Online and Approximate Network Construction from Bounded Connectivity Constraints

Abstract

The Network Construction problem, studied by Angluin et al., Hosoda et al., and others, asks for a minimum-cost network satisfying a set of connectivity constraints which specify subsets of the vertices in the network that have to form connected subgraphs. More formally, given a set [Formula: see text] of vertices, construction costs for all possible edges between pairs of vertices from [Formula: see text], and a sequence [Formula: see text] of connectivity constraints, the objective is to find a set [Formula: see text] of edges such that each [Formula: see text] induces a connected subgraph of the graph [Formula: see text] and the total cost of [Formula: see text] is minimized. First, we study the online version where every constraint must be satisfied immediately after its arrival and edges that have already been added can never be removed. We give an [Formula: see text]-competitive and [Formula: see text]-competitive polynomial-time algorithms, where [Formula: see text] is an upper bound on the size of constraints, while [Formula: see text] denote the number of constraints and the number of vertices, respectively. On the other hand, we observe that an [Formula: see text]-competitive lower bound as well as an [Formula: see text]-competitive lower bound in the cost-uniform case are implied by the known lower bounds for unbounded constraints. For the cost-uniform case with unbounded constraints, we provide an [Formula: see text]-competitive upper bound with high probability. The latter bound is against an oblivious adversary while our other randomized competitive bounds are against an adaptive adversary. Next, we discuss a hybrid approximation method for the (offline) Network Construction problem combining an approximation algorithm of Hosoda et al. with one of Angluin et al. and an application of the hybrid method to bioinformatics. Finally, we consider a natural strengthening of the connectivity requirements in the Network Construction problem, where each constraint has to induce a subgraph (of the constructed graph) of diameter at most [Formula: see text]. Among other things, we provide a polynomial-time [Formula: see text]-approximation algorithm for the Network Construction problem with the [Formula: see text]-diameter requirements, when each constraint has at most [Formula: see text] vertices, and show the APX-completeness of this variant.