The S-labeling problem: An algorithmic tour

• Published in: Discrete Applied Mathematics 2018-09, Vol.246, p.49-61
• Main Authors: Fertin, Guillaume, Rusu, Irena, Vialette, Stéphane
• Format: Article
• Language: English
• Subjects: Algorithm; Applied mathematics; Caterpillars; Computer Science; Data Structures and Algorithms; Graph algorithms; Graph labeling; Graphs; Greedy algorithms; Labeling; Lower bounds
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2017.07.036

Given a graph G=(V,E) of order n and maximum degree Δ, the NP-complete S-labeling problem consists in finding a labeling of G, i.e. a bijective mapping ϕ:V→{1,2…n}, such that SLϕ(G)=∑uv∈Emin{ϕ(u),ϕ(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 SLϕ(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 ϕ of G such that SLϕ(G)≤|E|+k is solvable in O∗(22k(2k)!) 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.