On the Maximum Connectivity Improvement Problem

In this paper, we define a new problem called the Maximum Connectivity Improvement (MCI) problem: given a directed graph \(G = (V,E)\), a weight function \(w:V \rightarrow \mathbb {N}_{\ge 0}\), a profit function \(p:V \rightarrow \mathbb {N}_{\ge 0}\), and an integer B, find a set S of at most B edges not in E that maximises \(f(S)=\sum _{v\in V}w_v\cdot p(R(v,S))\), where p(R(v, S)) is the sum of the profits of the nodes reachable from node v when the edges in S are added to G. We first show that we can focus on Directed Acyclic Graphs (DAG) without loss of generality. We prove that the MCI problem on DAG is \( NP \)-Hard to approximate to within a factor greater than \(1-1/e\) even if we restrict to graphs with a single source or a single sink, and MCI remains \( NP \)-Complete if we further restrict to unitary weights. We devise a polynomial time algorithm based on dynamic programming to solve the MCI problem on trees with a single source. We propose a polynomial time greedy algorithm that guarantees \((1-1/e)\)-approximation ratio on DAGs with a single source or a single sink.