Abstract
Zero-suppressed binary decision diagrams (ZDDs) are important data structures that are used in a number of combinatorial optimization settings. This paper explores a ZDD characterization for the maximal independent sets of a graph; a necessary and sufficient condition for when nodes in the ZDD can be merged is provided, and vertex orderings of the graph are studied to determine which orderings produce smaller ZDDs. A bound on the width of the maximal independent set ZDD is obtained, relating it to the Fibonacci numbers. Finally, computational results are reported.
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References
Akers SB (1978) Binary decision diagrams. IEEE Trans Comput 100(6):509–516
Behle M (2007) Binary decision diagrams and integer programming. PhD thesis, Universistät des Saarlands
Behle M, Eisenbrand F (2007) 0/1 vertex and facet enumeration with BDDs. In: Workshop on algorithm engineering and experiments (ALENEX)
Bergman D, Cire AA, Hoeve WV, Hooker JN (2012) Variable ordering for the application of BDDs to the maximum independent set problem. In: Beldiceanu N, Jussien N, Pinson E (eds) Lecture notes in computer science., Integration of AI and OR techniques in contraint programming for combinatorial optimzation problems, no. 7298Springer, Berlin, pp 34–49
Bergman D, Cire AA, Hoeve WV, Hooker JN (2013) Optimization bounds from binary decision diagrams. INFORMS J Comput
Bollig B, Wegener I (1996) Improving the variable ordering of OBDDs is NP-complete. IEEE Trans Comput 45(9):993–1002
Bron C, Kerbosch J (1973) Algorithm 457: finding all cliques of an undirected graph. Commun ACM 16(9):575–577
Bryant R (1986) Graph-based algorithms for boolean function manipulation. IEEE Trans Comput C–35(8):677–691
Bryant RE (1992) Symbolic boolean manipulation with ordered binary-decision diagrams. ACM Comput Surv (CSUR) 24(3):293–318
Coudert O (1997) Solving graph optimization problems with ZBDDs. In: European design and test conference, pp 224–228
Eppstein D, Strash D (2011) Listing all maximal cliques in large sparse real-world graphs. In: Pardalos PM, Rebennack S (eds) Lecture notes in computer science., Experimental algorithms, no. 6630Springer, Berlin, pp 364–375
Hadžić T, Hooker J (2008) Postoptimality analysis for integer programming using binary decision diagrams. Tech. Rep. 167, Tepper School of Business
Held S, Cook W, Sewell EC (2012) Maximum-weight stable sets and safe lower bounds for graph coloring. Math Program Comput 4(4):363–381
Johnson DS, Trick MA (1996) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, 11–13 Oct 1993
Lee C (1959) Representation of switching circuits by binary-decision programs. Bell Syst Tech J 38(4):985–999
Minato S (1993) Zero-suppressed BDDs for set manipulation in combinatorial problems. In: 30th Conference on design automation, pp 272–277
Minato S (2001) Zero-suppressed BDDs and their applications. Int J Softw Tools Technol Transf 3(2):156–170
Morrison DR, Sewell EC, Jacobson SH (2014) Solving the pricing problem in a generic branch-and-price algorithm using zero-suppressed binary decision diagrams arXiv:1401.5820 [cs.DS], http://arxiv.org/abs/1401.5820. Accessed 04 Mar 2014
Sedgewick R, Wayne K (2011) Algorithms. Addison-Wesley Professional
Tomita E, Tanaka A, Takahashi H (2006) The worst-case time complexity for generating all maximal cliques and computational experiments. Theor Comput Sci 363(1):28–42
Trick MA (2005) Computational series: graph coloring and its generalizations. http://mat.gsia.cmu.edu/COLOR02/. Accessed 04 Mar 2014
Weisstein EW (2013) Binet’s fibonacci number formula—from wolfram MathWorld. http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html. Accessed 04 Mar 2014
Acknowledgments
The authors would like to thank two anonymous referees and the associate editor for their detailed review of our paper, which resulted in a significantly improved draft. The computational results reported were obtained at the Simulation and Optimization Laboratory at the University of Illinois, Urbana-Champaign. This research has been supported in part by the Air Force Office of Scientific Research (FA9550-10-1-0387), as well as fellowships through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program and National Science Foundation Graduate Research Fellowship Program. Additionally, the third author was supported in part by (while serving at) the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government.
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Morrison, D.R., Sewell, E.C. & Jacobson, S.H. Characteristics of the maximal independent set ZDD. J Comb Optim 28, 121–139 (2014). https://doi.org/10.1007/s10878-014-9722-4
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DOI: https://doi.org/10.1007/s10878-014-9722-4