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Combinatorial and Algorithmic aspects of Reconfiguration

Valentin Bartier 1 
1 G-SCOP_OC - Optimisation Combinatoire
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : Combinatorial reconfiguration problems consist in finding step-by-step transformations between two feasible solutions of an optimization problem, called the source problem, in such a way that all intermediate solutions are also feasible. Each step of the transformation consists in applying an atomic modification of the current solution. Such a modification is called a reconfiguration operation. Reconfiguration problems model dynamic situations where a given solution already in place has to be modified for a more desirable one while maintaining some particular properties throughout the transformation.Given a source problem and a reconfiguration operation we can define the reconfiguration graph as follows: it is the graph which vertex set is the set of all feasible solutions of the source problem, and such that two solutions are adjacent if and only if one can be obtained from the other by applying exactly one reconfiguration operation.In this thesis, we study two fundamental questions in reconfiguration. The first one is the reachability question: given an initial feasible solution of a problem, a target feasible solution of the same problem, and a reconfiguration operation, is it possible to transform the initial one into the target one by successively applying the given reconfiguration operation, and such that each intermediate solution is also feasible? Equivalently speaking, is there a path between the initial solution and the target solution in the corresponding reconfiguration graph? Such a transformation is called a valid transformation.The second question we study concerns the diameter of the reconfiguration graph: Suppose that the reconfiguration graph is connected, in other words that there exists a valid transformation between any two feasible solutions of the source problem. What is the diameter of the reconfiguration graph? Equivalently speaking, what is the maximum number of reconfiguration operation to apply to transform any solution into any other? In this thesis, we focus on reconfiguration problems on graphs. We first investigate the graph recoloring problem. In this problem, we consider a graph and proper colorings of this graph, and the reconfiguration operation consists in modifying the color of one vertex. The number of colors that can be used in a transformation is limited. We focus on the question of finding transformations of linear size in the order of the input graph. We consider in particular chordal graphs and graphs of tree width two, and improve the best known bounds on the number of colors needed to obtain such transformations.The second part of this thesis is devoted to the independent set reconfiguration problem. We see an independent set as a number of tokens that are placed on the vertices of the independent set. We consider two reconfiguration operations. First, the token jumping operation: a token can be moved anywhere on the graph at each step. The second one is the sliding operation: a token can only move to a neighbor of the vertex it lies on.We focus on the complexity of the reachability problem defined by these operations. We first prove that the problem is PSPACE-complete on H-free graphs for both operations, unless H belongs to a finite set of graphs mathcal{H} that we describe, and investigate the complexity of the problem on H-free graph for some H in mathcal{H}. We then study the impact of girth on the parameterized complexity of the problem for both operations, and give a precise limit between tractability and intractability in classes related to bipartite graph for the token sliding operation. We then introduce a new model of independent set reconfiguration which we call galactic token sliding as a tool for parameterized reductions and use it to show that the token sliding reachability problem is fixed-parameter tractable when parameterized by the size of the independent sets on graphs of bounded-degree.
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Valentin Bartier. Combinatorial and Algorithmic aspects of Reconfiguration. Computational Complexity [cs.CC]. Université Grenoble Alpes [2020-..], 2021. English. ⟨NNT : 2021GRALM055⟩. ⟨tel-03643495⟩

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