The Distance Orientation Problem

Abstract

The Distance Orientation Problem (DOP) is formulated as follows: Given a graph with positive weights on its edges, are there weights for the vertices, such that for every edge x y it holds that the absolute difference between the weights of x and y is equal to the weight of x y ? This problem can also be formulated as a problem of finding a special orientation of G and was motivated by an application in Shape from Shading, a method in the field of Computer Vision. We present a linear-time algorithm for complete 3-cover graphs, a generalization of chordal graphs and planar triangulations. For outerplanar graphs we show that the DOP is fixed-parameter tractable and present a pseudo-polynomial time algorithm for integral edge weights. Both algorithms use the idea that the existence of feasible weights for the vertices of an outerplanar graph can be decided by only looking at the edge weights of each face of its outerplane embedding separately. We show that this property does not hold for any embedding of a planar graph which is not outerplanar. Furthermore, we prove that the DOP is strongly NP -complete for grid graphs, i.e., there is no pseudo-polynomial algorithm to solve it unless P = NP .