Algorithmic Operations Research Vol.5 (2010) 49–64 Probabilistic optimization in graph-problems

We study a probabilistic optimization model for graph-problems under vertex-uncertainty. We assume that any vertex vi of the input-graph G(V,E) has only a probability pi to be present in the final graph to be optimized (i.e., the final instance for the problem tackled will be only a sub-graph of the initial graph). Under this model, the original “deterministic” problem gives rise to a new (deterministic) problem on the same input-graph G, having the same set of feasible solutions as the former one, but its objective function can be very different from the original one, the set of its optimal solutions too. Moreover, this objective function is a sum of 2|V | terms; hence, its computation is not immediately polynomial. We give sufficient conditions for large classes of graph-problems under which objective functions of their probabilistic counterparts are polynomially computable and optimal solutions are well-characterized. Finally, we apply these general results to natural and well-known combinatorial problems that belong to the classes considered. Key words: graph, complexity, approximation, a priori optimization 1