Some lower bounds on sparse outer approximations of polytopes

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid inequalities. As an extension to this work, we study the following less idealized questions in this paper: (1) Are there integer programs, such that sparse inequalities do not approximate the integer hull well even when added to a linear programming relaxation? (2) Are there polytopes, where the quality of approximation by sparse inequalities cannot be significantly improved by adding a budgeted number of arbitrary (possibly dense) valid inequalities? (3) Are there polytopes that are difficult to approximate under every rotation? (4) Are there polytopes that are difficult to approximate in all directions using sparse inequalities? We answer each of the above questions in the positive

Sparsity in integer programming

Sparse input data is data in which most of the data coefficients are zero. Many areas of scientific computing and optimization have been very successful in harnessing the effect of sparsity of input data to improve the efficacy of algorithms. Surprisingly, the use of sparsity of input data is a very under explored direction of research in the context of Integer Programming. Harnessing the sparsity present in the underlying linear relaxation, using decomposition/reformulation techniques and complexity results for approximation algorithms correspond to most of the previous results in this area. In this thesis, we deal with understanding the effect of sparsity in Integer Programming. We study how to approximate polytopes using sparse cuts under various settings. We propose a variant on feasibility pump that automatically detects and harnesses sparsity. We study the ratio of the number of integral extreme points to the total number of extreme points for a family of random polytopes as a function of sparsity. Finally, we discuss the strength of multi-row aggregation cuts in the context of sign-pattern integer programs.Ph.D