Binary decision diagrams for computing the non-dominated set

Transcription

1 Binary decision diagrams for computing the non-dominated set July 13, 2015 Antti Toppila and Ahti Salo 27th European Conference on Operational Research, July 2015, University of Strathclyde, Glasgow, Scotland

2 Portfolio selection with interval-values 1/2 Projects = 1,.., Cost of project Value of project Value uncertain: Plausible lower and upper bound Feasible values = Project value Project cost

3 Portfolio selection with interval-values 2/2 Portfolio 0,1 =1if project in portfolio =0otherwise Cost of portfolio = Feasible portfolios = 0,1 Portfolio value = Portfolio value 1 2 1&2 3 2&3 1&2&3 1&3 Portfolio cost B

4 Non-dominated set Portfolio dominates portfolio ( ) iff ) for all ) for some Non-dominated (ND) portfolios = { such that } Portfolio value Infeasible = dominance = ND set Portfolio cost B

5 Computing the non-dominated set 1/2 Let 0,1 Contains all ND portfolios Check all pairs Discarding procedure: Do for all : If or ( and ) All infeasible or dominated portfolios removed from

6 Computing the non-dominated set 2/2 Checking 4 pairs of portfolios impossible for large Algorithm by Liesiö et al. 2007, 2008 Prescreens a substantially smaller to begin with Reduces the number of pairs to check using transitivity Works well when not too large If prescreen prematurely terminated, may not include any/all ND portfolios Memory for storing the matrix may exceed available capacity Discarding procedure may fail if too many pairs to check

7 Our contribution Binary decision diagrams (Bryant 1986) for storing Possible to store otherwise too large to be stored in computer memory as a binary matrix Discarding procedure where premature termination yields a superset of Computation can be continued later on New method for reducing the number of pairs to check Extended from mixed integer linear programming (MILP) (Fischetti et al. 2010)

8 Storing sets using Binary Decision Diagrams (BDDs) BDD = compressed binary decision tree (BDT) Identical nodes merged Redundant nodes eliminated Truthtable = BDT Redundant = BDD 0 1 0,1 Identical =1 =0

9 Efficient set operations using BDDs Set operations using BDDs Worst case computational time BDD 1 has nodes BDD 2 has nodes Intersection ) Member of ) Cardinality, complement ) Allows to construct BDDs of sets without a truthtable Theory & implementation: Bryant 1986, Brace =

10 Example: Compression of a ND set Let = 60, =0, =1, = 843 Projects costs in table (see fig.) Every maximal portfolio is ND Cost = = matrix Ádding 10 rows/s takes ~1 day Storage ~5000 Tb (in theory) BDD Generated in ~8 min (regular laptop) Storage nodes ~100 Mb (actual) Project

11 Discarding sets of projects using BDDs: Project dominance Project dominates project iff and more valuable and at most as costly as Feasible portfolio that excludes and includes is dominated Exchange for to get the dominating portfolio Thus all ND portfolios must be in = 0,1 = 1 or =0 1 0 BDD of Discard process:

12 Value Example 1/ Project 30 projects of equal cost Total of 190 project dominances Project 1 dominates project 3, 7, and 10-30, etc. Each project dominance yields a set within which ND portfolios are Each set stored as BDD The intersection of these sets computed using BDD set operations

13 Example 2/2 For clarity, node 0 and arcs to it removed from BDD BDD of the intersection of the 190 sets 97 nodes Corresponds to portfolios % of all 2 10 portfolios

14 Discussion: Discarding process using BDDs We implemented the discarding process using Branchand-Bound (B&B) Sytematically cuts sets of dominated portfolios Many of these methods BDD variants of those presented by Liesiö et al Developed a method for reducing the number of pairs to check Extension of the Fischetti-Toth dominance procedure (1988) Preliminary results suggest that large ND sets can computed using this implementation

15 Conclusions Storage as BDD opens new possiblities Deriving stronger bounds using information from the BDD Optimizing over BDD set = solving shortest path problem Use of implied dominance checks makes it possible to pairwise compare possibly huge ND sets Possible future research directions Zero-suppressed BDDs for improved storage (Minato 2007) Additive linear value functions (Liesiö et al. 2007) Uncertainty set defined by linear inequalities (Liesiö et al. 2007) Cost efficiency (Liesiö et al. 2008)

16 References Brace KS, Rudell RL, Bryant RE, Efficient implementation of a BDD package. Design Automation Conference, Proceedings 27th ACM/IEEE, 40-45, Jun 24-28, 1990 Bryant RE. Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers 100(8): , 1986 Fischetti M, Salvagnin D. Pruning moves. INFORMS Journal on Computing 22(1): , 2010 Fischetti M, Toth P. A new dominance procedure for combinatorial optimization problems. Operations Research Letters 7(4): ,1988 Liesiö J, Mild P, Salo A. Preference programming for robust portfolio modeling and project selection. European Journal of Operational Research 181(3): , 2007 Liesiö J, Mild P, Salo A. Robust portfolio modeling with incomplete cost information and project interdependencies. European Journal of Operational Research 190(3): , 2008 Minato S. Zero-suppressed BDDs for set manipulation in combinatorial problems. Proceedings of 30th Conference on Design Automation, , June 1993 Pisinger D. Where are the hard knapsack problems? Computers & Operations Research 32(9) , 2005