The Subset Interconnection Design problem with Subset Uncertainty

Subset Interconnection Design is an NP −hard problem which draws its motivation from various applications, such as the design of scalable overlay networks, the inference of inter-molecular protein connectivity, the design of vacuum systems and several other problems in the industry. In that problem, we are given a set of vertices V and a collection S of subsets of V and we want to find an edge set E of minimum cardinality such that every subset Si of the collection S induces a connected subgraph of G = (V,E). In this work, we concern ourselves with a generalization of the problem, which arises mainly from computational structural biology and where our data is covered by Subset Uncertainty. Under this scope, V is partitioned into a set of types T , and we are not aware of the exact composition of each subset Si. We only know the number of vertices of each type that it is composed by. So, apart from having to find the edge set E of minimum cardinality respecting the constraints, we also have to find a collection S with such composition, that will lead us to the optimal edge set. In the context of Subset Uncertainty we start by formally defining the problem and consider its hardness. Following, we analyse the problem and the structure of the solution to give bounds for the solution. Later on, we model the problem using an MILP approach. Moreover, we present and analyse an approximation algorithm. Finally, we devise heuristics based on LP-relaxation and conclude by making a series of experiments on different datasets comparing all the aforementioned methods, and presenting the results.