arXiv subjects

Stephan Patterson

Publications and source records attributed to Stephan Patterson.

4 recordsLinked to original sources

arXiv2022

An Integer Program for Pricing Support Points of Exact Barycenters

The computation of exact barycenters for a set of discrete measures is of interest in applications where sparse solutions are desired, and to assess the quality of solutions returned by approximate algorithms and heuristics. The task is known to be NP-hard for growing dimension and, even in low dimensions, extremely challenging in practice due to an exponential scaling of the linear programming formulations associated with the search for sparse solutions. A common approach to facilitate practical computations is an approximation based on the choice of a small, fixed set $S_0$ of support points, or a fixed set $S^*_0$ of combinations of support points from the measures, that may be assigned mass. Through a combination of linear and integer programming techniques, we model an integer program to compute additional combinations, and in turn support points, that, when added to $S^*_0$ or $S_0$, allow for a better approximation of the underlying exact barycenter problem. The approach improves on the scalability of a classical column generation approach: instead of a pricing problem that has to evaluate exponentially many reduced cost values, we solve a mixed-integer program of quadratic size. The properties of the model, and practical computations, reveal a tailored branch-and-bound routine as a good solution strategy.

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arXiv2019

On the Computational Complexity of Finding a Sparse Wasserstein Barycenter

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of probability measures with finite support. In this paper, we show that finding a barycenter of sparse support is hard, even in dimension 2 and for only 3 measures. We prove this claim by showing that a special case of an intimately related decision problem SCMP -- does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? -- is NP-hard for all rational data. Our proof is based on a reduction from planar 3-dimensional matching and follows a strategy laid out by Spieksma and Woeginger (1996) for a reduction to planar, minimum circumference 3-dimensional matching. While we closely mirror the actual steps of their proof, the arguments themselves differ fundamentally due to the complex nature of the discrete barycenter problem. Containment of SCMP in NP will remain open. We prove that, for a given measure, sparsity and cost of an optimal transport to a set of measures can be verified in polynomial time in the size of a bit encoding of the measure. However, the encoding size of a barycenter may be exponential in the encoding size of the underlying measures.

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arXiv2019

A Column Generation Approach to the Discrete Barycenter Problem

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation. In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig-Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.

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arXiv2018

Improved Linear Programs for Discrete Barycenters

Discrete barycenters are the optimal solutions to mass transport problems for a set of discrete measures. Such transport problems arise in many applications of operations research and statistics. The best known algorithms for exact barycenters are based on linear programming, but these programs scale exponentially in the number of measures, making them prohibitive for practical purposes. In this paper, we improve on these algorithms. First, by using the optimality conditions to restrict the search space, we provide a reduced linear program that contains dramatically fewer variables compared to previous formulations. Second, we recall a proof from the literature, which lends itself to a linear program that has not been considered for computations. We show that this second formulation is the best model for data in general position. Third, we combine the two programs into a single hybrid model that retains the best properties of both formulations for partially structured data. We study these models through an analysis of their scaling in size, the hardness of the required preprocessing, and computational experiments. In doing so, we show that each of the improved linear programs becomes the best model for different types of data.

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