Schema and ontology matching is a process of establishing correspondences between schema attributes and ontology concepts, for the purpose of data integration. Various commercial and academic tools have been developed to support this task. These tools provide impressive results on some datasets. However, as the matching is inherently uncertain, the developed heuristic techniques give rise to results that are not completely correct. In practice, post-matching human expert effort is needed to obtain a correct set of correspondences. We study this post-matching phase with the goal of reducing the costly human effort. We formally model this human-assisted phase and introduce a process of {\em matching reconciliation} that incrementally leads to identifying the correct correspondences. We achieve the goal of reducing the involved human effort, by exploiting a {\em network} of schemas that are matched against each other. We express the fundamental matching constraints present in the network in a declarative formalism, Answer Set Programming that in turn enables to reason about necessary user input. We demonstrate experimentally that our reasoning and heuristic techniques can indeed substantially reduce the necessary human involvement.