Polyhedral Optimization Lecture 5 – Part 1 M. Pawan Kumar Slides available online

Submodular Functions Examples Outline

Submodular Function Set S Function f over power set of S f(T) + f(U) ≥ f(T ∪ U) + f(T ∩ U) for all T, U ⊆ S

Supermodular Function Set S Function f over power set of S f(T) + f(U) ≤ f(T ∪ U) + f(T ∩ U) for all T, U ⊆ S

Modular Function Set S Function f over power set of S f(T) + f(U) = f(T ∪ U) + f(T ∩ U) for all T, U ⊆ S

Modular Function f(T) = ∑ s ∈ T w(s) + K Is f modular? All modular functions have above form? YES Prove at home

Diminishing Returns Define d f (s|T) = f(T ∪ {s}) - f(T) Gain by adding s to T If f is submodular, d f (s|T) is non-increasing

Diminishing Returns f(U ∪ {s}) + f(U ∪ {t}) ≥ f(U) + f(U ∪ {s,t}) for all U ⊆ S and distinct s,t ∈ S\U Necessary condition for submodularityProof? Gain by adding s to T Define d f (s|T) = f(T ∪ {s}) - f(T)

Diminishing Returns Sufficient condition for submodularityProof? Gain by adding s to T f(U ∪ {s}) + f(U ∪ {t}) ≥ f(U) + f(U ∪ {s,t}) for all U ⊆ S and distinct s,t ∈ S\U Define d f (s|T) = f(T ∪ {s}) - f(T)

Proof Sketch Consider T, U ⊆ S We have to prove f(T) + f(U) ≥ f(T ∪ U) + f(T ∩ U) We will use mathematical induction on |TΔU|

Proof Sketch |TΔU| = 1 Proof follows trivially Either U ⊆ T or T ⊆ U T ∪ U = U and T ∩ U = TLet T ⊆ U

Proof Sketch |TΔU| = 2 If U ⊆ T or T ⊆ U, then proof follows trivially If not, then simply use the condition f(U ∪ {s}) + f(U ∪ {t}) ≥ f(U) + f(U ∪ {s,t}) for all U ⊆ S and distinct s,t ∈ S\U

Proof Sketch |TΔU| ≥ 3 Assume, wlog, |T \ U| ≥ 2 |T Δ ((T \{t}) ∪ U)| < |T Δ U| Let t ∈ T\U Why? f(T ∪ U) - f(T) ≤ f((T\{t}) ∪ U) - f(T\{t}) Induction assumption

Proof Sketch |TΔU| ≥ 3 Assume, wlog, |T \ U| ≥ 2 |(T\{t}) Δ U| < |T Δ U| Let t ∈ T\U Why? f((T\{t}) ∪ U) - f(T\{t}) ≤ f(U) - f(T ∩ U) Induction assumption

Proof Sketch |TΔU| ≥ 3 f(T ∪ U) - f(T) ≤ f(U) - f(T ∩ U) Hence Proved

Submodular Functions Examples Outline

Matroids We have already seen the proof Matroid M = (S, I ) f = r M Minimum of f? Submodular 0 f is non-decreasing

Matroids We have already seen the proof Matroid M = (S, I ) f = r M Minimum of f? Submodular 0 f(T) ≤ f(U), for all T ⊆ U

Submodular Functions Examples –Matroid Intersection –Directed Graph Cuts –Set Unions Outline

Matroid Intersection Minimum of f? Matroid M 1 = (S, I 1 ) f(U) = r 1 (U) + r 2 (S\U) Matroid M 2 = (S, I 2 ) Proof?Submodular Largest common independent set Matroid Intersection Theorem

Submodular Functions Examples –Matroid Intersection –Directed Graph Cuts –Set Unions Outline

Directed Graph Cuts Minimum of f? Digraph G = (V, A) f(U) = ∑ a ∈ out-arcs(U) c(a) S = V Proof? Submodular Non-negative capacity c(a) of arc a ∈ A Is f non-decreasing? 0 NO

Directed Graph Cuts Minimum of f over U ⊆ S\{t} such that s ∈ U? Digraph G = (V, A) f(U) = ∑ a ∈ out-arcs(U) c(a) S = V Proof? Submodular Non-negative capacity c(a) of arc a ∈ A Minimum s-t cut = Maximum s-t flow

Submodular Functions Examples –Matroid Intersection –Directed Graph Cuts –Set Unions Outline

Set Unions T 1, T 2, …, T n ⊆ T f(U) = ∑ s ∈ U’ w(s), U’ = ∪ i ∈ U T i S = {1, 2, … n} Submodular Non-negative weight w(s) of element s ∈ T Minimum of f? Is f non-decreasing? 0 YES Proof?