Minimization Satoru Iwata (University of Tokyo) Submodular - PowerPoint PPT Presentation

Submodular Function Minimization Satoru Iwata (University of Tokyo)

Submodular Function Minimization   ( ) 0 f Assumption: X Minimization Evaluation Algorithm Oracle ( X ) f  Minimizer min{ ( ) | } ? f Y Y V

Submodular Function Minimization Grötschel, Lovász, Schrijver (1981, 1988) Ellipsoid Method  5 ( log ) Cunningham (1985) O n M    7 8 7 ( ) O n n ( log ) O n n Iwata, Fleischer, Fujishige (2000) Schrijver (2000) Fleischer, Iwata (2000) Iwata (2002) Fully Combinatorial Iwata (2003) Orlin (2007)     4 5 5 6 ( ) (( ) log ) O n n O n n M Iwata, Orlin (2009)

Submodular Functions and Polyhedra    ( ) 0 V f : 2 f R        ( ) ( ) ( ) ( ), , f X f Y f X Y f X Y X Y V Submodular Polyhedron      V ( ) { | , , ( ) ( )} P f x x R Y V x Y f Y Base Polyhedron    ( ) { | ( ), ( ) ( )} B f x x P f x V f V

Min-Max Theorem Edmonds (1970) Theorem    min ( ) max{ ( ) | ( ), 0 } f Y z V z P f z  Y V    max{ ( ) | ( )} x V x B f   ( ) : min{ 0 , ( )} x v x v   ( ) ( ) ( ) z V z Y f Y    ( ) ( ) ( ) x V x Y f Y

Combinatorial Approach    min ( ) max{ ( ) | ( )} f Y x V x B f  Y V

Min-Max Theorem    min ( ) max{ ( ) | ( ), 0 } f Y z V z P f z  Y V    max{ ( ) | ( )} x V x B f    ( ) : min{ ( ) | } f X f Y Y X Submodular   ( ) ( ) P f P f       ( ) ( ) ( ) min ( ) z B f z V f V f Y Y  V

Combinatorial Approach    min ( ) max{ ( ) | ( )} f Y x V x B f  Y V Convex Combination    x i y i  i I y i  ( f ) : Extreme Base Generated by B the Greedy Algorithm with an . Linear Ordering in L V i

IFF Scaling Algorithm   1 ( ) M f  X Submodular     :      2 2 n n     ( ) ( ) | | | \ | Cut Function f X f X X V X      : Flow in the Complete Digraph ( , ) u v x    i y i  - Augmenting Path  i I s t  x    z S T z  Increase ( ) V        { | ( ) } { | ( ) } S v z v T v z v

IFF Scaling Algorithm Double-Exchange S T No Path from to       : ( ) y y W  i i u v : { | : Reachable from } v v S      : min{ , } i       W : ( ) x x u v      ( , ) : ( , ) u v u v u v         S T i i v u L i u v Saturating Nonsaturating

IFF Scaling Algorithm    ( ) ( ), y i W f W i I T S No Path from to     No Active Triple ( ) ( ) ( ) ( , , ) i u v x W y W f W i i  i I     ( ) ( ) z V f W n W     2 ( ) ( ) x V f W n S 1 T   Min ( W ) : f 2 n  L 5 ( log ) O n M i

Submodular Function Minimization Grötschel, Lovász, Schrijver (1981, 1988) Ellipsoid Method  5 ( log ) Cunningham (1985) O n M    7 8 7 ( ) O n n ( log ) O n n Iwata, Fleischer, Fujishige (2000) Schrijver (2000) Fleischer, Iwata (2000) Iwata (2002) Fully Combinatorial Iwata (2003) Orlin (2007)     4 5 5 6 ( ) (( ) log ) O n n O n n M Iwata, Orlin (2009)

Fully Combinatorial Algorithm Addition, Subtraction, Comparison Oracle Call (Function Evaluation) Compute      , R Z Multiplication     /      q , R R Compute Division • Neglect the Gaussian Elimination Step. • Use Nonnegative Integer Combination Instead of Convex Combination. • Choose a Step Length Appropriately.

Finding All Minimizers   , Y : , : X X Y X Y Minimizer Minimizer The set of minimizers forms a distributive lattice. [G.Birkhoff] Any distributive lattice can be represented as the set of ideals of a partial ordered set.

Partial Order of an Extreme Base y  Extreme ( f ) : B Bixby, Cunningham, Topkis (1985) v v  v ( ) ( ) : y X f X Tight X 2 n 1    Represent all tight sets ( ( )) ( ( )), y L v f L v v V ( y ) G v . u ( u ) : Maximal Ideal Excluding H u  Test if is tight. ( ) { } H u v ( u ) H v u If not, then .

Finding All Minimizers Extreme Base ( ) ( x ) : G y G Superposition of i y i  ( f ) B SCC Decomposition Partial Order (DAG)  ( ) 0 x v ( ) G y i Convex Combination x   ( ) 0 x v   i y i  i I     ( 0 , ) i I i  ( ) 0 x v