Consensus halving is PPA-complete

Filos-Ratsikas, Aris; Goldberg, Paul W.

doi:10.1145/3188745.3188880

Cited by 27 publications

(110 citation statements)

References 43 publications

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“…they reduce to each other in polynomial time. Therefore, by [29] and the "in PPA" result proven in Section B.1, we immediately get the following corollary. If we knew that k-Necklace-splitting belonged to PPA for other values of k, we could of course make the blanket statement "Necklace-splitting is PPA-complete".…”

Section: Problems and Resultsmentioning

confidence: 55%

“…We present and discuss our main results, and in Section 2.1 we give an overview of the proof and new techniques, in particular with respect to the precursors [28,29] to this paper.…”

Section: Problems and Resultsmentioning

confidence: 99%

“…This is not known for PPAD-complete problems. For more details, and the significance of PPA-completeness, we refer the reader to the related discussion in [29]. PPA is the class of problems reducible to Leaf (Definition 1), and a PPA-complete problem is polynomial-time equivalent to Leaf.…”

Section: Introductionmentioning

confidence: 99%

“…(That is a result of Bonsma et al [10], for the "paint shop problem with words", equivalent to 2-thief Necklace-splitting with 2 beads of each colour. )In [29], we showed Necklace-splitting to be computationally equivalent to ε-Consensushalving for inverse-polynomial precision parameter ε, but the PPA-completeness of ε-Consensushalving was only shown for inverse-exponential ε.[29] established PPAD-hardness of Necklacesplitting, applying the main result of [28]. In this paper, we prove that ε-Consensus-halving is…”

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confidence: 95%

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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: Problems and Resultsmentioning

confidence: 55%

“…We present and discuss our main results, and in Section 2.1 we give an overview of the proof and new techniques, in particular with respect to the precursors [28,29] to this paper.…”

Section: Problems and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 95%

See 2 more Smart Citations

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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“…Many local optimisation problems have been proved PLS-complete [24,31,26,14,13]. Recently, the first natural complete problems were found for PPA [16,17] and PPP [33]. The famous Factoring problem has been partially related to PPA and PPP [22].…”

Section: Introductionmentioning

confidence: 99%

The Classes PPA-k: Existence from Arguments Modulo k

Hollender

2019

Web and Internet Economics

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The complexity classes PPA-k, k ≥ 2, have recently emerged as the main candidates for capturing the complexity of important problems in fair division, in particular Alon's Necklace-Splitting problem with k thieves. Indeed, the problem with two thieves has been shown complete for PPA = PPA-2. In this work, we present structural results which provide a solid foundation for the further study of these classes. Namely, we investigate the classes PPA-k in terms of (i) equivalent definitions, (ii) inner structure, (iii) relationship to each other and to other TFNP classes, and (iv) closure under Turing reductions.

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