Flow metrics

Abstract

We introduce ow metrics as a relaxation of path metrics (i.e. linear orderings). They are deÿned by polynomial-sized linear programs and have interesting properties including spreading. We use them to obtain relaxations for several NP-hard linear ordering problems such as minimum linear arrangement and minimum pathwidth. Our approach has the advantage of achieving the best-known approximation guarantees for these problems using the same relaxation and essentially the same rounding algorithm for all the problems while varying only the objective function from problem to problem. This is in contrast to the current state of the literature where each problem either has a new relaxation or a new rounding or both. We also characterize a natural projection of the ow polyhedron.

Exploiting geometric properties in combinatorial optimization = Benutten van geomwetrische eigenschappen in combinatiorische optimalisatie

“…Integer Linear programming is widely used to obtain good exact and approximation algorithms for different problems, see i.e. [24,44,72,75,78]. The advantage of this approach is that it is very generic and, therefore, we can apply many different well-developed LP-techniques to obtain better quality solutions or to speed-up the algorithm.…”

“…In this thesis we briefly show that the integrality gap of this formulation is at least Ω( √ n). We improve this formulation by merging it with the flow metric approach by Bornstein and Vempala [24]. The second ILP, the Tree Drawing formulation, is the outcome of this thesis.…”

“…In this subsection we give a brief introduction to the flow metric technique introduced by Bornstein and Vempala [24] where the notion flow metric refers to the relaxation of the path metric (i.e. linear ordering).…”

“…In other words, g u,v w represents the flow from u to v that goes through the vertex w. Definition 6.1.1. [24] A flow metric is a solution to the following linear program (F P ), where f and g are flow variables as defined above.…”

“…An important property of a flow metric is that it satisfies the one-dimensional spreading constraints as stated in the following theorem: Theorem 6.1.2. [24] For any subset S ⊆ V of vertices,…”