Connected Dominating Sets

PROBLEM DEFINITION Consider a graph G = (V,E). A subset C of V is called a dominating set if every vertex is either in C or adjacent to a vertex in C. If, furthermore, the subgraph induced by C is connected, then C is called a connected dominating set. A connected dominating set with a minimum cardinality is called a minimum connected dominating set (MCDS). Computing a MCDS is an NP-hard problem and there is no polynomial-time approximation with performance ratio ρH(∆) for ρ < 1 unless NP ⊆ DTIME(n ln ) where H is the harmonic function and ∆ is the maximum degree of the input graph [10]. A unit disk is a disk with radius one. A unit disk graph (UDG) is associated with a set of unit disks in the Euclidean plane. Each node is at the center of a unit disk. An edge exists between two nodes u and v if and only if |uv| ≤ 1 where |uv| is the Euclidean distance between u and v. This means that two nodes u and v are connected with an edge if and only if u’s disk covers v and v’s disk covers u. Computing an MCDS in a unit disk graph is still NP-hard. How hard is it to construct a good approximation for MCDS in unit disk graphs? Cheng et al. [5] answered this question by presenting a polynomial-time approximation scheme.