Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

Abstract

For an undirected graph G = (V, E) and a k-non-negative integer vector p = (p1, . . . , p k ), a mapping l : V → N ∪ {0} is called an L(p)-labeling of G if |l(u) − l(v)| ≥ p d for any two distinct vertices u, v ∈ V with distance d, and the maximum value of {l(v) | v ∈ V } is called the span of l. Originally, L(p)-labeling of G for p = (2, 1) is introduced in the context of frequency assignment in radio networks, where 'close' transmitters must receive different frequencies and 'very close' transmitters must receive frequencies that are at least two frequencies apart so that they can avoid interference. L(p)-Labeling is the problem of finding the minimum span λp among L(p)labelings of G, which is NP-hard for every non-zero p. L(p)-Labeling is well studied for specific p's; in particular, many (exact or approximation) algorithms for general graphs or restricted classes of graphs are proposed for p = (2, 1) or more generally p = (p, q). Unfortunately, most algorithms strongly depend on the values of p, and it is not apparent to extend algorithms for p to ones for another p ′ in general. In this paper, we give a simple polynomial-time reduction of L(p)-Labeling on graphs with a small diameter to Metric (Path) TSP, which enables us to use numerous results on (Metric) TSP. On the practical side, we can utilize various high-performance heuristics for TSP, such as Concordo and LKH, to solve our problem. On the theoretical side, we can see that the problem for any p under this framework is 1.5-approximable, and it can be solved by the Held-Karp algorithm in O(2 n n 2 ) time, where n is the number of vertices, and so on.