Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Polyhedral Diameters and Applications to Optimization

The Simplex method is the most popular algorithm for solving linear programs (LPs). Geometrically, it moves from an initial vertex solution to an improving neighboring one, selected according to a pivot rule. Despite decades of study, it is still not known whether there exists a pivot rule that makes the Simplex method run in polynomial time. Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called the circuits, that is a superset of the edges of a polytope. The number of circuit augmentations has been of interest as a proxy for the number of steps in the Simplex method, and the circuit-diameter of polyhedra has been studied as a lower bound to the combinatorial diameter of polyhedra. We show that in the circuit-augmentation framework the Greatest Improvement and Dantzig pivot rules are NP-hard, even for 0/1 LPs. On the other hand, the Steepest Descent pivot rule can be carried out in polynomial time in the 0/1 setting, and the number of circuit augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1 LPs. We introduce a new rule in the circuit-augmentation framework which we call Asymmetric Steepest Descent. We show both that it can be computed in polynomial time and that it reaches an optimal solution of an LP in a polynomial number of augmentations. It was not previously known that such a rule was possible. We further show a weakly-polynomial bound on the circuit diameter of rational polyhedra. We next show that the circuit-augmentation framework can be exploited to make novel conclusions about the classical Simplex method itself: In particular, as a byproduct of our circuit results, we prove that (i) computing the shortest (monotone) path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for 0/1-LPs (i.e., those whose vertex solutions are in {0, 1}^n), a monotone path of strongly-polynomial length can be constructed using steepest improving edges. Inspired by this, we further examine the lengths of other monotone paths generated by some local decision rules – which we call edge rules – including two modifications of the classical Shadow Vertex pivot rule. We leverage the techniques we use for analyzing edge rules to devise pivot rules for the Simplex method on 0/1-LPs that generate the same monotone paths as their edge rule counterparts. In particular, this shows that there exist pivot rules for the Simplex method on 0/1 LPs that reach an optimal solution by performing only a polynomial number of non-degenerate pivots, answering an open question in the literature

Geometric aspects of linear programming : shadow paths, central paths, and a cutting plane method

Most everyday algorithms are well-understood; predictions made theoretically about them closely match what we observe in practice. This is not the case for all algorithms, and some algorithms are still poorly understood on a theoretical level. This is the case for many algorithms used for solving optimization problems from operations reserach. Solving such optimization problems is essential in many industries and is done every day. One important example of such optimization problems are Linear Programming problems. There are a couple of different algorithms that are popular in practice, among which is one which has been in use for almost 80 years. Nonetheless, our theoretical understanding of these algorithms is limited. This thesis makes progress towards a better understanding of these key algorithms for lineair programming, among which are the simplex method, interior point methods, and cutting plane methods