Mapping and Combining Combinatorial Problems into Energy Landscapes via Pseudo-Boolean Constraints Priscila M.V. Lima 1 , Glaucia C. Pereira 2 , M. Mariela M. Morveli-Espinoza 2 , and Felipe M.G. Fran¸ ca 2 1 NCE/Instituto de Matem´atica, UFRJ, Rio de Janeiro, Brazil priscila@nce.ufrj.br http://www.geti.dcc.ufrj.br/ 2 COPPE – Sistemas e Computa¸c˜ ao, UFRJ, Rio de Janeiro, Brazil {gpereira, mme, felipe}@cos.ufrj.br http://www.cos.ufrj.br/∼felipe Abstract. This paper introduces a novel approach to the specification of hard combinatorial problems as pseudo-Boolean constraints. It is shown (i) how this set of constraints defines an energy landscape representing the space state of solutions of the target problem, and (ii) how easy is to combine different problems into new ones mostly via the union of the corresponding constraints. Graph colouring and Traveling Sales- person Problem (TSP) were chosen as the basic problems from which new combinations were investigated. Higher-order Hopfield networks of stochastic neurons were adopted as search engines in order to solve the mapped problems. Keywords: Higher-order Networks; Graph Colouring; Pseudo-Boolean Constraints; Satisfiability; Simulated Annealing; TSP. 1 Introduction The ability to learn associative behaviour through examples is a desirable fea- ture in an adaptive system. Nevertheless, it would not be practical to acquire, through examples, certain pieces of knowledge that had already been learnt by other systems. Besides, sometimes it is easier to describe a problem via its con- straints to an artificial neural network (ANN) such that the set of its global energy minima corresponds to the set of solutions to the problem in question. For example, an explanation of how the Traveling Salesperson Problem (TSP) can be defined as a set of mathematical constraints that are solvable by an ANN can be found in [6] and [5]. Alternatively, constraints may be essentially logical, constituting a kind of description or specification of a suitable solution for a problem being modeled. A problem that apparently does not involve optimizing a cost function is that of finding a model for a logical sentence. In propositional logic, that would consist M. De Gregorio et al. (Eds.): BVAI 2005, LNCS 3704, pp. 308–317, 2005. c  Springer-Verlag Berlin Heidelberg 2005