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Investigating decomposition methods for the maximum common subgraph and sum colouring problems

  • MINOT, Maël
Publication Date
Dec 19, 2017
Kaleidoscope Open Archive
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The objective of this thesis is, from a general standpoint, to design and evaluate decomposition methods for solving constrained optimisation problems. Two optimisation problems in particular are considered: the maximum common induced subgraph problem, in which the largest common part between two graphs is to be found, and the sum colouring problem, where a graph must be coloured in a way that minimises a sum of weights induced by the employed colours.The maximum common subgraph (MCIS) problem is notably difficult, with a strong applicability in domains such as biology, chemistry and image processing, where the need to measure the similarity between structured objects represented by graphs may arise. The outstanding difficulty of this problem makes it strongly advisable to employ a decomposition method, possibly coupled with a parallelisation of the solution process. However, existing decomposition methods are not well suited to solve the MCIS problem: some lead to a poor balance between subproblems, while others, like tree decomposition, are downright inapplicable.To enable the structural decomposition of such problems, Chmeiss et al. proposed an approach, TR-decomposition, acting at a low level: the microstructure of the problem. This approach had yet to be applied to the MCIS problem. We evaluate it in this context, aiming at reducing the size of the search space while also enabling parallelisation. Moreover, we introduce a post-decomposition step that focuses on alleviating redundancies between subproblems.The second problem that caught our interest is the sum colouring problem. It is an NP-hard variant of the widely known classical graph colouring problem. As in most colouring problems, it basically consists in assigning colours to the vertices of a given graph while making sure no neighbour vertices use the same colour. In the sum colouring problem, however, each colour is associated with a weight. The objective is to minimise the sum of the weights of the colours used by every vertex. This leads to generally harder instances than the classical colouring problem, which simply requires to use as few colours as possible.Only a few exact methods have been proposed for this problem. Among them stand notably a constraint programming (CP) model, a branch and bound approach, as well as an integer linear programming (ILP) model.We led an in-depth investigation of CP's capabilities to solve the sum colouring problem, while also looking into ways to make it more efficient. Additionally, we evaluated a combination of integer linear programming and constraint programming, with the intention of conciliating the strong points of these highly complementary approaches. We took inspiration from the classical backtracking bounded by tree decomposition (BTD) approach. We employ a tree decomposition with a strictly bounded height. Constraint programming is used to enumerate consistent assignment of the root cluster. For each of these assignments, one subproblem is trivially obtained for each leaf cluster. Subproblems are then independently solved to optimality using ILP.We then derive profit from the complementarity of our approaches by developing a portfolio approach. The resulting solver is able to choose one of the considered approaches automatically by relying on a number of features extracted from each instance.

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