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,.. lambda(h,k)}, such that any two vertices v(i) and v(j) receive colors c(v(i)) and c(v(j)) satisfying the following conditions: i) if v(i) and v(j) are adjacent then vertical bar c(v(i)) - c(v(j))vertical bar >= h; ii) if v(i) and v(j) are at distance two then vertical bar c(v(i)) - c(v(j))vertical bar >= k. The aim of the L(h, k)-labelling problem is to minimize lambda(h,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 DIP-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 vertex- and 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 (4)/D-3(2), where, D is equal to the minimum even value bounding the minimum of the maximum degrees of the two partitions. One of the above coloring algorithms is in fact an approximating edge-coloring algorithm for hypergraphs of maximum dimension d, i.e. the maximum edge cardinality, with performance ratio d. Furthermore, we consider a different approximation technique based on the reduction of the L(h, k)-labelling problem to the vertex-coloring of the square of a graph. Using this approach we derive an approximation algorithm with performance ratio bounded by min(h, 2k)root n + o(k root n), assuming h >= k. Hence, the first technique is competitive when D O(n(1/4)) These algorithms match with a result in [2] stating that L(1,1) labelling n-vertex bipartite graphs is hard to approximate within(n1/2-)epsilon, for any epsilon > 0, unless NP=ZPP. We then extend the latter approximation strategy to s-partite graphs, obtaining a (min(h, sk)root n + o(sk root n))-approximation ratio, and to general graphs deriving an (h root n + o(h root n))-approximation algorithm, assuming h >= k. Finally, we prove that the L(h, k)-labelling problem is not easier than coloring the square of a graph.