dynamic flows dynamic transshipment & evolving graphs 2/28/2012 tcs group seminar 1

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Dynamic FlowsDynamic Transshipment&Evolving Graphs

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Seminar outline

Earliest Arrival Flows• reminder & example• evacuation problems

Dynamic Transshipment & Evolving Graphs

Lexicographically Maximal Flows

Push-Relabel framework

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Earliest Arrival Flows

•Example

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•Time-expanded Graph▫(Ford-Fulkerson ’58)

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t=1t=2t=3t=4t=5

t=7

t=0

t=6

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1. Compute distance labels in residual graph

1. it defines a cut

2. no augmenting path can arrive before

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t=1t=2t=3t=4t=5

t=7

t=0

t=6

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0 1 2 3

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2. Add shortest path

3. Repeat

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t=7

t=0

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0 3 4 70 1 2 3

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2. Add shortest path

3. Repeat

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t=1t=2t=3t=4t=5

t=7

t=0

t=6

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0 3 4 7

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•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...

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Seminar outline


Dynamic Transshipment & Evolving Graphs• definitions• equivalence• submodularity



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Dynamic transshipment

•Several sources with a fixed supply•Several sinks with a fixed demand

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S2

A B S-S1

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Evolving graph

•Edges have a schedule [t1;t2], [t3,t4],...

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Dynamic transshipment= Flow in evolving graph

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S2

A B S-S1

[-2;0]

[-3;0]

S

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Flow in evolving graph= dynamic transshipment

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[t1;t2]

•similar to capacitated max flow = uncapacitated transshipment in static graphs.

demand: t2-t1delay: T-t2

supply: t2-t1delay: t1

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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.

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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)

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Seminar outline



Lexicographically Maximal Flows• definition• algorithm• building a solution


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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)

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Lexicographically maximaldynamic flows1. Put distance labels at 0 for sources and

at T for sinks2. 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}

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Lexicographically maximaldynamic 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)

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Solution with submodular oracle• Do a complex dichotomic search with

the help of the oracle in order to1. restrict the capacities of edges that exit the

sources/enter the sinks2. order the sources and sinks

• The obtained lexicographically maximal dynamic flow answers the dynamic transshipment problem

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Seminar outline




Push-relabel framework• similarities & problems• fractional solution• integral solution

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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

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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

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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 ???

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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)

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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.

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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.

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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

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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 solutions2. using unique potentials and restricted

capacities

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dynamic flows dynamic transshipment & evolving graphs 2/28/2012 tcs group seminar 1

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