Approximability and inapproximability of the minimum certificate dispersal problem

Given an n-vertex directed graph G=(V,E) and a set R⊆V×V of requests, we consider assigning a set of edges to each vertex in G so that for every request (u,v) in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinalities of the edge sets assigned to each vertex. This problem has been shown to be NP-hard in general, though it is polynomially solvable for some restricted classes of graphs and restricted request structures, such as bidirectional trees with requests of all pairs of vertices. In this paper, we give an advanced investigation about the difficulty of MCD by focusing on the relationship between its (in)approximability and request structures. We first show that MCD with general R has Θ(log n) lower and upper bounds on approximation ratio under the assumption P≠NP. We then assume R forms a clique structure, called Subset-Full, which is a natural setting in the context of the application. Interestingly, under this natural setting, MCD becomes 2-approximable, though it has still no polynomial time approximation algorithm whose factor is better than 677/676 unless P=NP. Finally, we show that this approximation ratio can be improved to 3/2 for the undirected variant of MCD with Subset-Full.