"Combinatorial Bootstrap Inference IN in Prtially Identified Incomplete Structural Models"

We propose a computationally feasible inference method infinite games of complete information. Galichon and Henry (2011) and Beresteanu, Molchanov, and Molinari (2011) show that such models are equivalent to a collection of moment inequalities that increases exponentially with the number of discrete outcomes. We propose an equivalent characterization based on classical combinatorial optimization methods that alleviates this computational burden and allows the construction of confidence regions with an effcient combinatorial bootstrap procedure that runs in linear computing time. The method can also be applied to the empirical analysis of cooperative and noncooperative games, instrumental variable models of discrete choice and revealed preference analysis. We propose an application to the determinants of long term elderly care choices.

Scheduling MapReduce Jobs under Multi-Round Precedences

We consider non-preemptive scheduling of MapReduce jobs with multiple tasks in the practical scenario where each job requires several map-reduce rounds. We seek to minimize the average weighted completion time and consider scheduling on identical and unrelated parallel processors. For identical processors, we present LP-based O(1)-approximation algorithms. For unrelated processors, the approximation ratio naturally depends on the maximum number of rounds of any job. Since the number of rounds per job in typical MapReduce algorithms is a small constant, our scheduling algorithms achieve a small approximation ratio in practice. For the single-round case, we substantially improve on previously best known approximation guarantees for both identical and unrelated processors. Moreover, we conduct an experimental analysis and compare the performance of our algorithms against a fast heuristic and a lower bound on the optimal solution, thus demonstrating their promising practical performance

On Budget-Feasible Mechanism Design for Symmetric Submodular Objectives

We study a class of procurement auctions with a budget constraint, where an auctioneer is interested in buying resources or services from a set of agents. Ideally, the auctioneer would like to select a subset of the resources so as to maximize his valuation function, without exceeding a given budget. As the resources are owned by strategic agents however, our overall goal is to design mechanisms that are truthful, budget-feasible, and obtain a good approximation to the optimal value. Budget-feasibility creates additional challenges, making several approaches inapplicable in this setting. Previous results on budget-feasible mechanisms have considered mostly monotone valuation functions. In this work, we mainly focus on symmetric submodular valuations, a prominent class of non-monotone submodular functions that includes cut functions. We begin first with a purely algorithmic result, obtaining a $\frac{2e}{e-1}$-approximation for maximizing symmetric submodular functions under a budget constraint. We view this as a standalone result of independent interest, as it is the best known factor achieved by a deterministic algorithm. We then proceed to propose truthful, budget feasible mechanisms (both deterministic and randomized), paying particular attention on the Budgeted Max Cut problem. Our results significantly improve the known approximation ratios for these objectives, while establishing polynomial running time for cases where only exponential mechanisms were known. At the heart of our approach lies an appropriate combination of local search algorithms with results for monotone submodular valuations, applied to the derived local optima.Comment: A conference version appears in WINE 201

A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs