Given a graph of order and maximum degree , the NP-complete -labeling problem consists in finding a labeling of , i.e. a bijective mapping , such that is minimized. In this paper, we study the -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 , 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 of such that is solvable in time, thus fixed-parameterized tractable in . We finally show that the -Labeling problem is polynomial-time solvable for two classes of graphs, namely split graphs and (sets of) caterpillars.
A preliminary version of this work appeared in Proceedings of the 26th International Workshop on Combinatorial Algorithms (IWOCA 2015) (Fertin et al., 2015) [10] and in Proceedings of the 5th European Conference on Combinatorics, Graph Theory and Applications(EuroComb 2009) (Fertin and Vialette, 2009) [11].