Online and Approximate Network Construction from Bounded Connectivity Constraints

Abstract

The Network Construction problem, studied by Angluin et al., Hodosa 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 V of vertices, construction costs for all possible edges between pairs of vertices from V, and a sequence $S_1, S_2, \ldots \subseteq V$ of connectivity constraints, the objective is to find a set E of edges such that each $S_i$ induces a connected subgraph of the graph (V, E) and the total cost of E 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 $O(B^2\log n)$-competitive and $O((B+\log r)\log n)$-competitive polynomial-time algorithms along with an $\varOmega (B)$-competitive lower bound, where B is an upper bound on the size of constraints, while $r,\ n$ denote the number of constraints and the number of vertices, respectively. In the cost-uniform case, we provide an $\varOmega (\sqrt{B})$-competitive lower bound and an $O(\sqrt{n} (\log n + \log r))$-competitive upper bound with high probability, when constraints are unbounded. All our randomized competitive bounds are against an adaptive adversary, except for the last one which is against an oblivious 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 is supposed to induce a subgraph (of the constructed graph) of diameter at most d. Among other things, we provide a polynomial-time $(\left( {\begin{array}{c}B\\ 2\end{array}}\right) -B+2)\left( {\begin{array}{c}B\\ 2\end{array}}\right)$-approximation algorithm for the Network Construction problem with the d-diameter requirements.

Research supported in part by VR grant 2017-03750 (Swedish Research Council).