On the S-Labeling problem

Let G be a graph of order n and size m. A labeling of G is a bijective mapping θ:V(G)→{1,2,…,n}, and we call Θ(G) the set of all labelings of G. For any graph G and any labeling θ∈Θ(G), let SL(G,θ)=∑e∈E(G) min{θ(u):u∈e}. In this paper, we consider the S-Labeling problem, defined as follows: Given a graph G, find a labeling θ∈Θ(G) that minimizes SL(G,θ). The S-Labeling problem has been shown to be NP-complete [S. Vialette, Packing of (0, 1)-matrices, Theoretical Informatics and Applications RAIRO 40 (2006), no. 4, 519–536]. We prove here basic properties of any optimal S-labeling of a graph G, and relate it to the Vertex Cover problem. Then, we derive bounds for SL(G,θ), and we give approximation ratios for different families of graphs. We finally show that the S-Labeling problem is polynomial-time solvable for split graphs.Due to space constraints, proofs are totally absent from this paper. They will be available in its journal version.