1. 1 LP, extended maxflow, TRW OR: How to understand Vladimirs most recent work Ramin Zabih Cornell University

2. 2 Outline Linear programming and duality Extended maxflow TRW Some slides from Ziv Bar-Yossef (Technion) Much content from Vladimir Kolmogorov

3. 3 Linear Programming Fast algorithms - O(n 3 ) or so Numerous interesting special cases can be encoded as LPs Many ways to convert between LPs –Sometimes you can transform into a fast special case Can sometimes solve integer programs –Integrality gap: IP versus LP solution

4. 4 Example

5. 5

6. 6 Solving LP Graphically Feasible region Vertices Objective function Optimal solution

7. 7 Notes on solutions Maximum is always at a vertex –Simplex algorithm Certain constraints are tight –So you can check if the solution is integral!

8. 8 Convert any LP to Standard Form

9. 9 Bounding the Optimal Solution Suppose we wish to find an upper bound on the optimal solution Multiply first constraint by 3: z 90 Add first and second constraints: z 54 What linear combination gives the best upper bound?

10. 10 Bound Optimal Solution by LP Primal Dual

11. 11 Primal vs. Dual Lemma: The dual of the dual is the primal. Theorem (Weak Duality) For any feasible solution x for the primal (cost vector c), and for any feasible solution y for the dual (cost b), Theorem (Strong Duality) If the primal has an optimal solution x *, then the dual also has an optimal solution y *, and

12. 12 RDZs favorite dual

13. 13 Notes Solutions to dual are not obviously cuts –Why are the y I,j values 0 or 1? Any flow · any cut –We can find the biggest flow fast By strong duality this is the value of the cheapest cut –But how do we actually find the cut? –How about in generality?

14. 14 Complementary slackness Easy to go from dual optimum solution to primal optimal solution –No guarantees if dual solution isnt optimal Dual variable y j is zero iff jth constraint is slack (not tight) –So you know which constraint lines intersect at the optimum!

15. 15 LP for energy minimization We can formulate E(x) as an IP –Since IPs are NP-hard –We replace x p by indicator variables x p;k which are like probabilities (non negative, sum to 1) Similarly, need x pq;ij for adjacent pixels Parameters are costs –We can then relax this IP to an LP and solve it If integer solutions, we are done Note that we first need to actually solve the LP!

16. 16 Primal LP [Chekuri00]

17. 17 Importance of this LP This LP is vital for both extended maxflow (quadratic pseudo-boolean optimization) and TRW(-S) –Both work on its dual So by solving the dual we are maximizing a lower bound on the energy

18. 18 Extended maxflow vs TRW Maxflow is only for binary labels –I.e., for the expansion move algorithm Maxflow solves the LP exactly Solutions are [1 0] or [0 1] or [.5.5] No ambiguities if problem is regular Automatically does flipping If not regular, solution is persistent

19. 19 Extended maxflow non-negative - lower bound on the energy: maximize Maximize lower bound on E

20. 20 TRW idea Decompose graph into a collection of trees with different parameterizations Weak tree agreement (WTA): –Consider all trees incorporating a node. Their parameterizations with cost 0 all agree. Normal form tree parameterization –node params are min marginals (after BP) Goal is normal form and WTA

21. 21 TRW(-S) properties WTA and normal form holds at the dual solution (max ( )) –For binary labels only! TRW(-S) tries to achieve this –TRW doesnt always converge to a solution where this is true (TRW-S does) –What it converges for isnt necessarily the dual solution (except for 2 labels) –The primal solution may not be integral