A central goal in the field of optimization is to develop effective procedures for decision-making in the presence of constraints. These constraints are often imposed by real-world data, but it is frequently the case that the relevant data is not completely known to the agent performing the optimization; it is natural to model such settings using probability distributions associated with the data. In such analyses, the constraints associated with the optimization problem are now themselves "stochastic," and a standard goal is to maximize the likelihood of satisfying all the constraints while incurring minimum cost. (As a motivating example, an airline may wish to operate as few flights as possible while ensuring that with 99% probability, no passenger is bumped.) Apart from modeling uncertainty, optimization with stochastic constraints also provides a way to succinctly model constraints whose standard description is very large; design problems in voting theory, where there are very many voters, are examples of this kind. This project studies both of these kinds of problems, called stochastic design problems, from a unified new perspective based on techniques from computational complexity theory. The project also trains graduate students who will achieve fluency both in complexity theory and in optimization, and will promote cross-disciplinary activities between operations research and theoretical computer science.
The motivating insight which underlies this project is that Boolean function analysis -- a topic at the intersection of harmonic analysis, probability theory, and complexity theory -- provides a useful suite of techniques for stochastic design problems. The investigators will study two broad topics. The first one is on chance-constrained optimization: In problems of this sort, one is given a set of stochastic constraints and the aim is to satisfy all the constraints with at least a certain fixed threshold probability. While previous work on such problems has typically achieved computationally efficient algorithms by relaxing the actual set of constraints, the investigators will focus on algorithms which exactly satisfy the original given set of stochastic constraints. This line of work will address the chance-constrained versions of fundamental optimization problems such as bin packing, knapsack, and linear programming. The second broad topic is that of inverse problems in social choice theory: Game theorists use so-called "power indices" to measure the influence of voters in voting schemes. A basic algorithmic problem is to design efficient algorithms for the inverse problem, in which, given a set of prescribed power indices, the goal is to construct a voting game with these indices. The investigators will study questions such as (a) to what extent is a given voting scheme specified by its power indices? (b) what is the complexity of exactly reconstructing an unknown target voting game given its power indices? (c) when and to what extent is reconstruction possible in a partial information setting?
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
