Identifying codes in hereditary classes of graphs and VC-dimension

Nicolas Bousquet (1, 2) , Aurélie Lagoutte (3) , Zhentao Li (4) , Aline Parreau (5, 6) , Stéphan Thomassé (3)

1 Department of Mathematics and Statistics [Montréal]
2 ALGCO - Algorithmes, Graphes et Combinatoire
3 MC2 - Modèles de calcul, Complexité, Combinatoire
4 ENS-PSL - École normale supérieure - Paris
5 Département de Mathématiques [Liège]
6 GOAL - Graphes, AlgOrithmes et AppLications

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes of graphs closed under induced subgraphs. Our dichotomy is derived from the VC-dimension of the considered class C, that is the maximum VC-dimension over the hypergraphs formed by the closed neighbourhoods of elements of C. We show that hereditary classes with infinite VC-dimension have infinitely many graphs with an identifying code of size logarithmic in the number of vertices while classes with finite VC-dimension have a polynomial lower bound. We then turn to approximation algorithms. We show that the problem of finding a smallest identifying code in a given graph from some class is log-APX-hard for any hereditary class of infinite VC-dimension. For hereditary classes of finite VC-dimension, the only known previous results show that we can approximate the identifying code problem within a constant factor in some particular classes, e.g. line graphs, planar graphs and unit interval graphs. We prove that it can be approximate within a factor 6 for interval graphs. In contrast, we show that on C_4-free bipartite graphs (a class of finite VC-dimension) it cannot be approximated to within a factor of c.log(|V|) for some c>0.

Mots clés: VC-dimension; approximation; Identifying codes; hereditary class; interval graph

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Nicolas Bousquet, Aurélie Lagoutte, Zhentao Li, Aline Parreau, Stéphan Thomassé. Identifying codes in hereditary classes of graphs and VC-dimension. SIAM Journal on Discrete Mathematics, 2015, 29 (4), pp.2047-2064. ⟨10.1137/14097879X⟩. ⟨hal-01038012v2⟩