Algorithmic Aspects of the S-Labeling Problem

Given a graph \(G = (V,E)\) of order n and maximum degree \(\varDelta \), the NP-complete S-labeling problem consists in finding a labeling of G, i.e. a bijective mapping \(\phi : V \rightarrow \{1, 2 \ldots n\}\), such that \(\mathrm{SL}_{\phi }(G)=\sum _{\{u,v\} \in E} \min \{\phi (u), \phi (v)\}\) is minimized. A preliminary study of the S-labeling problem has been undertaken in [9]; here, we prolongate this study, and focus more specifically on algorithmic results concerning the problem. We first give intrinsic properties of optimal labelings, which will prove useful for our algorithmic study. We then show that the S-Labeling problem is polynomial-time solvable for (sets of) caterpillars. We also provide upper and lower bounds on \(\mathrm{SL}_{\phi }(G)\), that in turn allow us to determine polynomial-time approximation algorithms for different classes of graphs such as regular graphs, connected graphs and forests, but also for general graphs. Concerning exact algorithms, we show that the problem is solvable in \(O^*(1.44225^{n\varDelta })\) time, and that deciding whether there exist a labeling \(\phi \) of G such that \(\mathrm{SL}_{\phi }(G) \le |E| + k\) is solvable in \(O^*(2^{2\sqrt{k}}\ (2 \sqrt{k})!)\).