Searching for realizations of finite metric spaces in tight spans

Abstract

An important problem that commonly arises in areas such as internet trafficflow 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 T D . The tight span T D is a canonical polytopal complex that can be associated to D, and our approach explores parts of T D 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 l 1 -distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.

Optimal realizations of two-dimensional, totally-decomposable metrics

“…Apart from having an intrinsic mathematical interest, if this conjecture were true, it could provide new strategies for computing optimal realisations, as it would provide a "search space" (albeit a rather large one in general) in which to systematically search for optimal realisation [12].…”

“…In fact this immediately follows from a somewhat stronger theorem that we shall prove (Theorem 4.1), which shows that a certain special type of optimal realisation of a two-dimensional, totally-decomposable metric d can be found as a homeomorphic subgraph of (G d , w ∞ ). Note also that Theorem 1.2 implies that the optimal realisation problem for l 1 -planar metrics is equivalent to the Minimum Manhattan Network (MMN) problem; since the MMN problem is NP-hard [4], the optimal realisation problem for two-dimensional metrics is also NP-hard (see [12,Section 5] for more details and some algorithmic consequences).…”

Optimal realizations and the block decomposition of a finite metric space

“…In another direction, it could be of interest to understand more deeply how the structure of the tight span of a metric is related to the structure of optimal realizations of the metric. For example, it is shown in [16] that optimal realizations of a totally split decomposable metric D whose tight span is 2-dimensional can always be found in F ≤1 (D), a fact which is used in [17] to help compute optimal realizations of such metrics D. In general, clarifying the precise relationship between optimal realizations of D and F ≤1 (D) remains an important open problem.…”