Randomized Strategies for Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory and optimization. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update (MWU) method. The direct application of the MWU method does not yield a strict multiplicative approximation algorithm but yield one with an additional additive error term. A key technique to overcome the issue is to introduce a new concept called (η,γ)-reductions for objective functions with parameters η and γ. We show that our scheme outputs a nearly α-approximate solution if there exists an α-approximation algorithm for a subproblem defined by (η,γ)-reductions. This improves approximation ratios in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.

Scalable Robust Kidney Exchange

In barter exchanges, participants directly trade their endowed goods in a constrained economic setting without money. Transactions in barter exchanges are often facilitated via a central clearinghouse that must match participants even in the face of uncertainty—over participants, existence and quality of potential trades, and so on. Leveraging robust combinatorial optimization techniques, we address uncertainty in kidney exchange, a real-world barter market where patients swap (in)compatible paired donors. We provide two scalable robust methods to handle two distinct types of uncertainty in kidney exchange—over the quality and the existence of a potential match. The latter case directly addresses a weakness in all stochastic-optimization-based methods to the kidney exchange clearing problem, which all necessarily require explicit estimates of the probability of a transaction existing—a still-unsolved problem in this nascent market. We also propose a novel, scalable kidney exchange formulation that eliminates the need for an exponential-time constraint generation process in competing formulations, maintains provable optimality, and serves as a subsolver for our robust approach. For each type of uncertainty we demonstrate the benefits of robustness on real data from a large, fielded kidney exchange in the United States. We conclude by drawing parallels between robustness and notions of fairness in the kidney exchange setting.