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On packing, colouring and identification problems

Abstract : In this thesis we study three theoretical computer science problems, namely the orthogonal packing problem (OPP for short), strong edge-colouring and identifying codes. OPP consists in testing whether a set of rectangular items can be packed in a rectangular container without overlapping and without exceeding the borders of this container. An additional constraint is that the rotation of the items is not allowed. The problem is NP-hard even when the problem is reduced to packing squares in a square. We propose an exact algorithm for solving OPP efficiently using the characterization of the problem by interval graphs proposed by Fekete and Schepers. For this purpose we use some compact representation of interval graphs - MPQ-trees. We show experimental results of our approach by comparing them to the results of other algorithms known in the literature. We observe promising gains. The study of strong edge-colouring and identifying codes is focused on the structural and computational aspects of these combinatorial problems. In the case of strong edge-colouring we are interested in the families of planar graphs and subcubic graphs. We show optimal upper bounds for the strong chromatic index of subcubic graphs as a function of the maximum average degree. We also show that every planar subcubic graph without induced cycles of length 4 and 5 can be strong edge-coloured with at most nine colours. Finally, we confirm the difficulty of the problem by showing that it remains NP-complete even in some restricted classes of planar subcubic graphs. For the subject of identifying codes we propose a characterization of non-trivial graphs having maximum identifying code number, that is n-1, where n is the number of vertices. We study the case of line graphs and prove lower and upper bounds for identifying code number in this class. At last we investigate the complexity of the corresponding decision problem and show the existence of a linear algorithm for computing the identifying code number of the line graph L(G) where G has the size of the tree-width bounded by a constant. On the other hand, we show that the identifying code problem is NP-complete in various subclasses of planar graphs.
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Submitted on : Monday, March 18, 2013 - 5:39:56 PM
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Petru Valicov. On packing, colouring and identification problems. Data Structures and Algorithms [cs.DS]. Université Sciences et Technologies - Bordeaux I, 2012. English. ⟨tel-00801982⟩

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