Three-Dimensional Channel Routing is in N P

It is mentioned in our previous paper [4] that the 3-D channel routing is NP-complete, and the proof of the NPhardness is outlined there. The purpose of this paper is to show that the 3-D channel routing is in NP , complementing the previous paper. The 3-D channel is a 3-D grid G consisting of columns, rows, and layers which are rectilinear grid planes defined by fixing x-, y-, and z-coordinates at integers, respectively. The numbers of columns, rows, and layers are called the width, depth, and height of G, respectively. (See Fig. 1.) G is called a (W,D,H)-channel if the width is W , depth is D, and height is H . A vertex of G is a grid point with integer coordinates. We assume without loss of generality that the vertex set of a (W,D,H)-channel is {(x, y, z)|x ∈ [W ], y ∈ [D], z ∈ [H ]}, where [n] = {1, 2, . . . , n} for a positive integer n. Layers defined by z = H and z = 1 are called the top and bottom layers, respectively. A terminal is a vertex of G located in the top or bottom layer. A net is a set of terminals to be connected. A net containing k terminals is called a k-net. The object of the 3-D channel routing problem is to connect the terminals in each net with a tree inG using as few layers as possible and as short wires as possible in such a way that trees spanning distinct nets are vertex-disjoint. A set of nets is said to be routable in G if G has vertex-disjoint trees spanning the nets. A set of nets of a (W,D,H)-channel is said to be routable with height H if it is routable in the (W,D,H)channel. We consider the following decision problem.