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Polyhedral OptimizationLecture 5 – Part 1
M. Pawan Kumar
Slides available online http://cvn.ecp.fr/personnel/pawan/
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| = 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
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 ⊆
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
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