Complexity and Approximability of Problems Related to Integer Programming

Many problems of both practical and theoretical importance concern themselves not only with finding a solution but also finding a ‘best’ solution in some sense. As a concrete example, let us consider the travelling salesperson problem (TSP): a salesperson has to visit some specified towns on a tour starting and ending in her/his home town; in which order should the towns be visited in order to minimize the length of the tour? We see that this problem actually becomes meaningless if we only ask the question whether a tour exists or not (under some plausible assumptions on the available communication system). This kind of problems are often referred to as combinatorial optimization problems. They are typically computationally hard (NP-hard) to solve exactly but they can sometimes be efficiently approximated. Let us consider TSP once again: if the distances satisfy the triangle inequality, then the exact problem is NP-hard but a tour that is at most 1.5 longer than the optimal tour can be found in low-order polynomial time. If the distances are Euclidean, then the exact problem is still NP-hard but the optimal tour can be approximated within 1+ ǫ for every ǫ > 0, cf. [1]. Thus, useful results can indeed be obtained by using approximate methods. In this project, we concentrate on optimisation problems that are (more or less) closely related to integer programming, i.e. the optimisation problem max{cx | Ax ≤ b, x ∈ N} where A is an m × n rational matrix, b is a rational m-vector, and c is a rational n-vector. Integer programming has been used for solving an enormous amount of problems ranging from network planning to optimal control to scheduling to coding theory, and it is an archetypical example of an NP-hard problem. This problem has been intensively studied from various viewpoints and a good overview of this research can be found in [9].