The algebraic approaches to graph transformation are based on the concept of gluing of graphs, modelled by pushouts in suitable categories of graphs and graph morphisms. This allows one not only to give an explicit algebraic or set theoretical description of the constructions, but also to use concepts and results from category theory in order to build up a rich theory and to give elegant proofs even in complex situations. In this chapter we start with an overwiev of the basic notions common to the two algebraic approaches, the double-pushout (DPO) approach and the singlepushout (SPO) approach; next we present the classical theory and some recent development of the double-pushout approach. The next chapter is devoted instead to the single-pushout approach, and it is closed by a comparison between the two approaches.

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The introduction of digital terrestrial broadcasting all over Europe requires a complete and challenging replanning of in-place analog systems. However, an abrupt migration of resources (transmitters and frequencies) from analog to digital networks cannot be accomplished because the analog services must be preserved temporarily. Hence, a multiobjective problem arises, in which several networks sharing a common set of resources have to be designed. This problem is referred to as the network packing problem. In Italy, this problem is particularly challenging because of a large number of transmitters, orographical features, and strict requirements imposed by Italian law. In this paper, we report our experience in developing solution methods at the major Italian broadcaster Radiotelevisione Italiana (RAI S.p.A.). We propose a two-stage heuristic. In the first stage, emission powers are assigned to each network separately. In the second stage, frequencies are assigned to all networks so as to minimize the loss from mutual interference. A software tool incorporating our methodology is currently in use at RAI to help discover and select high-quality alternatives for the deployment of digital equipment.Subject classifications: communications: frequency and power assignment; integer programming, heuristic: neighborhood search. Area of review: OR Practice.

The task of scheduling communications between satellites and ground control stations is getting more and more critical since an increasing number of satellites must be controlled by a small set of stations. In such a congested scenario, the current practice, in which experts build hand-made schedules, often leaves a large number of communication requests unserved. We report on our experience in the design of an optimization-based support tool at the European Space Agency. We propose a tight time-indexed formulation of the problem able to include several complex technological constraints. A non-standard Lagrangian heuristic is then devised which provides near-optimal solutions of a set of largescale test problems arising in the forthcoming GALILEO constellation. The heuristic shows numerical stability and robustness adequate for practical implementation. The resulting tool is used by the Italian reference operator for GALILEO system management and is currently under testing at the European Space Agency.