Exploring Structure and Reformulations in Different Integer Programming Algorithms

In this thesis we consider four topics all related to using problem reformulations in order to solve integer programs, i.e. optimization problems in which the decision variables must be integer.
We first consider the polyhedral approach. We start by addressing the ques- tion of lifting valid inequalities, i.e. finding a valid inequality for a set Y from the knowledge of a valid inequality for a lower-dimensional restriction X of Y . We simplify and clarify the presentation of the procedure. This allows us to derive conditions under which the computation of the lifting is tractable.
The second topic is the study of valid inequalities for the single node flow set. The single node flow set is the problem obtained by considering one node of a fixed charge network flow problem. We derive valid inequalities for this set and various generalizations. Our approach is a systematic procedure using only basic tools of integer programming: fixing and complementing variables, mixed- integer rounding and lifting. The method allows us to explain and generate a large range of inequalities describing the convex hull of such sets.
The last two topics are based on non-standard approaches for integer pro- gramming. We first show how the group relaxation approach can be used to provide reformulations for the integral basis method. This is based on a study of extended formulations for the group problem. We present four extended for- mulations and show that the projections of three of these formulations provide the convex hull of the original group problem. Initial computational tests of the approach are also reported.
Finally we consider a problem that is difficult for the standard branch-and- bound approach even for small instances. A reformulation based on lattice basis reduction is known to be more effective. However the step to compute the re- duced basis is O(n4) and becomes a bottleneck for small to medium instances. By using the structure of the problem, we show that we can decompose the problem and obtain the basis by taking the kronecker product of two smaller bases easier to compute. Furthermore, if the two small bases are reduced, the kronecker product is also reduced up to a reordering of the vectors. Computa- tional results show the gain from such an approach.