Hardness and Approximation of Octilinear Steiner Trees

Abstract

Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or ±45° diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the so-called X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NP-completeness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a planar graph of size [Formula: see text] which contains a (1 + ε)–approximation of a minimum octilinear Steiner tree for every ε > 0 and n = |K|. Hence, we can apply any α–approximation algorithm for the Steiner tree problem in graphs (for planar graphs, Borradaile et al. very recently presented a polynomial time approximation scheme) and achieve an (α + ε)–approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons).

Optimal Interconnection Trees in the Plane

“…Until recently, NP-hardness had been established for only two cases, both of which are examples of -geometry Steiner tree problems. These two cases are the rectilinear metric ( D 2) [169] and the octilinear metric ( D 4) [288]. Here we extend these results to all -geometry Steiner tree problems with > 2, using a restriction of the terminals to parallel lines very similar to that employed in Sect.…”

“…3.4. Also of note is a graph-based approximation algorithm for the octilinear obstacle-avoiding Steiner tree problem proposed by Müller-Hannemann and Schulze [288]. superficial way.…”

Approximation of Octilinear Steiner Trees Constrained by Hard and Soft Obstacles

“…The rectilinear and the Euclidean Steiner tree problem have been shown to be NP-Hard in [GJ77] and [GGJ77], respectively. Quite recently, we have been able to prove that the octilinear Steiner tree problem is also NP-hard in the strong sense [MS05].…”

“…Hence, this transformation is not polynomial. Müller-Hannemann and Schulze [MS05] recently constructed a graph of size O(n 2 /ε 2 ) which contains for every ε > 0 a (1 + ε)-approximation for the case without obstacles and with hard obstacles.…”