Additionally, in diametral path graphs, we reduce the approximate MIS problem to finding the solution with the constraint of bounded diameter. By using a property of the separator, we then obtain a polynomial-time algorithm to find a MIS in diametral path graphs with bounded degree. In this thesis, we are going to introduce a separator construction for a diametral path graph. However, many efficient algorithms and approximation schemes to resolve the MIS problem are found in some special graph classes. Most scientists believe that there is no polynomial-time algorithm for this problem in general graphs. Constructing an algorithm to find a maximum independent set (MIS) is one of the first NP-Hard problems which have been studied for a long time. Nowadays, graph algorithms have been applied to many practical issues such as networking, very large-scale integration (VLSI), and transport systems, etc. Our algorithms are numerically illustrated on a subway network design problem and a facility location problem. The second algorithm considers the special case of s-t paths and leads to a fully-polynomial time approximation scheme. We study in details the resulting approximation ratio, which depends on the structure of the metric space and of the feasible subgraphs. The first one relies on solving a nominal counterpart of the problem considering pairwise worst-case distances. We also propose two types of polynomial-time approximation algorithms. In view of this, we propose en exact solution algorithm combining integer programming formulations with a cutting plane algorithm, identifying the cases where the separation problem can be solved efficiently. We first prove that these problems are NP-hard even when the feasible subgraphs consist either of all spanning trees or of all s-t paths. The objective is to minimize the sum of the distances in the chosen subgraph for the worst positions of the vertices in their uncertainty sets. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets. Many discrete optimization problems amount to select a feasible subgraph of least weight.
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