Title:

Coloured graph models : associating emitters and ships

This thesis examines graphtheoretic approaches to problems arising in emitter toplatform association, for example in associating emitters and ships. The first part of this thesis focuses on emitters that are observed by two different sensors, which can only determine the bearing the observed signal was emitted from. The aim is to pair observations from the two sensors, that originate from the same emitter. In this thesis it is shown that different types of observations can be represented as one of c colours and each observed bearing as a row or a column in an n by n grid. Given the horizontal and vertical projections of a coloured grid, the aim is to reconstruct the original layout of the coloured squares on the grid. Deciding whether two sets of positive natural numbers are the horizontal and vertical projections of a coloured grid is NPcomplete for two or more colours [24, 35, 47] and has close connections to colour degree matrix problems. Necessary and sufficient conditions are known for a demand matrix to be a colour degree matrix of an edgecoloured forest [9, 22]. In this thesis the first step beyond forests is taken: necessary and sufficient conditions for a demand matrix to be realisable by a graph with at most one cycle are proved. As part of the proof some directly forced structures are discovered, that is, structures that must exist in every realisation. Moreover, corresponding results for multigraphs and pseudoforests with at most k cyclic components are presented. This part concludes with O (n2 ) time algorithms to check these conditions and return a witness if one of the conditions is violated. Finally O(n3) time algorithms to find a realisation, if one exists, are described. The second part of this thesis introduces the coloured Lmodel. This is an original idealised graph model, developed to explore the combinatorial properties of the emittertoplatform association problem, referred to as the reverse radar problem in this thesis. The aim is to decide which groups of observed radars originate from the same ship, taking into account which combinations of radar models are known to be carried by different types of ships. It is shown what exactly makes finding a solution to this idealised version of the reverse radar problem NPhard, and that there are tractable cases equivalent to finding (weighted) matchings in related graphs, which have bounded pathwidth. Restricting to graphs with bounded pathwidth is a reasonable simplification, as signals come in along bearings (i.e. along a cycle) and a cycle has pathwidth two. New algorithms to find (weighted) matchings in graphs with bounded pathwidth and treewidth are presented, which can be used to solve the reverse radar problem in this model. Finally, the likelihood and twotime models are introduced, which complement the coloured Lmodel by generalising measurement errors and by introducing time. It is shown that the problem remains NPhard, even for very simple cases and some tractable cases are described.
