Consider a set of oligomers listing the subunits involved in sub-complexes of a macro-molecular assembly, obtained e.g. using native mass spectrometry or affinity purification. Given these oligomers, connectivity inference (CI) consists of finding the most plausible contacts between these subunits, and minimum connectivity inference (MCI) is the variant consisting of finding a set of contacts of smallest cardinality. MCI problems avoid speculating on the total number of contacts, but yield a subset of all contacts and do not allow exploiting a priori information on the likelihood of individual contacts. In this context, we present two novel algorithms, ALGO-MILP-W and ALGO-MILP-WB.The former solves the minimum weight connectivity inference (MWCI), an optimization problem whose criterion mixes the number of contacts and their likelihood. The latter uses the former in a bootstrap fashion, to improve the sensitivity and the specificity of solution sets.Experiments on the yeast exosome, for which both a high resolution crystal structure and a large set of oligomers is known, show that our algorithms predict contacts with high specificity and sensitivity, yielding a very significant improvement over previous work.The software accompanying this paper is made available, and should prove of ubiquitous interest whenever connectivity inference from oligomers is faced.