1. Dynamic Flows Dynamic Transshipment & Evolving Graphs 2/28/2012 TCS Group Seminar 1

2. Seminar outline Earliest Arrival Flows reminder & example evacuation problems Dynamic Transshipment & Evolving GraphsLexicographically Maximal FlowsPush-Relabel framework 2/28/2012TCS Group Seminar 2

3. Earliest Arrival Flows Example 2/28/2012TCS Group Seminar 3 S+ABS-

4. Earliest Arrival Flows Time-expanded Graph ▫(Ford-Fulkerson ’58) 2/28/2012TCS Group Seminar 4 S+ABS- t=1 t=2 t=3 t=4 t=5 t=7 t=0 t=6

5. Earliest Arrival Flows 1.Compute distance labels in residual graph 1.it defines a cut 2.no augmenting path can arrive before 2/28/2012TCS Group Seminar 5 S+ABS- t=1 t=2 t=3 t=4 t=5 t=7 t=0 t=6 S+ABS- 0123

6. Earliest Arrival Flows 1.Compute distance labels in residual graph 2.Add shortest path 3.Repeat 2/28/2012TCS Group Seminar 6 S+ABS- t=1 t=2 t=3 t=4 t=5 t=7 t=0 t=6 S+ABS- 03470123

7. Earliest Arrival Flows 1.Compute distance labels in residual graph 2.Add shortest path 3.Repeat 2/28/2012TCS Group Seminar 7 S+ABS- t=1 t=2 t=3 t=4 t=5 t=7 t=0 t=6 S+ABS- 0347

8. Earliest Arrival Flows Several sources: evacuation problem See: works from Skutella, Minieka, and students. Maybe interesting for extracting maximum information from a short-lived WSN ▫battlezone ▫vulcano, nuclear reactor... 2/28/2012TCS Group Seminar 8

9. Seminar outline Earliest Arrival FlowsDynamic Transshipment & Evolving Graphs definitions equivalence submodularity Lexicographically Maximal FlowsPush-Relabel framework 2/28/2012TCS Group Seminar 9

10. Dynamic transshipment Several sources with a fixed supply Several sinks with a fixed demand 2/28/2012TCS Group Seminar 10 S2 ABS-S1

11. Evolving graph Edges have a schedule [t1;t2], [t3,t4],... 2/28/2012TCS Group Seminar 11 S+ABS-

12. Dynamic transshipment = Flow in evolving graph 2/28/2012TCS Group Seminar 12 S2 ABS-S1 [-2;0] [-3;0] S

13. Flow in evolving graph = dynamic transshipment 2/28/2012TCS Group Seminar 13 [t1;t2] similar to capacitated max flow = uncapacitated transshipment in static graphs. demand: t2-t1 delay: T-t2 supply: t2-t1 delay: t1

14. Submodularity The dual of the dynamic transshipment problem is to find a subset of sources/sinks and a minimum cut in the time-expanded graph between those subsets. ▫(solve a min-cost flow for each sources/sinks subset) The min-cut function is submodular on sources/sinks subsets. 2/28/2012TCS Group Seminar 14

15. Submodularity Minimizing a submodular function can be done with a variant of the Ellipsoid method ▫convex function on convex sets ▫P, but not practical Test feasibility of dynamic transshipment with a submodular oracle (Hoppe&Tardos ’95) 2/28/2012TCS Group Seminar 15

16. Seminar outline Earliest Arrival FlowsDynamic Transshipment & Evolving GraphsLexicographically Maximal Flows definition algorithm building a solution Push-Relabel framework 2/28/2012TCS Group Seminar 16

17. Lexicographically Maximal Flows Given a sequence of sources/sinks ▫(a,b,c,d,e...) A lexicographically maximal flow maximizes the amount of flow ▫from a to (b,c,d,e...) ▫from (a,b) to (c,d,e...) ▫from (a,b,c) to (d,e...) ▫etc. It exists and is easily computable (Megiddo ’74) 2/28/2012TCS Group Seminar 17

18. Lexicographically maximal dynamic flows 1.Put distance labels at 0 for sources and at T for sinks 2.Compute min-cost flow (= max dynamic flow) from {a,b,...,x,y} to {z} 3.Compute min-cost augmenting flow from {a,b,...,x} to {y,z}................................................................................ 27.Compute min-cost augmenting flow from {a} to {b,...,x,y,z} 2/28/2012TCS Group Seminar 18

19. Lexicographically maximal dynamic flows At each step, the subset of sources decreases ▫distance labels can only increase ▫augmenting flows yield a valid dynamic solution The labels at a given step indicate a minimum cut for the current subset of sources ▫the final solution saturates that cut ▫the actual proof is rather technical (see Hoppe&Tardos ’00) 2/28/2012TCS Group Seminar 19

20. Solution with submodular oracle Do a complex dichotomic search with the help of the oracle in order to 1.restrict the capacities of edges that exit the sources/enter the sinks 2.order the sources and sinks The obtained lexicographically maximal dynamic flow answers the dynamic transshipment problem 2/28/2012TCS Group Seminar 20

21. Seminar outline Earliest Arrival FlowsDynamic Transshipment & Evolving GraphsLexicographically Maximal FlowsPush-relabel framework similarities & problems fractional solution integral solution 2/28/2012TCS Group Seminar 21

22. Push-relabel similarities & problems A lexicographically maximal flow is actually a giant saturating push with labels ▫a:26, b:25, c:24, d:23,...., z:1 Idea: dynamic push-relabel algorithm Difficulties: ▫non-saturating pushes ▫several vertices at the same level 2/28/2012TCS Group Seminar 22

23. Push-relabel similarities & problems Non-saturating push problem ▫Having a minimum cost flow (= maximum dynamic flow) is vital for coherent distance labels and coherent solution Same level problem: ▫pushing from a vertex may send flow to other vertices at same level a,b,{c,d,e},f 2/28/2012TCS Group Seminar 23

24. Push-relabel framework All sources & sinks start at potential 0. The algorithm maintains a lexicographically maximal dynamic flow from potential 26 down to potential 0 When a node has excess flow, increase its potential by 1 and ▫recompute the lex-max dynamic flow ▫(1 min-cost flow computation) ▫= saturating push What if it’s too much ??? 2/28/2012TCS Group Seminar 24

25. Fractional push-relabel Fractional push ▫a node is at potential P in 0.72 of the solution and at potential P+1 in 0.28 of the solution When a node has excess flow, try to increase its potential to a full number. ▫if it still has excess flow, fine. ▫if it has a deficit, make a linear combination of (full push/no action) to have zero excess Nodes on a same level: ▫find a linear combination for all nodes (doable) 2/28/2012TCS Group Seminar 25

26. Fractional push-relabel Min-cost flow corresponding to potential P/P+1 is not affected by fractions of other potentials ▫(a,{a,b(0.27)},{a,b,c(0.3),d(0.5)},{a,b,c,d,e}) At a given level, try to push all potentials to full number: ▫(a,{a,b(0.27)},{a,b,c,d},{a,b,c,d,e}) ▫effect is c:+4 unit, d -1 unit, e-3 units ▫select a fraction so that c and d are non negative, and c or d is at zero ▫push the other node alone. 2/28/2012TCS Group Seminar 26

27. Staggered push-relabel Natural approach for integral solutions: ▫dichotomic search on source/sink capacities (i.e. size of the hose) A node has full capacity at potential P, and partial capacity at potential P+1: ▫(a, {a,b},{a,b,c(partial)},{a,b,c,d}) Problem with multiple nodes at same level: ▫multiple dichotomic search is actually exponential. 2/28/2012TCS Group Seminar 27

28. Staggered push-relabel Assign unique potentials to each node: ▫a: 0, 52, 104, ▫b: 0,1, 51, 53, 103, ▫c: 0, 2, 50, 54, 102, Maximum number of pushes unchanged ▫(still 26 per node) Saturating push: increase node level. Non-saturating push: increase node capacity on top level 2/28/2012TCS Group Seminar 28

29. Conclusion It is possible to augment/modify a dynamic flow under the condition of strictly increasing distance labels. A lex-max dynamic flow is actually a configuration in a push/relabel scheme. Non-saturating push can be done while maintaining feasibility by: 1.using fractional solutions 2.using unique potentials and restricted capacities 2/28/2012TCS Group Seminar 29