Statistics and Computing 14: 181–198, 2004 C  2004 Kluwer Academic Publishers. Manufactured in The Netherlands. Node deletion sequences in inﬂuence diagrams using genetic algorithms M. G ´ OMEZ ∗ and C. BIELZA † ∗ Department of Computer Science and Artiﬁcial Intelligence, University of Granada, Spain mgomez@decsai.ugr.es † Department of Artiﬁcial Intelligence, Technical University of Madrid, Spain mcbielza@ﬁ.upm.es Received August 2002 and accepted November 2003 Inﬂuence diagrams are powerful tools for representing and solving complex inference and decision- making problems under uncertainty. They are directed acyclic graphs with nodes and arcs that have a precise meaning. The algorithm for evaluating an inﬂuence diagram deletes nodes from the graph in a particular order given by the position of each node and its arcs with respect to the value node. In many cases, however, there is more than one possible node deletion sequence. They all lead to the optimal solution of the problem, but may involve different computational efforts, which is a primary issue when facing real-size models. Finding the optimal deletion sequence is a NP-hard problem. The proposals given in the literature have proven to require complex transformations of the inﬂuence diagram. In this paper, we present a genetic algorithm-based approach, which merely has to be added to the inﬂuence diagram evaluation algorithm we use, and whose codiﬁcation is straightforward. The experiments, varying parameters like crossover and mutation operators, population sizes and mutation rates, are analysed statistically, showing favourable results over existing heuristics. Keywords: decision-making under uncertainty, inﬂuence diagrams, genetic algorithms, NP-hard problems, node deletion sequence, statistical analysis 1. Introduction An inﬂuence diagram (Howard and Matheson 1984) is a directed acyclic graph that is very commonly used in decision-making and inference problems. It has three types of nodes: (1) de- cision nodes (rectangular) representing decisions to be made; (2) chance nodes (circular) representing uncertainties modelled by probability distributions; and (3) a value node (diamond- shaped) with no successors, representing the (expected) utilities that model decision-maker’s preferences. The arcs have differ- ent meanings depending upon which node they are directed to: the arcs to chance nodes or the value node indicate probabilistic dependence and functional dependence, respectively, while the arcs pointing at a decision node indicate the information known at the time of making that decision. The former are called con- ditional arcs and are what we are interested in here. The latter are only informational. The basic operations of the Shachter’s evaluation algorithm (Shachter 1986) are chance node removal (by computing the expected utility) and decision node removal (by maximizing the expected utility), carried out provided that the chance/decision nodes are predecessors of the value node. A chance node can be removed if its only successor is the value node. As a conse- quence, the latter inherits all the removed node predecessors. To eliminate a decision node that precedes the value node, it is as- sumed that there are no barren (or sink) nodes and that all other conditional predecessors of the value node are informational pre- decessors of the decision node. The value node inherits no new conditional predecessors. When it is possible to remove neither a chance nor a decision node, the algorithm ﬁnds an arc between two chance nodes that can be reverted –via Bayes’ formula–, each node inheriting the predecessors of the other one. This transformation will make it possible to ﬁnally ﬁnd (perhaps after various arc reversals) a chance node to remove. All these opera- tions are combined sequentially to give the standard algorithm. Let G = ( N , A) denote the inﬂuence diagram with nodes N and arcs A. Let D and C be the sets in N of decision and chance nodes, respectively, and v be the value node. Given a node i , the sets C (i ), I (i ), S(i ) denote its conditional predecessors, its informational predecessors and its direct successors, 0960-3174 C  2004 Kluwer Academic Publishers