Branch-and-bound trees, integrality gaps and online optimization : a tale of algorithms and randomness

Optimization problems are everywhere: from creating high school timetables to designing computer chips. Many such problems can be reduced to one single generic type of problem, called a mixed-integer-program (MIP). For this reason, computer programs known as MIP-solvers that are able to solve such problems can be used to solve a wide variety of other problems. Hence, MIP-solvers are widely employed in practice, both in industry and academia. Although there are many different MIP-solvers, they all generally apply the same algorithm: the branch-and-bound algorithm. We know that the running time of branch-and-bound can be exponential in the worst case, but in practice much lower running times are often observed. Up untill recently, no good theoretical explanation for this phenomenon was known.

In this thesis, we make a step towards resolving this paradox, by doing a theoretical average-case analysis of the branch-and-bound algorithm. As part of this analysis we show that for several classes of randomly generated problems, the running time of the branch-and-bound algorithm is only polynomial in the problem size. We also design a new node selection rule for the branch-and-bound algorithm that performs that performs better than previously known rules in the theoretical 'explorable heap'-model. Furthermore, we provide new results about the online hypergraph matching problem We also provide an experimental analysis of supplementary techniques for an exact version of the branch-and-bound algorithm.