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Flow networks

Flow networks. 2. 5. 1. How much flow can we push through from s to t ? (Numbers are capacities.). 4. 7. 3. 2. 5. 2. Flow networks. 5. 1. s. 4. 7. t. 3. 2. 5. Def:

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Flow networks

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  1. Flow networks 2 5 1 How much flow can we push through from s to t ? (Numbers are capacities.) 4 7 3 2 5

  2. 2 Flow networks 5 1 s 4 7 t 3 2 5 • Def: • A flow network is a directed graph G=(V,E) where edges have capacities c:E->R+. There are two specified vertices s (source) and t (sink). • A flow f:E->R must satisfy: • Capacity constraint: for every edge e: f(e) · c(e) • Flow conservation: • for every v in V-{s,t}: e out of v f(e) = e into v f(e) • The value of the flow is: e out of s f(e) - e into s f(e)

  3. Maximum flow problem Input: a flow network G=(V,E), with capacities c, the source s and sink t Output: a maximum-value flow Algorithm ? 2 5 1 s 4 7 t 3 2 5

  4. Maximum flow problem Input: a flow network G=(V,E), with capacities c, the source s and sink t Output: a maximum-value flow Algorithm ? 2 5 5 s 4 7 t 3 6 5

  5. Maximum flow problem – Ford-Fulkerson “Def”: Given a flow f, an augmenting path is a path s=v1, v2, …, vk=t such that f(vi,vi+1) < c(vi,vi+1) for i=1,…,k-1 2 5 1 s 4 2 t 3 7 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  6. Maximum flow problem – Ford-Fulkerson “Def”: Given a flow f, an augmenting path is a path s=v1, v2, …, vk=t such that f(vi,vi+1) < c(vi,vi+1) for i=1,…,k-1 How to find augmenting paths ? 2 5 1 s 4 2 t 3 7 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  7. Maximum flow problem – Ford-Fulkerson “Def”: Given is G=(V,E), c, f. The residual graph has edges weighted by the residual capacities, i.e. cf(e) = c(e)-f(e) 2 5 1 s 4 2 t 3 7 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  8. Maximum flow problem – Ford-Fulkerson “Def”: Given is G=(V,E), c, f. The residual graph has edges weighted by the residual capacities, i.e. cf(e) = c(e)-f(e) Idea: Find an s-t path in the residual graph! 2 5 1 s 4 2 t 3 7 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  9. Maximum flow problem – Ford-Fulkerson Consider this input: 1000 1000 s 1 t 1000 1000 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  10. Maximum flow problem – Ford-Fulkerson Consider this input: Need to refine the definition of augmenting paths and residual graph. 1000 1000 s 1 t 1000 1000 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  11. Maximum flow problem – Ford-Fulkerson • Refined def: • Given is G=(V,E), c, f. The residual graph Gf=(V,E’) contains the following edges: • forward edge: if e 2 E and f(e) < c(e) then include e in E’ with weight • cf(e) = c(e)-f(e), • backward edge: if e=(u,v) 2 E with f(e)>0 then include (v,u) in E’ with weight • cf(v,u) = f(u,v). 1000 1000 s 1 t 1000 1000

  12. Maximum flow problem – Ford-Fulkerson • Ford-Fulkerson ( G=(V,E), c, s, t ) • For every edge e let f(e)=0 • 2. Construct the residual graph Gf • While exists s-t path in Gf do • Let p be an s-t path in Gf • Let d=mine in p cf(e) • For every e on p do • If e is a forward edge then • f(e)+=d • else • f(reverse(e))-=d • Update Gf (construct new Gf) • Return f

  13. Maximum flow problem – Ford-Fulkerson • Ford-Fulkerson ( G=(V,E), c, s, t ) • For every edge e let f(e)=0 • 2. Construct the residual graph Gf • While exists s-t path in Gf do • Let p be an s-t path in Gf • Let d=mine in p cf(e) • For every e on p do • If e is a forward edge then • f(e)+=d • else • f(reverse(e))-=d • Update Gf (construct new Gf) • Return f Running time:

  14. Maximum flow problem – Ford-Fulkerson 1000 1000 s 1 t 1000 1000 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  15. Maximum flow problem – Ford-Fulkerson Lemma: Ford-Fulkerson works. Def: Given G=(V,E), c. An s-t cut of G is a subset of vertices S s.t. s 2 S and t 2 SC. Its value is e out of S c(e) 2 5 1 s 4 7 t 3 2 5

  16. Maximum flow problem – Ford-Fulkerson Lemma: Ford-Fulkerson works. The Max-flow min-cut theorem: Let min-cut(G) be the minimal value of an s-t cut of G. Then: f is a maximum flow iff value(f)=min-cut(G) 2 5 1 s 4 7 t 3 2 5

  17. Maximum flow problem – Ford-Fulkerson Lemma: Ford-Fulkerson works. 2 5 1 s 4 7 t 3 2 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  18. Improving Ford-Fulkerson Can find better paths to reduce the running time? 2 5 1 s 4 7 t 3 2 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  19. Improving Ford-Fulkerson • Can find better paths to reduce the running time? • many ways, will discuss two: • Scaling paths • BFS 2 5 1 s 4 7 t 3 2 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  20. Improving Ford-Fulkerson • Can find better paths to reduce the running time? • many ways, will discuss two: • Scaling paths 2 5 1 s 4 7 t 3 2 5 • Ford-Fulkerson ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augmenting path p do • Augment flow f along p • Return f

  21. Improving Ford-Fulkerson • Can find better paths to reduce the running time? • many ways, will discuss two: • Scaling paths • BFS • Thm: Edmonds-Karp takes O(|V||E|) iterations. • Running time of Edmonds-Karp: 2 5 1 s 4 7 t 3 2 5 • Edmonds-Karp ( G=(V,E), c, s, t ) • Initialize flow f to 0 • While exists augm. path p (check with BFS) do • Augment flow f along p • Return f

  22. Applications of Network Flows • multiple sources, multiple sinks s1 t1 t2 s2 t3 s3 t4

  23. Applications of Network Flows • how to find minimum cut 2 5 1 s 4 7 t 3 7 5

  24. Applications of Network Flows • maximum number of edge-disjoint s-t paths s t

  25. Applications of Network Flows • maximum number of edge-disjoint s-t paths s t

  26. Applications of Network Flows • maximum bipartite matching

  27. Applications of Network Flows • maximum weighted (perfect) bipartite matching 7 5 2 3 4 4 1 6 3 8

  28. Applications of Network Flows • maximum weighted (perfect) bipartite matching 7 5 2 3 4 4 1 6 3 8

  29. Applications of Network Flows • maximum weighted (perfect) bipartite matching 7 5 2 3 4 4 1 6 3 8

  30. Introduction to Linear Programming • Consider the Diet problem: • n food items, m nutrients • for every nutrient: the daily quota bj • for each item: cost per pound ci • for every item and nutrient: how much of the nutrient in a pound of item: ai,j

  31. Introduction to Linear Programming A linear program looks like this: • Find x1, x2, …, xn which • minimize • c1x1 + c2x2 + … + cnxn • and satisfy these constraints: a1,1x1 + a2,1x2 + … + an,1xn >= b1 a1,2x1 + a2,2x2 + … + an,2xn >= b2 … a1,mx1 + a2,mx2 + … + am,nxn >= bm

  32. Introduction to Linear Programming A linear program in compressed form: Given a vector c in Rn, a vector b in Rm and a matrix A in Rm x n, find a vector x in Rn which satisfies xA >= b and maximizes cxT. Thm: Exists a polynomial-time algorithm solving linear programs. Caveat: Sometimes need integer programs (no algorithm for integer programs is likely to exist) !

  33. Vertex Cover via LP Given is a graph G=(V,E) and a number k. A vertex cover is a subset C of V such that every edge has an endpoint in C. The Vertex Cover problem asks if there is a vertex cover of size k.

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