Elsevier

Discrete Optimization

Volume 10, Issue 4, November 2013, Pages 310-319
Discrete Optimization

Searching for realizations of finite metric spaces in tight spans

https://doi.org/10.1016/j.disopt.2013.08.002Get rights and content
Under an Elsevier user license
open archive

Abstract

An important problem that commonly arises in areas such as internet traffic-flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs. Such a graph is called an optimal realization and finding such realizations is known to be NP-hard. Recently Varone presented a heuristic greedy algorithm for computing optimal realizations. Here we present an alternative heuristic that exploits the relationship between realizations of the metric D and its so-called tight span TD. The tight span TD is a canonical polytopal complex that can be associated to D, and our approach explores parts of TD for realizations in a way that is similar to the classical simplex algorithm. We also provide computational results illustrating the performance of our approach for different types of metrics, including l1-distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.

Keywords

Combinatorial optimization
Metric
Graph
Realization
Tight span

Cited by (0)