# The S-labeling problem: An algorithmic tour

Abstract : 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)=Sum_{uv \in 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) \leq |E|+k is solvable in O*(2^(2 sqrt(k)) (2 sqrt(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.
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https://hal.archives-ouvertes.fr/hal-01710032
Contributor : Stéphane Vialette <>
Submitted on : Thursday, February 15, 2018 - 3:16:50 PM
Last modification on : Thursday, February 27, 2020 - 1:05:04 AM

### Citation

Guillaume Fertin, Irena Rusu, Stéphane Vialette. The S-labeling problem: An algorithmic tour. Discrete Applied Mathematics, Elsevier, 2018, 246, pp.49-61. ⟨10.1016/j.dam.2017.07.036⟩. ⟨hal-01710032⟩

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