The complexity of splitting necklaces and bisecting ham sandwiches

Abstract

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 ...

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…Note that from the above discussion, we have the following "computational hierarchy": computing a solution to this variation of the octahedral Tucker problem is harder than computing a ham sandwich cut (note also that this implies that the latter is in PPA), which is harder than computing a solution to the fair splitting necklace problem. In particular, the Filos-Ratsikas and Goldberg's paper [163] proves that computing the ham sandwich cut is PPA-complete (see Section 4.2).…”

“…A well-known challenge of computational complexity was to decide whether the computation of a fair splitting is a PPA-complete problem [313]. This was just recently settled by Filos-Ratsikas and Goldberg [162,163]. The fact that splitting necklace is PPAcomplete implies that the algorithmic version of the octahedral Tucker is PPA-complete as well, but there is also a paper directly proving that the algorithmic Octahedral Tucker's lemma is PPA-complete [136] (see Section 2.4).…”

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

“…The -Consensus-Halving problem received considerable attention in the literature of computer science over the past few years, as it was proven to be the first "natural" PPA-complete problem [28], i.e., a problem that does not have a polynomial-sized circuit explicitly in its definition, answering a decade-old open question [33,41]. Additionally, Filos-Ratsikas and Goldberg [29], reduced from this problem to establish the PPA-completeness of Necklace Splitting with two thieves; these PPA-completeness results provided the first definitive evidence of intractability for these two classic problems, establishing for instance that solving them is at least as hard as finding a Nash equilibrium of a strategic game [18,19]. Filos-Ratsikas et al [30] improved on the results for the -Consensus-Halving problem, by showing that the problem remains PPA-complete, even if one restricts the attention to very small classes of agents' valuations, namely piecewise uniform valuations with only two valuation blocks.…”

“…This latter result falls under the general umbrella of imposing restrictions on the structure of the problem, to explore if the computational hardness persists or whether we can obtain polynomialtime algorithms. Filos-Ratsikas et al [30] applied this approach along the axis of the valuation functions, while considering a general number of agents, similarly to [28,29]. In this paper, we take a different approach, and we restrict the number of agents to be constant.…”

The Classes PPA-k: Existence from Arguments Modulo k

“…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].…”