Matching demand (riders) to supply (drivers) efficiently is a fundamental problem for ridesharing platforms who need to match the riders (almost) as soon as the request arrives with only partial knowledge about future ride requests. A myopic approach that computes an optimal matching for current requests ignoring future uncertainty can be highly sub-optimal. In this paper, we consider a two-stage robust optimization framework for this matching problem where future demand uncertainty is modeled using a set of demand scenarios (specified explicitly or implicitly). The goal is to match the current request to drivers (in the first stage) so that the cost of first stage matching and the worst case cost over all scenarios for the second stage matching is minimized. We show that the two-stage robust matching is NP-hard under various cost functions and present constant approximation algorithms for different settings of our two-stage problem. Furthermore, we test our algorithms on real-life taxi data from the city of Shenzhen and show that they substantially improve upon myopic solutions and reduce the maximum wait time of the second-stage riders by an average of 30% in our experimental results.

We consider the problem of designing video games (modeled here by choosing the structure of a linear program solved by players) so that players with different resources play diverse strategies. In particular, game designers hope to avoid scenarios where players use the same "weapons" or "tactics" even as they progress through the game. We model this design question as a choice over the constraint matrix A and cost vector c that seeks to maximize the number of possible supports of unique optimal solutions (what we call loadouts) of Linear Programs max{c x | Ax ≤ b, x ≥ 0} with nonnegative data considered over all resource vectors b.We provide an upper bound on the optimal number of loadouts and provide a family of constructions that have an asymptotically optimal number of loadouts. The upper bound is based on a connection between our problem and the study of triangulations of point sets arising from polyhedral combinatorics, and specifically the combinatorics of the cyclic polytope. Our asymptotically optimal construction also draws inspiration from the properties of the cyclic polytope. Our construction provides practical guidance to game designers seeking to offer a diversity of play for their plays.

We study the problem of learning a hypergraph via edge detecting queries. In this problem, a learner queries subsets of vertices of a hidden hypergraph and observes whether these subsets contain an edge or not. In general, learning a hypergraph with m edges of maximum size d requires Ω((2m/d) d/2 ) queries [7]. In this paper, we aim to identify families of hypergraphs that can be learned without suffering from a query complexity that grows exponentially in the size of the edges.We show that hypermatchings and low-degree near-uniform hypergraphs with n vertices are learnable with poly(n) queries. For learning hypermatchings (hypergraphs of maximum degree 1), we give an O(log 3 n)-round algorithm with O(n log 5 n) queries. We complement this upper bound by showing that there are no algorithms with poly(n) queries that learn hypermatchings in o(log log n) adaptive rounds. For hypergraphs with maximum degree ∆ and edge size ratio ρ, we give a non-adaptive algorithm with O((2n) ρ∆+1 log 2 n) queries. To the best of our knowledge, these are the first algorithms with poly(n, m) query complexity for learning non-trivial families of hypergraphs that have a super-constant number of edges of super-constant size.

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We study the d-Uniform Hypergraph Matching (d-UHM) problem: given an n-vertex hypergraph G where every hyperedge is of size d, find a maximum cardinality set of disjoint hyperedges. For d ≥ 3, the problem of finding the maximum matching is N P-complete, and was one of Karp's 21 N P-complete problems. In this paper we are interested in the problem of finding matchings in hypergraphs in the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In this model, we present the first three parallel algorithms for d-Uniform Hypergraph Matching, and we analyse them in terms of resources such as memory usage, rounds of communication needed, and approximation ratio. The highlights include: