Binary stretch embedding of weighted graphs

Abstract

In this paper, we introduce and study the problem of binary stretch embedding of edge-weighted graphs in both integer and fractional settings. Roughly speaking, the binary stretch embedding problem for a weighted graph G is to find a mapping from the vertex set of G, to the vertices of a hypercube graph such that the distance between every pair of the vertices is not reduced under the mapping, hence the name binary stretch embedding. The minimum dimension of a hypercube for which such a stretch embedding exists is called the binary addressing number of G. We show that the binary addressing number of weighted graphs is the optimum value of an integer program. The optimum value for the corresponding linear relaxation problem is called the fractional binary addressing number of G. This embedding type problem is closely related to the well-known addressing problem of Graham and Pollak and isometric hypercube embedding problem of Firsov. Using tools and techniques such as Hadamard codes and the linear programming theory help us to find upper and lower bounds, approximations, or exact values of the binary addressing number and the fractional variant of graphs. As an application of our results, we derive improved upper bounds or exact values of the maximum size of Lee metric codes of certain parameters.