Pseudo-Boolean Optimization Wooram Heo Applied Algorithm Lab., KAIST 2018-12-03

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 2018-12-03

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 2018-12-03

Definitions and Notations Multi-linear polynomials representation (1) Posiform representation (2) 2018-12-03

Representations of PB function 2018-12-03

Representations of PB function 2018-12-03

Representations of PB function 2018-12-03

Representations of PB function 2018-12-03

Rounding and derandomization 2018-12-03

Rounding and derandomization 2018-12-03

Rounding and derandomization 2018-12-03

Rounding and derandomization 2018-12-03

Local optima Observation 2018-12-03

Local optima 2018-12-03

Local optima 2018-12-03

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 2018-12-03

Reductions to Quadratic Optimization 2018-12-03

Reductions to Quadratic Optimization 2018-12-03

Reductions to Quadratic Optimization 2018-12-03

Reductions to Quadratic Optimization Cannot reduce at a time 3 or more variables Finding a better selection procedure for pairs is NP-hard 2018-12-03

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) 2018-12-03

Basic Algorithm 2018-12-03

Basic Algorithm 2018-12-03

Basic Algorithm 2018-12-03