My team has pursued work on four separate topics.
The first topic has been to more deeply understand fundamental systems for proving infeasibility of integer programs. This is a topic with deep connections to the field of proof complexity, which seeks to understand how difficult it is to demonstrate the infeasibility of unsatisfiable logical formulae using different systems of logical deduction. In this regard, we showed that a powerful integer programming strategy, known as general branching and cutting plane proofs, can be used to provide much shorter proofs of infeasibility for so-called Tseitin formulas than was expected to be possible. Our work refuted a conjecture due to Cook, Coullard and Turan from the 1987 and was recently awarded the best paper award at the 2020 conference on computational complexity (CCC 2020).
The second topic was to explore the power of interior point methods (IPM) for solving linear programs exactly. Interior points methods are the most theoretical efficient methods for solving linear programs, though has mostly been theoretically analyzed from the perspective of obtaining approximate instead of exact solutions. Exact solutions to linear programming approximations of integer programs (IP) are crucially needed by modern IP solvers, and thus understanding how to efficiently extract exact solutions from an IPM is of great importance. Our contribution in this line has been to improve the "numerical complexity" of an IPM solver. More precisely, we developed a novel IPM whose running time for producing exact solutions can be controlled by very refined measure of an LPs numerical complexity, which was conjectured to be sufficient by Monteiro and Tsuchiya in 2003. Pushing past numerical complexity measures, we then developed a new characterization of the complexity of IPMs a geometric notion known as straight-line complexity. This has enabled the first strongly polynomial algorithm for solving linear programs expressible with at most 2 variables per inequality, resolving a 40-year old open problem of Megiddo.
The third topic was to develop a new theoretical framework for an integer programming technique known as cutting planes. Cutting planes provide a way of automatically strengthening linear programming relaxations of integer programs and are crucial for the efficiency of modern IP solvers. From the theoretical perspective however, even proving basic convergence results for cutting plane methods has proven very challenging. Our work is addressing this difficulty by providing a novel framework for selecting cutting planes which automatically ensures convergence.
The last topic has examined the efficiency of the classical branch and bound algorithm for solving random integer programs. In this context, we showed that integer programs with Gaussian data can are solved in polynomial time as long as the number of constraints is fixed. We achieve this by building on recent work of Dey, Dubey and Molinaro (SODA 2021), who showed branch & bound performs well on random packing problems for which the gap between the value of the program and that of its linear programming relaxation is small. In particular, we first show that this gap is indeed small in the Gaussian case, and we generalize their result to cover a very wide class of random integer programs, which we call random log-concave integer programs. Using techniques from discrepancy theory, we then further generalized these results to integer programs whose entries have discrete distributions (e.g. coefficients in 0,1,-1), which are far more common in practice. This has resulted in a flexible theory for explaining the average case performance of branch & bound, which has helped bridge the gap between theory and practice significantly.
The results stated above were all presented at top venues in the areas of theoretical computer science and mathematical optimization.
