Complexity vs. optimality: Unraveling source-destination connection in uncertain graphs

Abstract

Determination of source-destination connectivity in networks has long been a fundamental problem, where most existing works are based on deterministic graphs that overlook the inherent uncertainty in network links. To overcome such limitation, this paper models the network as an uncertain graph where each edge e exists independently with some probability p(e). The problem examined is that of determining whether a given pair of nodes, a source s and a destination t, are connected by a path or separated by a cut. Assuming that during each determining process we are associated with an underlying graph, the existence of each edge can be unraveled through edge testing at a cost of c(e). Our goal is to find an optimal strategy incurring the minimum expected testing cost with the expectation taken over all possible underlying graphs that form a product distribution. Formulating it into a combinatorial optimization problem, we first characterize the computational complexity of optimally determining source-destination connectivity in uncertain graphs. Specifically, through proving the NP-hardness of two closely related problems, we show that, contrary to its counterpart in deterministic graphs, this problem cannot be solved in polynomial time unless P=NP. Driven by the necessity of designing an exact algorithm, we then apply the Markov Decision Process framework to give a dynamic programming algorithm that derives the optimal strategies. As the exact algorithm may have prohibitive time complexity in practical situations, we further propose two more efficient approximation schemes compromising the optimality. The first one is a simple greedy approach with linear approximation ratio. Interestingly, we show that naive as it is, it has comparable performance than some other seemingly more sophisticated algorithms. Second, by harnessing the sub-modularity of the problem, we further design a more elaborate algorithm with better approximation ratio. The effectiveness of the proposed algorithms are justified through extensive simulations on three real network datasets, from which we demonstrate that the proposed algorithms yield strategies with smaller expected cost than conventional heuristics.