On Radiocoloring Hierarchically Specified Planar Graphs: PSPACE-Completeness and Approximations

Hierarchical speci cations of graphs have been widely used in many important applications, such as VLSI design, parallel programming and software engineering. A well known hierarchical speci cation model, considered in this work, is that of Lengauer [9, 10], referred to as L-speci cations. In this paper we discuss a restriction on the Lspeci cations resulting to graphs which we call Well-Separated (WS). This class is characterized by a polynomial time (to the size of the speci cation of the graph) testable combinatorial property. In this work we study the Radiocoloring Problem (RCP) on WS Lspeci ed 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 di erent 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, G, where G has the same vertex set as G and there is an edge between any two vertices of G if their distance in G is at most 2. We rst show that RCP is PSPACE-complete for WS L-speci ed hierarchical planar graphs. Second, we present a polynomial time 3approximation algorithm as well as a more eÆcient 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]. ? This work has been partially supported by the EU IST/FET projects ALCOM-FT, FLAGS, CRESCCO and EU/RTN Project ARACNE. Part of the last author's work was done during his visit at Max-Planck-Institute f ur Informatik (MPI).