤ n.In Definition 2, k is thought of as the number of thieves who desire to split the necklace in such a way that the beads of each colour are equally shared. In this paper, usually we have k = 2 and we refer to this special case as Necklace-splitting.Definition 3 (Discrete Ham Sandwich) In the Discrete Ham Sandwich problem, there are n sets of points in n dimensions having integer coordinates (equivalently one could use rationals). A solution consists of a hyperplane that splits each set of points into subsets of equal size (if any points lie on the plane, we are allowed ...

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Section: Introductionmentioning

confidence: 97%

The complexity of splitting necklaces and bisecting ham sandwiches

Filos-Ratsikas

Goldberg

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…One may, for instance, explore necklace splittings with the added constraint that adjacent pieces of the necklace cannot be claimed by certain pairs of thieves; for example, Asada et al [32] prove that four thieves on a circle can share the beads of the necklace, with the restriction that the two pairs of nonadjacent thieves will not receive adjacent pieces of the necklace. There are also several nice high-dimensional generalizations of (convex) splitting of booty; see [70,133] and the references therein.…”

Section: Necklace Splittingmentioning

confidence: 99%