The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography 1 Tiziana Calamoneri Department of Computer Science “Sapienza” University of Rome - Italy via Salaria 113, 00198 Roma, Italy. Email: [email protected] Given any fixed nonnegative integer values h and k, the L(h, k)- labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with minimum span. The L(h, k)-labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author. Keywords: L(h, k)-labelling; frequency assignment; radiocoloring; λ-coloring; distance-2-coloring; D2-vertex coloring Received February 2011; revised 00 Month 2011 1. INTRODUCTION follows: One of the key topics in graph theory is graph Definition 1.1. Given a graph G = coloring. Fascinating generalizations of the no- (V, E) and two nonnegative integers h and tion of graph coloring are motivated by prob- k, an L(h, k)-labelling is an assignment of lems of channel assignment in wireless commu- nonnegative integers to the nodes of G such nications, traffic phasing, fleet maintenance, that adjacent nodes are labelled using colors task assignment, and other applications. (See at least h apart, and nodes having a common [156] for a survey.) neighbour are labelled using colors at least While in the classical vertex coloring k apart. The aim of the L(h, k)-labelling problem [105] a condition is imposed only on problem is to minimize the span σh,k(G), colors of adjacent nodes, many generalizations i.e. the difference between the largest and the require colors to respect stronger conditions, smallest used colors. The minimum span over e.g. restrictions are imposed on colors both of all possible labelling functions is denoted by adjacent nodes and of nodes at distance 2 in λh,k(G) and is called λh,k-number of G. the graph. This paper will focus on a specific graph Observe that this definition imposes a coloring generalization that arose from a condition on labels of nodes connected by a channel assignment problem in radio networks 2 length path instead of using the concept [90]: the L(h, k)-labelling problem, defined as of distance 2, that is very common in the The Computer Journal, Vol. ??, No. ??, ???? 2 T. Calamoneri literature. The reason is that this definition L(p, q)-coloring problem, distance-2-coloring works both when h k and when h < k. and D2-vertex coloring problem (when h = The present formulation≥ allows the nodes of a k = 1), radiocoloring problem and λ-coloring triangle to be labelled with three colors at least problem (when h = 2 and k = 1). max h, k apart from each other, although Many variants of the problem have been { } they are at mutual distance 1; when h k introduced in the literature, as well: instead of ≥ the two definitions coincide. minimizing the span, seek the L(h, k)-labelling Furthermore, as the smallest used color that minimizes the order, i.e. the number of is usually 0, an L(h, k)-labelling with span effectively used colors [90]; given a span σ, σh,k(G) can use σh,k(G) + 1 different colors; decide whether it is possible to L(h, k)-label this feature is slightly counter-intuitive, but is the input graph using all colors between 0 and kept for historical reasons. σ (no-hole L(h, k)-labelling) [157]; consider the The notion of L(h, k)-labelling was intro- color set as a cyclic interval, i.e. the distance duced by Griggs and Yeh in the special case between two labels i, j 0, 1,...,σ defined h = 2 and k = 1 [89, 184] in connection as min i j , σ + 1 ∈i { j [101];} use a with the problem of assigning frequencies in a more general{| − | model in− which | − |} the labels and multihop radio network (for a survey on the separations are real numbers [86]; generalize class of frequency assignment problems, see the problem to the case when the metric is e.g. [2, 58, 123, 144]), although it has been described by a graph H (H(h, k)-labelling) previously mentioned by Roberts [155] in his [68]; consider the precoloring extension, where summary on T -colorings and investigated in some nodes of the graph are given as already the special case h = 1 and k = 1 as a com- (pre)colored, and the question is if this binatorial problem and hence without any ref- precoloring can be extended to a proper erence to channel assignment (see for instance coloring of the entire graph using a given [180]). number of colors [71]; consider a one-to- After its definition, the L(h, k)-labelling one L(h, k)-labelling (L′(h, k)-labelling) [42]; problem has been used to model several L(h, k)-label a digraph, where the distance problems, for certain values of h and k. from a node x to a node y is the length of Some examples are the following: Bertossi a shortest dipath from x to y [40]; study and Bonuccelli [16] introduced a kind of another parameter, called edge-span, defined integer ”control code” assignment in packet as the minimum, over all feasible labellings, radio networks to avoid hidden collisions, of the max f(u) f(v) : (u, v) E(G) equivalent to the L(0, 1)-labelling problem; [185]; impose{| the labelling− | to be balanced,∈ i.e.} channel assignment in optical cluster based all colors must be used more or less the same networks [10] can be seen either as the L(0, 1)- number of times (equitable coloring) [133]. or as the L(1, 1)-labelling problem, depending Some of these generalizations are considered on the fact that the clusters can contain one in [183]. ore more nodes; more in general, channel assignment problems, with a channel defined The extent of the literature and the huge as a frequency, a time slot, a control code, number of papers concerning the L(h, k)- etc., can be modeled by an L(h, k)-labelling labelling problem have been the main moti- problem, for convenient values of h and k. vation of the surveys [2, 29, 183], each one ap- Besides the practical aspects, also purely proaching the problem from a different point theoretical questions are very interesting. of view (operative research, graph algorithms These are only some reasons why there is and extremal combinatorial, respectively), but considerable literature devoted to the study they are all published at least five years ago. of the L(h, k)-labelling problem, following Since a substantial progress has been achieved many different approaches, including graph in the last years, the author thinks that an theory and combinatorics [156, 169], simulated updated survey and annotated bibliography annealing [54, 141], genetic algorithms [50, would be useful. The present paper is an up- 132], tabu search [39], and neural networks date of [29]. [76, 131]. In all these contexts, the problem In this work, the case k = 0, for any fixed h, has been called with different names; among is not considered as this problem becomes the others, we recall: L(h, k)-labelling problem, classical vertex coloring problem. Instead, a The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 3 particular accent is posed on the special cases Studying a completely different problem h =0, 1, 2 and k = 1: the first one is equivalent (Hessian matrices of certain non linear func- to the problem of optimally coloring the square tions), McCormick [142] gives a greedy algo- of the input graph and the second one has been rithm that guarantees a O(√n)-approximation considered in the seminal works by Roberts, for coloring the square of a graph. The algo- Griggs and Yeh. Both these problems have rithm is based on the greedy technique: con- been intensively studied. sider the nodes in any order, then the color The decision version of the L(h, k)-labelling assigned to node vi is the smallest color that problem has been proved to be NP-complete, has not been used by any node which is at even under restrictive hypotheses. Section 2 distance at most 2 from vi; the performance lists these results. In Section 3 some general ratio is obtained by simple considerations on the degree of G and of its square. lower and upper bounds on the value of λh,k are summarized. Approaching an equivalent scheduling prob- For some special classes of graphs a labelling lem, Ramanathan and Lloyd [152] present an can be computed efficiently, while for other approximation algorithm with a performance classes of graphs only approximate algorithms guarantee of O(θ), where θ is the thickness of are known. Both these kinds of results are the graph. Intuitively, the thickness of a graph described in Section 4. measures ”its nearness to planarity”. More formally, the thickness of a graph G = (V, E) In the rest of this paper we will consider is the minimum number of subsets into which simple and loopless graphs with n nodes, the edge set E must be partitioned so that each maximum degree ∆, chromatic number χ(G), subset in the partition forms a planar graph on clique number ω(G) and girth (i.e.