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In this paper we try to describe the main characters of Heuristics ‘derived’ from Nature, a border area between Operations Research and Artificial Intelligence, with applications to graph optimization problems. These algorithms take inspiration from physics, biology, social sciences, and use a certain amount of repeated trials, given by one or more ‘agents’ operating with a mechanism of competition‐cooperation. Two introductory sections, devoted respectively to a presentation of some general concepts and to a tentative classification of Heuristics from Nature open the work. The paper is then composed of six review sections: each of them concerns a heuristic and its application to an NP‐hard combinatorial optimization problem. We consider the following topics: genetic algorithms with timetable problems, simulated annealing with dial‐a‐ride problems, sampling and clustering with communication spanning tree problems, tabu search with job‐shop‐scheduling problems, neural nets with location problems, ant system with travelling salesman and quadratic assignment problems.

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Feedback problems consist of removing a minimal number ofvertices of a directed or undirected graph in order to make it acyclic. The problem is known to be NPcomplete. In this paper we consider the variant on undirected graphs. The polyhedral structure of the Feedback V ertex Set polytope is studied. We prove that this polytope is full dimensional and show that some inequalities are facet de ning. We describe a new large class of valid constraints, the subset inequalities. A branch-and-cut algorithm for the exact solution of the problem is then outlined, and separation algorithms for the inequalities studied in the paper are proposed. A Local Search heuristic is described next. Finally we create a library of 1400 random generated instances with the geometric structure suggested by the applications, and we computationally compare the two algorithmic approaches on our library.