We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is polynomial-time equivalent to NECKLACE SPLITTING, which establishes PPAD-hardness for NECKLACE SPLITTING, and suggests that it is also PPA-complete.
Abstract. We study the problem of approximate social welfare maximization (without money) in onesided matching problems when agents have unrestricted cardinal preferences over a finite set of items. Random priority is a very well-known truthful-in-expectation mechanism for the problem. We prove that the approximation ratio of random priority is Θ(n −1/2 ) while no truthful-in-expectation mechanism can achieve an approximation ratio better than O(n −1/2 ), where n is the number of agents and items. Furthermore, we prove that the approximation ratio of all ordinal (not necessarily truthful-in-expectation) mechanisms is upper bounded by O(n −1/2 ), indicating that random priority is asymptotically the best truthful-in-expectation mechanism and the best ordinal mechanism for the problem.
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 ...
We study the problem of allocating impressions to sellers in ecommerce websites, such as Amazon, eBay or Taobao, aiming to maximize the total revenue generated by the platform. We employ a general framework of reinforcement mechanism design, which uses deep reinforcement learning to design efficient algorithms, taking the strategic behaviour of the sellers into account. Specifically, we model the impression allocation problem as a Markov decision process, where the states encode the history of impressions, prices, transactions and generated revenue and the actions are the possible impression allocations in each round. To tackle the problem of continuity and high-dimensionality of states and actions, we adopt the ideas of the DDPG algorithm to design an actor-critic policy gradient algorithm which takes advantage of the problem domain in order to achieve convergence and stability.We evaluate our proposed algorithm, coined IA(GRU), by comparing it against DDPG, as well as several natural heuristics, under different rationality models for the sellers -we assume that sellers follow well-known no-regret type strategies which may vary in their degree of sophistication. We find that IA(GRU) outperforms all algorithms in terms of the total revenue.
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