Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Abstract

An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum LocatingDominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.

Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

“…Locating-Dominating Set was rst proved to be NP-complete in [7], a result extended to bipartite graphs in [5]. This was improved to pla-nar bipartite unit disk graphs [29] and to planar bipartite subcubic graphs [14].…”

“…On the positive side, Locating-Dominating Set is constant-factor approximable for bounded degree graphs [20], line graphs [14,15], interval graphs [4] and is linear-time solvable for graphs of bounded clique-width (using Courcelle's theorem [8]). Furthermore, an explicit linear-time algorithm solving LocatingDominating Set on trees is known [33].…”

Centroidal bases in graphs

“…The positive approximation bound follows, as Minimum Locating‐Dominating Set is well known to be ‐approximable, see for example Gravier et al . Moreover, it follows from a reduction for Minimum Identifying Code in the first author's thesis [5, section 6.4] and a lemma from Gravier et al (see also Foucaud ) that Minimum Locating‐Dominating Set is NP‐hard to approximate within a factor of for graphs having a vertex adjacent to all other vertices. This proves the nonapproximability bound.…”

Parameterized and approximation complexity of Partial VC Dimension

“…We now study the four dual parameterizations for Partial VC Dimension, that is, parameters |E|−k, |X|−k, |E|−ℓ and |X|−ℓ. Note that Distinguishing Transversal is NP-hard for neighborhood hypergraphs of graphs (see for example [35]). This corresponds to Partial VC Dimension with |X| = |E| = ℓ and hence Partial VC Dimension is not in XP for parameters |E| − ℓ and |X| − ℓ.…”