An Effective SAT Encoding for Magic Labeling

This work presents a Boolean satisfiability (SAT) encoding for a special problem from combinatorial optimization. In the last years much progress has been made in the optimization of practical SAT solvers (see the SAT competition [5]). This has made SAT encodings for combinatorial problems highly attractive. In this work we propose an encoding for the combinatorial problem Magic Labeling which has important applications in the field of wireless networks [4]. It is defined as follows. Let an undirected, unweighted graph G = (V,E) be given with vertex set V and edge set E, where |V | = n and |E| = m. A labeling is a one-to-one mapping λ : V ∪E → {1, 2, . . . ,m+ n}. Define the weight ω(e) of an edge e ∈ E as the sum of the label of e and of the labels of its two endpoints. An edge-magic total labeling (EMTL) is a labeling λ for which a constant h ∈ N exists such that ω(e) = h for each edge e ∈ E. Similarly, define the weight ω(v) of a vertex v ∈ V as the sum of the label of v and of the labels of all edges incident to v. A vertex-magic total labeling (VMTL) is a labeling λ for which a constant k ∈ N exists such that ω(v) = k for each vertex v ∈ V . Finally, a totally magic labeling (TML) is a labeling λ for which (not necessarily equal) constants h, k ∈ N exist such that λ is edge-magic with constant h and vertex-magic with constant k. h and k are called magic constants. A vertex v ∈ V and an edge e ∈ E are denoted neighboring, if e is incident to v. Note that different EMTLs exist for the same graph, and the same holds for VMTLs. Surveys of results for magic graphs are given in [4]. We consider the following three problems for a given graph: (1) Does an EMTL exist with given magic constant h ∈ N? (2) Does a VMTL exist with given magic constant k ∈ N? (3) Does a TML exist with given magic constants h, k ∈ N?