Polyhedral OptimizationLecture 5 – Part 1

M. Pawan Kumar

pawan.kumar@ecp.fr

Slides available online http://cvn.ecp.fr/personnel/pawan/

• 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

YES

Prove at home

Diminishing Returns

Define df(s|T) = f(T {s}) - f(T) ∪

Gain by adding s to T

If f is submodular, df (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 submodularity Proof?

Gain by adding s to T

Define df(s|T) = f(T {s}) - f(T) ∪

Diminishing Returns

Sufficient condition for submodularity Proof?

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 df(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 = T∪Let 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 = rM

Minimum of f?

Submodular

0

f is non-decreasing

Matroids

We have already seen the proof

Matroid M = (S, I)

f = rM

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 M1 = (S, I1)

f(U) = r1(U) + r2(S\U)

Matroid M2 = (S, I2)

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

T1, T2, …, Tn T⊆

f(U) = ∑s U’ ∈ w(s), U’ = ∪i U∈ Ti

S = {1, 2, … n}

Submodular

Non-negative weight w(s) of element s T∈

Minimum of f?

Is f non-decreasing?

0

YES

Proof?