The S-LABELING problem: An algorithmic tour

Given a graph G = (V, E) of order n and maximum degree Delta, the NP-complete S-LABELING problem consists in finding a labeling of G, i.e. a bijective mapping phi : V -> {1, 2...n) such that SL phi(G) = Sigma(uv is an element of E) min{phi(u), phi(v)} is minimized. In this paper, we study the S-LABELING problem, with a particular focus on algorithmic issues. We first give intrinsic properties of optimal labelings, which will prove useful for our algorithmic study. We then provide lower bounds on SL phi(G), together with a generic greedy algorithm, which collectively allow us to approximate the problem in several classes of graphs in particular, we obtain constant approximation ratios for regular graphs and bounded degree graphs. We also show that deciding whether there exists a labeling phi of G such that SL phi(G) <= vertical bar E vertical bar + k is solvable in O*(2(2 root k) (2 root k)!) time, thus fixed-parameterized tractable in k. We finally show that the S-LABELING problem is polynomial-time solvable for two classes of graphs, namely split graphs and (sets of) caterpillars. (C) 2017 Published by Elsevier B.V.