Abstract
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.
This work was supported by NSF under grant CMMI-1130012 and AFOSR under grant FA-95501110180.
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References
Akers, S.B.: Binary decision diagrams. IEEE Transactions on Computers C-27, 509–516 (1978)
Andersen, H.R., Hadzic, T., Hooker, J.N., Tiedemann, P.: A Constraint Store Based on Multivalued Decision Diagrams. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 118–132. Springer, Heidelberg (2007)
Becker, B., Behle, M., Eisenbrand, F., Wimmer, R.: BDDs in a Branch and Cut Framework. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 452–463. Springer, Heidelberg (2005)
Behle, M., Eisenbrand, F.: 0/1 vertex and facet enumeration with bdds. In: ALENEX. SIAM (2007)
Bergman, D., van Hoeve, W.-J., Hooker, J.N.: Manipulating MDD Relaxations for Combinatorial Optimization. In: Achterberg, T., Beck, J.C. (eds.) CPAIOR 2011. LNCS, vol. 6697, pp. 20–35. Springer, Heidelberg (2011)
Bollig, Wegener: Improving the variable ordering of OBDDs is NP-complete. IEEETC: IEEE Transactions on Computers 45 (1996)
Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers C-35, 677–691 (1986)
Calkin, N.J., Wilf, H.S.: The number of independent sets in a grid graph. SIAM J. Discrete Math. 11(1), 54–60 (1998)
Dyer, M.E., Frieze, A.M., Jerrum, M.: On counting independent sets in sparse graphs. SIAM J. Comput. 31(5), 1527–1541 (2002)
Ebendt, R., Gunther, W., Drechsler, R.: An improved branch and bound algorithm for exact BDD minimization. IEEE Trans. on CAD of Integrated Circuits and Systems 22(12), 1657–1663 (2003)
Forbes, F., Ycart, B.: Counting stable sets on cartesian products of graphs. Discrete Mathematics 186(1-3), 105–116 (1998)
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)
Hadzic, T., Hooker, J.N.: Cost-bounded binary decision diagrams for 0-1 programming. Technical report, Carnegie Mellon University (2007)
Hadzic, T., Hooker, J.N., O’Sullivan, B., Tiedemann, P.: Approximate Compilation of Constraints into Multivalued Decision Diagrams. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 448–462. Springer, Heidelberg (2008)
Hoda, S., van Hoeve, W.-J., Hooker, J.N.: A Systematic Approach to MDD-Based Constraint Programming. In: Cohen, D. (ed.) CP 2010. LNCS, vol. 6308, pp. 266–280. Springer, Heidelberg (2010)
Jordan, C.: Sur les assemblages de lignes. J. Reine Angew Math. 70, 185–190 (1869)
Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Systems Technical Journal 38, 985–999 (1959)
Zhao, Y.: The number of independent sets in a regular graph. Combinatorics, Probability & Computing 19(2), 315–320 (2010)
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Bergman, D., Cire, A.A., van Hoeve, WJ., Hooker, J.N. (2012). Variable Ordering for the Application of BDDs to the Maximum Independent Set Problem. In: Beldiceanu, N., Jussien, N., Pinson, É. (eds) Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems. CPAIOR 2012. Lecture Notes in Computer Science, vol 7298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29828-8_3
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