Submodular Minimisation using Graph Cuts

Transcription

1 Submodular Minimisation using Graph Cuts Pankaj Pansari 18 April,

2 Overview Graph construction to minimise special class of submodular functions For this special class, submodular minimisation translates to constrained modular minimisation Given a submodular function via an oracle, NP-hard to determine if graph-representable 2

3 Outline Introduction Graph Construction and NP-hardness Modular Minimisation and Minimum st-cuts 3

4 Part I - Introduction Submodular, modular and supermodular functions Cut functions, st-cut functions, minimum st-cuts Lattice 4

5 Combinatorial Optimization All discrete optimization problems are of the form: max{f (X ) : X F} min{f (X ) : X F} where F is a discrete set of feasible solutions and f is a set function, that is, f : 2 F R. We can try to deal with each problem individually, or capture some properties of f, F that make it tractable 5

6 Submodular Functions Equivalent defintions: 1. Define the marginal value of element j: f X (j) = f (X {j}) f (X ) f is submodular if X Y, j / Y : f X (j) f Y (j) 2. The set function f is submodular if for any X, Y f (X Y ) + f (X Y ) f (X ) + f (Y ) 6

7 Modular Functions Equivalent defintions: 1. f is modular if X Y, j / Y : f X (j) = f Y (j) 2. f is modular if for any X, Y f (X Y ) + f (X Y ) = f (X ) + f (Y ) 7

8 Supermodular Functions Equivalent defintions: 1. f is supermodular if X Y, j / Y : f X (j) f Y (j) 2. f is supermodular if for any X, Y f (X Y ) + f (X Y ) f (X ) + f (Y ) 8

9 An Observation The whole is the sum of its parts Let f (φ) = 0: Modular: equal to f (A) = f (i) i A Submodular: less than f (A) i A f (i) Supermodular: greater than 9

10 10

11 11 Lattice A family F = F 2 S A family F is a lattice if it is closed under union and intersection A, B F = A B F and A B F Example: Let S = {a, b, c} F {φ, {a}, {b}, {a, b}} {φ, {a, b}, {c}, {a, b, c}} {{a}, {c}, {a, c}} Lattice? Yes Yes No Derive lattices for S = {a, b, c, d}.

12 12 Proper Lattice F is a proper lattice if F = φ and F = S = φ, S F F {φ, {a}, {b}, {a, b}} {{a}, {a, b}, {a, c}, {a, b, c}} Proper Lattice? Yes No

13 13 Oracle A black box which computes output f (x) for any input x

14 14 Cut Functions Given G = (V, A), a cut is a partition of V in to (X, X ), X V f (X ) = i X,j X a ij

15 Example 15

16 Cut functions are submodular (Proof on board) 16

17 17

18 18 Minimum Cut Trivial solution: f (φ) = 0 Need to enforce X, X to be non-empty Source {s} X, Sink {t} X

19 19 st-cut Functions {s} X, {t}in X f (X ) = i X,j X a ij

20 Minimum st-cut min f (X ) = i X,j X a ij such that {s} X, {t}in X Min cut value = 18 20

21 21 Submodular Minimisation Given set S and submodular function f, submodular minimisation is min f (A) A S Continuous Discrete Ellipsoid Minimum Cut O( S k log(max f (A) )) O( S 3 ) Min-norm point O( S 7 )

22 22 Outline Introduction Graph Construction and NP-hardness Modular Minimisation and Minimum st-cuts

23 23 Part II - Graph Construction and NP-hardness Note: f values available only via oracle

24 24 A Property of Cut Functions For any three disjoint subsets A, B, C of S f (A B C) = f (A B) + f (B C) + f (C A) f (A) f (B) f (C) + f (φ) (Proof outline on board) = f is determined by its values on sets of cardinality at most 2

25 25 Graph Construction Recipe to construct directed graph G = (V, A) V = S {s, t} To specify A 1. Source to vertex arcs a sv 2. Vertex to vertex arcs a uv 3. Vertex to sink arcs a vt Graph construction and example on board

26 26 To certify if a submodular function is cut function...is NP-hard = exponential oracle calls required to certify Proof on board

27 27 Outline Introduction Graph Construction and NP-hardness Modular Minimisation and Minimum st-cuts

28 Part III - Modular Minimisation and Minimum st-cuts 28

29 29 Modular min over a lattice Modular minimisation over power set 2 S is trivial over a lattice F is harder

30 30 Closure Given G = (V, A), X V is a closure if = No outcoming arcs from X δ(x ) = φ

31 31 Minimum Weight Closure Given G = (V, A) with weights w v for v V min v X w v s.t u(δ(x )) = φ

32 32 Modular min over a lattice as st-min cut Strategy: Modular min over a lattice Min weight closure Min weight closure Min st-cut

33 33 Modular min over a lattice Min weight closure On board

34 34 Modular min over a lattice as st-min cut Strategy: Modular min over a lattice Min weight closure Min weight closure Min st-cut

35 35 Min weight closure Min st-cut On board

36 36 st-min cut as modular min over a lattice On board

37 37 Reference William Cunningham, Minimum Cuts, Modular Functions, and Matroid Polyhedra, Networks, Vol. 15 (1985)