On Radiocoloring Hierarchically Specified Planar Graphs: -Completeness and Approximations

Hierarchical specifications of graphs have been widely used in many important applications, such as VLSI design, parallel programming and software engineering. A well known hierarchical specification model, considered in this work, is that of Lengauer [9, 10] referred to as L-specifications. In this paper we discuss a restriction on the L-specifications resulting to graphs which we call Well-Separated (WS). This class is characterized by a polynomial time (to the size of the specification of the graph) testable combinatorial property.In this work we study the Radiocoloring Problem (RCP) on WS L-specified hierarchical planar graphs. The optimization version of RCP studied here, consists in assigning colors to the vertices of a graph, such that any two vertices of distance at most two get different colors. The objective here is to minimize the number of colors used. This problem is equivalent to the problem of vertex coloring the square of a graph G, G2, where G2 has the same vertex set as G and there is an edge between any two vertices of G2 if their distance in G is at most 2.We first show that RCP is % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFpepucqWFse-ucqWF % pepucqWFaeFqcqWFce-qcqWFWesraaa!4989! $$ \mathcal{P}\mathcal{S}\mathcal{P}\mathcal{A}\mathcal{C}\mathcal{E} $$ -complete for WS L-specified hierarchical planar graphs. Second, we present a polynomial time 3-approximation algorithm as well as a more efficient 4-approximation algorithm for RCP on graphs of this class.We note that, the best currently known approximation ratio for the RCP on ordinary (non-hierarchical) planar graphs of general degree is 2 ([6, 1]). Note also that the only known results on any kind of coloring problems have been shown for another special kind of hierarchical graphs (unit disk graphs) achieving a 6-approximation solution [13].