Search strategies for hybrid search spaces

Recently, there has been much interest in enhancing purely combinatorial formalisms with numerical information. For example, planning formalisms can be enriched by taking resource constraints and probabilistic information into account. The mixed integer programming (MIP) paradigm from operations research provides a natural tool for solving optimization problems that combine such numeric and non-numeric information. The MIP approach relies heavily on linear program relaxations and branch-and-bound search. This is in contrast with depth-first or iterative deepening strategies generally used in AI. We provide a detailed characterization of the structure of the underlying search spaces as explored by these search strategies. Our analysis indicates that the traditional approach of identifying dominating search strategies for a given problem domain is inadequate. We show that much can be gained from combining search strategies for solving hard MIP problems, thereby leveraging the strength of different search strategies regarding both the combinatorial and numeric components of the problem.