ℓ1-embeddability of 2-dimensional ℓ1-rigid periodic graphs

Abstract

The ℓ1-embedding problem of a graph is the problem to find a map from its vertex set to â" d such that the length of the shortest path between any two vertices is equal to the ℓ1- distance between the mapping of the two vertices in â" d . The ℓ1-embedding problem partially contains the shortest path problem since an ℓ1-embedding provides the all-pairs shortest paths. While Höfting and Wanke showed that the shortest path problem is NP-hard, Chepoi, Deza, and Grishukhin showed a polynomial-time algorithm for the ℓ1-embedding of planar 2-dimensional periodic graphs. In this paper, we study the ℓ1-embedding problem on ℓ1- rigid 2-dimensional periodic graphs, for which there are finite representations of the ℓ1-embedding. The periodic graphs form a strictly larger class than planar ℓ1-embeddable 2-dimensional periodic graphs. Using the theory of geodesic fiber, which was originally proposed by Eon as an invariant of a periodic graph, we show an exponential-time algorithm for the ℓ1-embedding of ℓ1-rigid 2-dimensional periodic graphs, including the non-planar ones. Through Höfting and Wanke's formulation of the shortest path problem as an integer program, our algorithm also provides an algorithm for solving a special class of parametric integer programming. © 2014 Springer International Publishing Switzerland.