1. Pseudo-Boolean Optimization
Wooram Heo
Applied Algorithm Lab., KAIST

2. Introduction Set functions Example
Mapping from the family of subsets of a finite ground set to the set of reals
Example
Subset S of the finite ground set A = { 1, 2, …, n }
is the characteristic vector of S

3. Definitions and Notations
Characteristic vector of a subset S ,
is the characteristic vector of S,
Pseudo-Boolean function
One-to-one correspondence btw subsets and
These functions are in fact set functions

4. Definitions and Notations
Multi-linear polynomials representation
(1)
Posiform representation
(2)

5. Representations of PB function

6. Representations of PB function

7. Representations of PB function

8. Representations of PB function

9. Rounding and derandomization

10. Rounding and derandomization

11. Rounding and derandomization

12. Rounding and derandomization

13. Local optima
Observation

14. Local optima

15. Local optima

16. Local optima Finding a local minimum remains a difficult problem
Natural idea to find global minimum is to use larger neighborhoods
Most widely applied method is the tabu search
Convexity of continuous extensions of PBF

17. Reductions to Quadratic Optimization

18. Reductions to Quadratic Optimization

19. Reductions to Quadratic Optimization

20. Reductions to Quadratic Optimization
Cannot reduce at a time 3 or more variables
Finding a better selection procedure for pairs is NP-hard

21. Basic Algorithm General algorithm for finding the optimum
Based on the necessary condition of local optimality
Find an expression for a component in terms of the other components(eliminate a variable)

22. Basic Algorithm

23. Basic Algorithm

24. Basic Algorithm