Maximum Flow by Incremental Breadth First Search

1. Maximum Flow byIncrementalBreadth First Search Andrew V. Goldberg Microsoft Research Sagi Hed Tel Aviv University Haim Kaplan Tel Aviv University s Renato F. Werneck Microsoft Research Robert E. Tarjan Princeton University & HP Labs

2. Maximum Flow • Input: directed graph G=(V,E), vertices s, t є V and capacity assignment c(e) for e є E • Output: flow function f satisfying -conservation: for every v≠s,t Σ(u,v)єE f(u,v) = Σ(v,u)єE f(v,u) capacity: for every e f(e) ≤ c(e)with maximal |f|=sum of flow out of s (into t) • Well studied problem • Equivalent to the Minimum s-t Cut problem • Solution methods:Augmenting Path (and blocking flow), Network Simplex, Push-Relabel

3. Maximum Flowin Computer Vision • Graphs have specific structure • Regular low degree grids • Arc capacities: different models for grid arcs and s-t arcs

4. BK • Boykov and Kolmogorov developed an algorithm (BK) which is the fastest in practice on the vision instances[Boykov, Kolmogorov 04] • Used as the standard min-cut algorithm in computer vision • Usually outperforms Push-Relabel implementation by considerable factors • Problem: BK has no known polynomial time guarantee…Best bound is O(mnF) for integral capacities (F is the maximal flow value) • Indeed on some instances, BK performs poorly and is outperformed by Push-Relabel implementation

5. Our ContributionIBFS • We develop the IBFS algorithm –Incremental Breadth First Search • Has many similarities to BK • However, performs shortest path or nearly shortest path augmentations • Competative in practice to BKUsually outperforms BK by small factors • Has a polynomial worst case time guaranteeO(mn2)

6. Augmenting Path Algorithms • Augmenting path algorithms constantly maintain a flow function f, fconstantly increases. • When the algorithm terminates f is maximal • Augmentation: add (maximal) Xto flow along an s-t path • Residual graph: Gf = (V,Ef)Ef = {(u,v) | (u,v) є E V f(u,v) < c(u,v)} U {(v,u) | (u,v) є E V f(u,v) > 0}Extend f and c to f(v,u)=-f(u,v) and c(v,u)=0 for (u,v)є E • Ford & Falkerson: augmentations in Gf give maximal flow s

7. BK Overview • Maintain trees S, T in the residual graph • Iterate 3 phases: Growth, Augmentation, Adoption • Growth: grow S and T bi-directionally t s s S T

8. BK Overview • Augmentation: when the trees meet, we augment flow • Adoption: after an augmentation, we try to reconnect “orphaned” sub-trees s s t S T

9. IBFS Overview • We maintain S, T as BFS trees with heights ≈ Ds , Dt • Augment on shortest or nearly shortest paths (+1) s s shortest+1 t shortest S T

10. IBFS Overview • Adoption / how to rebuild the trees:If subtree reconnects at the same level, we’re done t s s S T Ds Dt

11. IBFS Overview • Otherwise: • Relabel: set label to lowest potential parent + 1 • Make children into orphan sub-trees s s t S T

12. IBFS Overview • BFS trees => worst case bound O(mn2)(must also maintain a current arc) t s s S T

13. IBFS vs. BK • Maintaining BFS trees=> more work rebuilding the trees after each augmentation • Shortest augmenting paths=> less work in each augmentation • Shortest augmenting paths lead to less augmentations=> growth steps • We get rid of the parent traversal step s

14. IBFS Experiments • Ran on computer vision instancespublic benchmark [http://vision.csd.uwo.ca/maxflow-data/]our own creation [http://www.cs.tau.ac.il/~sagihed/ibfs/] • BK implementation available publicly [http://vision.csd.uwo.ca/code/] • We compare to a modified version of BK, with the same low level optimizations as our own (≈ 20% faster) • IBFS outperforms BK on all but two instances:2 different capacity versions of the instance “bone” • Factors are mostly modest. For few they are large. s

15. IBFS Experiments • Operation Counts(per vertex) s

16. Thank you! s