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

Abstract

Given an undirected graph G, an L(h,k)-labelling of G assigns colors to vertices from the integer set {0, ..., λ _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 |c(v _i ) – c(v _j )| ≥ h; ii) if v _i and v _j are at distance two then |c(v _i ) – c(v _j )| ≥ k. The aim of the L(h,k)-labelling problem is to minimize λ _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 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 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 $\frac{4}{3} D^2$, 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)$\sqrt{n} + o(k + \sqrt{n})$, 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 n ^{1/2 − ε}, for any ε > 0, unless NP = ZPP.

We then extend the latter approximation strategy to s-partite graphs, obtaining a (min(h, sk)$\sqrt{n} + o(sk \sqrt{n})$)-approximation ratio, and to general graphs deriving an $(h\sqrt{n} + o(h\sqrt{n}))$-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.

Partially supported by the Italian Research Project PRIN 2003 ”Optimization, simulation and complexity of the design and management of communication networks”.