Polyhedral Optimization Lecture 5 – Part 1

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

2. Outline • Submodular Functions • Examples

3. 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

4. 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

5. 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

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

7. 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

8. Diminishing Returns Define df(s|T) = f(T ∪{s}) - f(T) 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 Necessary condition for submodularity Proof?

9. Diminishing Returns Define df(s|T) = f(T ∪{s}) - f(T) 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 Sufficient condition for submodularity Proof?

10. 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|

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

12. 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

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

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

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

16. Outline • Submodular Functions • Examples

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

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

19. Outline • Submodular Functions • Examples • Matroid Intersection • Directed Graph Cuts • Set Unions

20. Matroid Intersection Matroid M1 = (S,I1) Matroid M2 = (S,I2) f(U) = r1(U) + r2(S\U) Submodular Proof? Minimum of f? Largest common independent set MatroidIntersection Theorem

21. Outline • Submodular Functions • Examples • Matroid Intersection • Directed Graph Cuts • Set Unions

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

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

24. Outline • Submodular Functions • Examples • Matroid Intersection • Directed Graph Cuts • Set Unions

25. Set Unions T1, T2, …, Tn ⊆ T S = {1, 2, … n} Proof? f(U) = ∑s∈U’ w(s), U’ = ∪i∈U Ti Submodular Non-negative weight w(s) of element s ∈ T Minimum of f? 0 Is f non-decreasing? YES