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. space constraints, proofs are totally absent from this paper. They will be available in its journal version.