Variable Ordering for the Application of BDDs to the Maximum Independent Set Problem
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The ordering of variables can have a significant effect on the size of the reduced binary decision diagram (BDD) that represents the set of solutions to a combinatorial optimization problem. It also influences the quality of the objective function bound provided by a limited-width relaxation of the BDD. We investigate these effects for the maximum independent set problem. By identifying variable orderings for the BDD, we show that the width of an exact BDD can be given a theoretical upper bound for certain classes of graphs. In addition, we draw an interesting connection between the Fibonacci numbers and the width of exact BDDs for general graphs. We propose variable ordering heuristics inspired by these results, as well as a k-layer look-ahead heuristic applicable to any problem domain. We find experimentally that orderings that result in smaller exact BDDs have a strong tendency to produce tighter bounds in relaxation BDDs.
KeywordsVariable Ordering Fibonacci Number Binary Decision Diagram Constraint Optimization Problem Postoptimality Analysis
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- 4.Behle, M., Eisenbrand, F.: 0/1 vertex and facet enumeration with bdds. In: ALENEX. SIAM (2007)Google Scholar
- 6.Bollig, Wegener: Improving the variable ordering of OBDDs is NP-complete. IEEETC: IEEE Transactions on Computers 45 (1996)Google Scholar
- 12.Hadzic, T., Hooker, J.N.: Postoptimality analysis for integer programming using binary decision diagrams. Presented at GICOLAG Workshop (Global Optimization: Integrating Convexity, Optimization, Logic Programming, and Computational Algebraic Geometry), Vienna. Technical report, Carnegie Mellon University (2006)Google Scholar
- 13.Hadzic, T., Hooker, J.N.: Cost-bounded binary decision diagrams for 0-1 programming. Technical report, Carnegie Mellon University (2007)Google Scholar
- 17.Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Systems Technical Journal 38, 985–999 (1959)Google Scholar