On the Approximability of the L(h, k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

Given an undirected graph G, an L(h, k)-labelling of G assigns colors to vertices from the integer set {0, . . . , λh,k}, such that any two vertices vi and vj receive colors c(vi) and c(vj) satisfying the following conditions: i) if vi and vj are adjacent then |c(vi)− c(vj)| ≥ h; ii) if vi and vj are at distance two then |c(vi)− c(vj)| ≥ k. The aim of the L(h, k)-labelling problem is to minimize λh,k. In this paper we study the approximability of the L(h, k)labelling problem on bipartite graphs and extend the results to s-partite and general graphs. Indeed, the decision version of this problem is known to be NP-complete in general and, to our knowledge, there are no polynomial solutions, either exact or approximate, for bipartite graphs. Here, we state some results concerning the approximability of the L(h, k)-labelling problem for bipartite graphs, exploiting a novel technique, consisting in computing approximate vertexand edge-colorings of auxiliary graphs to deduce an L(h, k)-labelling for the input bipartite graph. We derive an approximation algorithm with performance ratio bounded by