Algorithms - Princeton University · ROBERT SEDGEWICK | KEVIN WAYNE Algorithms ‣ introduction ‣...

ROBERT SEDGEWICK | KEVIN WAYNE

F O U R T H E D I T I O N

Algorithms

http://algs4.cs.princeton.edu

Algorithms ROBERT SEDGEWICK | KEVIN WAYNE

6.4 MAXIMUM FLOW

‣ introduction

‣ Ford-Fulkerson algorithm

‣ maxflow-mincut theorem

‣ analysis of running time

‣ Java implementation

‣ applications







Algorithms

‣ introduction





‣ applications

6.4 MAXIMUM FLOW





Input. An edge-weighted digraph, source vertex s, and target vertex t.

Mincut problem

3

5s t

15

1015

16

9

15

6

8 10

154

4 10

10

each edge has apositive capacity

capacity

Def. A st-cut (cut) is a partition of the vertices into two disjoint sets,

with s in one set A and t in the other set B.

Def. Its capacity is the sum of the capacities of the edges from A to B.

Mincut problem

4

5s

15

10

t

capacity = 10 + 5 + 15 = 30




10

Mincut problem

5

8

don't count edgesfrom B to A

t

16capacity = 10 + 8 + 16 = 34

s




Minimum st-cut (mincut) problem. Find a cut of minimum capacity.

10

Mincut problem

6

s

10

t

capacity = 10 + 8 + 10 = 28

8

"Free world" goal. Cut supplies (if cold war turns into real war).

Figure 2From Harris and Ross [1955]: Schematic diagram of the railway network of the Western So-viet Union and Eastern European countries, with a maximum flow of value 163,000 tons fromRussia to Eastern Europe, and a cut of capacity 163,000 tons indicated as ‘The bottleneck’.

Mincut application (RAND 1950s)

7

rail network connecting Soviet Union with Eastern European countries(map declassified by Pentagon in 1999)

Potential mincut application (2010s)

Government-in-power’s goal. Cut off communication to set of people.

8

Input. An edge-weighted digraph, source vertex s, and target vertex t.

Maxflow problem

9

4

4 15

10

105s t

6

10

9

8

15

1015

16

15

capacity

each edge has apositive capacity

Def. An st-flow (flow) is an assignment of values to the edges such that:

・Capacity constraint: 0 ≤ edge's flow ≤ edge's capacity.

・Local equilibrium: inflow = outflow at every vertex (except s and t).

10

Maxflow problem

0 / 4

0 / 4 0 / 15

10 / 10

10 / 105 / 5 vs t

0 / 6

5 / 10

5 / 9

5 / 8

5 / 15

10 / 1010 / 15

10 / 16

inflow at v = 5 + 5 + 0 = 10

outflow at v = 10 + 0 = 10

flow capacity

0 / 15

11

Maxflow problem




Def. The value of a flow is the inflow at t.

0 / 4

10 / 10

10 / 105 / 5s t

5 / 10

5 / 9

5 / 8

5 / 15

10 / 1010 / 15

0 / 15

value = 5 + 10 + 10 = 25

0 / 4

0 / 6

10 / 16

0 / 15

we assume no edges point to s or from t

12

Maxflow problem




Def. The value of a flow is the inflow at t.

Maximum st-flow (maxflow) problem. Find a flow of maximum value.

0 / 4

10 / 10

10 / 105 / 5s

8 / 10

8 / 9

8 / 8

10 / 1013 / 15

0 / 15

value = 8 + 10 + 10 = 28

0 / 4

3 / 6

13 / 16

0 / 15

t

2 / 15

Soviet Union goal. Maximize flow of supplies to Eastern Europe.

Figure 2From Harris and Ross [1955]: Schematic diagram of the railway network of the Western So-viet Union and Eastern European countries, with a maximum flow of value 163,000 tons fromRussia to Eastern Europe, and a cut of capacity 163,000 tons indicated as ‘The bottleneck’.

Maxflow application (Tolstoǐ 1930s)

13

flow

capacity

rail network connecting Soviet Union with Eastern European countries(map declassified by Pentagon in 1999)

Potential maxflow application (2010s)

"Free world" goal. Maximize flow of information to specified set of people.

14

facebook graph

Input. A weighted digraph, source vertex s, and target vertex t. Mincut problem. Find a cut of minimum capacity.

Maxflow problem. Find a flow of maximum value.

Remarkable fact. These two problems are dual!

Summary

15

s

value of flow = 28

t

0 / 4

10 / 10

10 / 105 / 5

8 / 10

8 / 9

8 / 8

2 / 15

10 / 10

13 / 15

0 / 4

3 / 6

13 / 16

0 / 15

0 / 15

s

capacity of cut = 28

10

8

10

t



Algorithms

‣ introduction





‣ applications

6.4 MAXIMUM FLOW





Initialization. Start with 0 flow.

Ford-Fulkerson algorithm

17

s t

0 / 4

0 / 40 / 15

0 / 10 0 / 15

0 / 15

0 / 10

0 / 10

0 / 9

0 / 15

0 / 8

0 / 6

0 / 16

initialization

0 / 5 0

value of flow

0 / 10

flow capacity

Augmenting path. Find an undirected path from s to t such that:

・Can increase flow on forward edges (not full).

・Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

18

0 / 4

0 / 40 / 15

0 / 15

0 / 15 0 / 10

0 / 9

0 / 8

0 / 6

0 / 16

s t

0 / 10 0 / 15

0 / 10

0 / 10 0 / 15

0 / 10

10

10

10

—

—

—0 / 5

1st augmenting path

0 + 10 = 10

0 / 10

bottleneck capacity = 10





19

0 / 5

0 / 4

0 / 4

0 / 15

0 / 15

0 / 10

0 / 9

0 / 8

0 / 6

10 / 10 10 / 15

10 / 10s t

0 / 15 0 / 10

0 / 16

0 / 15 0 / 10

0 / 16

10

10

10

—

—

—

2nd augmenting path

10 + 10 = 20





20

0 / 4

0 / 4 0 / 150 / 6

10 / 10

10 / 10

10 / 15 10 / 10

10 / 16

s t0 / 5

0 / 10

0 / 9

0 / 8

10 / 15

0 / 5

0 / 10

0 / 9

0 / 8

10 / 15

3rd augmenting path

20 + 5 = 255—

5—

5—

5—

5—

0 / 15

backward edge(not empty)





21

0 / 4

0 / 4 0 / 15

10 / 10

10 / 105 / 5s t

0 / 6

5 / 10

5 / 9

5 / 8

0 / 6

5 / 10

5 / 9

5 / 8

5 / 155 / 15

10 / 1010 / 15

10 / 15

10 / 1610 / 16

4th augmenting path

25 + 3 = 288—

2—

8—

8—

13—

13—

3—

0 / 15


Termination. All paths from s to t are blocked by either a

・Full forward edge.

・Empty backward edge.


22

no more augmenting paths

0 / 4

0 / 4 0 / 15

10 / 10

10 / 105 / 5s t

3 / 6

8 / 10

8 / 9

8 / 8

2 / 15

10 / 1013 / 15

28

13 / 16

0 / 15

full forward edge

empty backward edge


Questions.

・How to compute a mincut?

・How to find an augmenting path?

・If FF terminates, does it always compute a maxflow?

・Does FF always terminate? If so, after how many augmentations?

23

Start with 0 flow.While there exists an augmenting path: - find an augmenting path - compute bottleneck capacity - increase flow on that path by bottleneck capacity




Algorithms

‣ introduction





‣ applications

6.4 MAXIMUM FLOW





Relationship between flows and cuts

Def. The net flow across a cut (A, B) is the sum of the flows on its edges

from A to B minus the sum of the flows on its edges from B to A.

25

0 / 4

10 / 10

10 / 105 / 5s t

5 / 10

5 / 9

5 / 8

5 / 15

10 / 1010 / 15

0 / 15

value of flow = 25

0 / 4

0 / 6

10 / 16

0 / 15

net flow across cut = 5 + 10 + 10 = 25




26

0 / 4

10 / 10

10 / 105 / 5s t

5 / 10

5 / 9

5 / 8

5 / 15

10 / 1010 / 15

0 / 150 / 4

0 / 6

10 / 16

0 / 15

value of flow = 25

net flow across cut = 10 + 5 + 10 = 25




27

0 / 4

10 / 10

10 / 105 / 5s t

5 / 10

5 / 9

5 / 8

5 / 15

10 / 1010 / 15

0 / 15

value of flow = 25

0 / 4

0 / 6

10 / 16

0 / 15

net flow across cut = (10 + 10 + 5 + 10 + 0 + 0) – (5 + 5 + 0 + 0) = 25

edges from B to A


Flow-value lemma. Let f be any flow and let (A, B) be any cut. Then, the net

flow across (A, B) equals the value of f.

Intuition. Conservation of flow.

Pf. By induction on the size of B.

・Base case: B = { t }.

・Induction step: remains true by local equilibrium when moving

any vertex from A to B.

Corollary. Outflow from s = inflow to t = value of flow.

28


Weak duality. Let f be any flow and let (A, B) be any cut.

Then, the value of the flow ≤ the capacity of the cut.

Pf. Value of flow f = net flow across cut (A, B) ≤ capacity of cut (A, B).

29

flow-value lemma flow bounded by capacity

s t

0 / 4

10 / 10

9 / 105 / 5

8 / 10

8 / 9

7 / 8

2 / 15

10 / 10

12 / 15

0 / 4

2 / 6

12 / 16

0 / 15

0 / 15

s

15

5

10

t

value of flow = 27 capacity of cut = 30

Maxflow-mincut theorem

Augmenting path theorem. A flow f is a maxflow iff no augmenting paths.

Maxflow-mincut theorem. Value of the maxflow = capacity of mincut.

Pf. The following three conditions are equivalent for any flow f : i. There exists a cut whose capacity equals the value of the flow f. ii. f is a maxflow.

iii. There is no augmenting path with respect to f.

[ i ⇒ ii ]

・Suppose that (A, B) is a cut with capacity equal to the value of f.

・Then, the value of any flow f ' ≤ capacity of (A, B) = value of f.

・Thus, f is a maxflow.

30

weak duality by assumption






[ ii ⇒ iii ] We prove contrapositive: ~iii ⇒ ~ii.

・Suppose that there is an augmenting path with respect to f.

・Can improve flow f by sending flow along this path.

・Thus, f is not a maxflow.

31






[ iii ⇒ i ]

Suppose that there is no augmenting path with respect to f.

・Let (A, B) be a cut where A is the set of vertices connected to s by an

undirected path with no full forward or empty backward edges.

・By definition of cut, s is in A.

・Since no augmenting path, t is in B.

・Capacity of cut = net flow across cut

= value of flow f.32

forward edges full; backward edges empty

flow-value lemma

To compute mincut (A, B) from maxflow f :

・By augmenting path theorem, no augmenting paths with respect to f.

・Compute A = set of vertices connected to s by an undirected path

with no full forward or empty backward edges.

Computing a mincut from a maxflow

33

forward edge(not full)


0 / 4

0 / 15

10 / 10

10 / 105 / 5 t

8 / 10

8 / 9

8 / 8

2 / 15

10 / 10

16 / 16

0 / 15

ss

13 / 15

13 / 156 / 66 / 6

3 / 43 / 4

full forward edge

empty backward edge

A



Algorithms

‣ introduction





‣ applications

6.4 MAXIMUM FLOW






Questions.

・How to compute a mincut? Easy. ✔

・How to find an augmenting path? BFS works well.

・If FF terminates, does it always compute a maxflow? Yes. ✔

・Does FF always terminate? If so, after how many augmentations?

35

yes, provided edge capacities are integers(or augmenting paths are chosen carefully)

requires clever analysis

Start with 0 flow.While there exists an augmenting path: - find an augmenting path - compute bottleneck capacity - increase flow on that path by bottleneck capacity


Ford-Fulkerson algorithm with integer capacities

Important special case. Edge capacities are integers between 1 and U.

Invariant. The flow is integer-valued throughout Ford-Fulkerson.

Pf. [by induction]

・Bottleneck capacity is an integer.

・Flow on an edge increases/decreases by bottleneck capacity.

Proposition. Number of augmentations ≤ the value of the maxflow.

Pf. Each augmentation increases the value by at least 1.

Integrality theorem. There exists an integer-valued maxflow.

Pf. Ford-Fulkerson terminates and maxflow that it finds is integer-valued.

36

flow on each edge is an integer

critical for some applications (stay tuned)

Bad news. Even when edge capacities are integers, number of augmenting

paths could be equal to the value of the maxflow.

Bad case for Ford-Fulkerson

37

t

s

0

1

00

0capacity

flow

100

100

100

100

0

initialize with 0 flow




38

0

0

t

s

0

1

0

0

1st iteration

1

1

1

100

100

100

100




39

1

1

1

2nd iteration

0

0

t

s

1

1

1

0

100

100

100

100




40

1

1

t

s

1

1

1

0

3rd iteration

2

2

1

100

100

100

100




41

2

2

1

4th iteration

1

1

t

s

1

2

2

0

100

100

100

100




42

. . .




43

t

s

99

1

99

0

199th iteration

100

100

1

99

99

100

100

100

100




44

100

100

1

200th iteration

99

99

t

s

1

100

100

0

100

100

100

100



Good news. This case is easily avoided. [use shortest/fattest path]


45

0

100

100

t

s

1

100

100

100

10010

0

100

can be exponential in input size

How to choose augmenting paths?

Use care when selecting augmenting paths.

・Some choices lead to exponential algorithms.

・Clever choices lead to polynomial algorithms.

46

augmenting path number of paths implementation

random path ≤ E U randomized queue

DFS path ≤ E U stack (DFS)

shortest path ≤ ½ E V queue (BFS)

fattest path ≤ E ln(E U) priority queue

digraph with V vertices, E edges, and integer capacities between 1 and U

How to choose augmenting paths?

Choose augmenting paths with:

・Shortest path: fewest number of edges.

・Fattest path: max bottleneck capacity.

47

Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

J A C K E D M O N D S

University of Waterloo, Waterloo, Ontario, Canada

AND

R I C H A R D M. K A R P

University of California, Berkeley, California

ABSTRACT. This paper presents new algori thms for the maximum flow problem, the Hitchcock t r anspo r t a t i on problem, and the general min imum-cos t flow problem. Upper bounds on the numbers of steps in these algori thms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algori thms.

Firs t , the paper s ta tes the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out t h a t an improper choice of flow augment ing pa ths can lead to severe computa t iona l difficulties. Then rules of choice t h a t avoid these difficulties are given. We show tha t , if each flow augmenta t ion is made along an augment ing pa th having a minimum number of arcs, then a maximum flow in an n-node network will be obta ined af te r no more than ~(n a - n) augmenta t ions ; and then we show tha t if each flow change is chosen to produce a maximum increase in the flow value then, provided the capacit ies are integral , a maximum flow will be de te rmined wi th in at most 1 + logM/(M--1) if(t, S) augmenta t ions , wheref*(t, s) is the value of the maximum flow and M is the maximum number of arcs across a cut.

Next a new algor i thm is given for the minimum-cos t flow problem, in which all shor tes t -pa th computa t ions are performed on networks wi th all weights nonnegat ive . In par t icular , this a lgor i thm solves the n X n ass igmnent problem in O(n 3) steps. Following t h a t we explore a " sca l ing" technique for solving a minimum-cost flow problem by t r ea t ing a sequence of derived problems wi th "scaled down" capacit ies. I t is shown tha t , using this technique, the solution of a I i i tchcock t r anspor t a t i on problem wi th m sources and n sinks, m ~ n, and maximum flow B, requires at most (n + 2) log2 (B/n) flow augmenta t ions . Similar results are also given for the general minimum-cost flow problem.

An abs t rac t s t a t ing the main results of the present paper was presented at the Calgary In te rna t iona l Conference on Combinator ia l S t ruc tures and Thei r Applicat ions, J u n e 1969. In a paper by l)inic (1970) a resul t closely related to the main resul t of Section 1.2 is obtained. Dinic shows tha t , in a network wi th n nodes and p arcs, a maximum flow can be computed in 0 (n2p) pr imi t ive operat ions by an a lgor i thm which augments along shor tes t augment ing paths.

KEY WOl¢l)S AND PHP~ASES: network flows, t r anspor ta t ion problem, analysis of algori thms

CR CATEGOI{.IES: 5.3, 5.4, 8.3

Copyr ight © 1972, Association for Comput ing Machinery , Inc. General permission to republish, bu t not for profit, all or par t of this mater ia l is granted,

provided t ha t reference is made to this publ ica t ion, to its date of issue, and to the fact tha t r epr in t ing privileges were granted by permission of the Association for Comput ing Machinery. Authors ' addresses : J . Edmonds, Depa r tmen t of Combinator ics and Optimizat ion, Univers i ty of Waterloo, Waterloo, Ontario, Canada; R. M. Karp, College of Engineering, Operations Research Center , Univers i ty of California, Berkeley, CA 94720; the l a t t e r au thor ' s research has been par t ia l ly suppor ted by the Nat iona l Science Founda t ion raider Gran t GP-15473 with the Univers i ty of California.

Jc~urnal of the Association for Computing Machinery, Vol. 19, No. 2, Apri| 1972. pp. 248-264.

Edmonds-Karp 1972 (USA) Dinic 1970 (Soviet Union)



Algorithms

‣ introduction





‣ applications

6.4 MAXIMUM FLOW





Flow edge data type. Associate flow fe and capacity ce with edge e = v→w.

Flow network data type. Must be able to process edge e = v→w in either

direction: include e in adjacency lists of both v and w.

Residual (spare) capacity.

・Forward edge: residual capacity = ce – fe.

・Backward edge: residual capacity = fe.

Augment flow.

・Forward edge: add ∆ .

・Backward edge: subtract ∆ .

Flow network representation

49

w7 / 9v

flow fe capacity ce

v w

2

7

residual capacityforward edge

backward edge

Residual network. A useful view of a flow network.

Key point. Augmenting paths in original network are in 1-1

correspondence with directed paths in residual network.

t


50

9 / 10

4 / 104 / 5

9 / 10

9 / 9

0 / 8

4 / 44 / 4 0 / 15

original network

s t

1

61

1

9

8

4415

residual network

4 4

99


s

forward edge(not full)

includes all edges with positive residual capacity

51

Flow edge API

public class FlowEdge public class FlowEdge public class FlowEdge

FlowEdge(int v, int w, double capacity) create a flow edge v→w

int from() vertex this edge points from

int to() vertex this edge points to

int other(int v) other endpoint

double capacity() capacity of this edge

double flow() flow in this edge

double residualCapacityTo(int v) residual capacity toward v

void addResidualFlowTo(int v, double delta) add delta flow toward v

w7 / 9v

flow fe capacity ce

v w

2

7


backward edge

52

Flow edge: Java implementation

public class FlowEdge{ private final int v, w; // from and to private final double capacity; // capacity private double flow; // flow

public FlowEdge(int v, int w, double capacity) { this.v = v; this.w = w; this.capacity = capacity; }

public int from() { return v; } public int to() { return w; } public double capacity() { return capacity; } public double flow() { return flow; }

public int other(int vertex) { if (vertex == v) return w; else if (vertex == w) return v; else throw new IllegalArgumentException(); }

public double residualCapacityTo(int vertex) {...} public void addResidualFlowTo(int vertex, double delta) {...}}

flow variable(mutable)

next slide

53

Flow edge: Java implementation (continued)

public double residualCapacityTo(int vertex) { if (vertex == v) return flow; else if (vertex == w) return capacity - flow; else throw new IllegalArgumentException(); }

public void addResidualFlowTo(int vertex, double delta) { if (vertex == v) flow -= delta; else if (vertex == w) flow += delta; else throw new IllegalArgumentException(); }

forward edge

backward edge

forward edge

backward edge

w7 / 9v

flow fe capacity ce

v w

2

7


backward edge

54

Conventions. Allow self-loops and parallel edges.

Flow network API

public class FlowNetwork public class FlowNetwork

FlowNetwork(int V)FlowNetwork(int V) create an empty flow network with V vertices

FlowNetwork(In in)FlowNetwork(In in) construct flow network input stream

void addEdge(FlowEdge e)addEdge(FlowEdge e) add flow edge e to this flow network

Iterable<FlowEdge> adj(int v)adj(int v) forward and backward edges incident to v

Iterable<FlowEdge> edges()edges() all edges in this flow network

int V()V() number of vertices

int E()E() number of edges

String toString()toString() string representation

Flow network: Java implementation

55

public class FlowNetwork{ private final int V; private Bag<FlowEdge>[] adj; public FlowNetwork(int V) { this.V = V; adj = (Bag<FlowEdge>[]) new Bag[V]; for (int v = 0; v < V; v++) adj[v] = new Bag<FlowEdge>(); }

public void addEdge(FlowEdge e) { int v = e.from(); int w = e.to(); adj[v].add(e); adj[w].add(e); }

public Iterable<FlowEdge> adj(int v) { return adj[v]; }}

same as EdgeWeightedGraph,but adjacency lists ofFlowEdges instead of Edges

add forward edgeadd backward edge

Maintain vertex-indexed array of FlowEdge lists (use Bag abstraction).

Note. Adjacency list includes edges with 0 residual capacity.

(residual network is represented implicitly)

Flow network: adjacency-lists representation

56


adj[]0

1

2

3

4

5

0 2 1.03.0 0 1 2.0 2.0

Bagobjects

4 5 1.03.0 3 5 2.0 2.0

4 5 1.03.0 2 4 1.0 1.0 1 4 1.0 0.0

3 5 2.02.0 2 3 1.0 0.0 1 3 3.0 2.0

2 4 1.01.0 2 3 1.0 0.0 0 2 3.0 1.0

1 4 0.01.0 1 3 3.0 2.0 0 1 2.0 2.0

references to the same FlowEdge object

6 80 1 2.00 2 3.01 3 3.01 4 1.02 3 1.02 4 1.03 5 2.04 5 3.0

tinyFN.txt

V

E

57

Finding a shortest augmenting path (cf. breadth-first search)

private boolean hasAugmentingPath(FlowNetwork G, int s, int t){ edgeTo = new FlowEdge[G.V()]; marked = new boolean[G.V()];

Queue<Integer> queue = new Queue<Integer>(); queue.enqueue(s); marked[s] = true; while (!queue.isEmpty()) { int v = queue.dequeue();

for (FlowEdge e : G.adj(v)) { int w = e.other(v); if (!marked[w] && (e.residualCapacityTo(w) > 0) ) { edgeTo[w] = e; marked[w] = true; queue.enqueue(w); } } }

return marked[t];}

save last edge on path to w; mark w;add w to the queue

found path from s to win the residual network?

is t reachable from s in residual network?

58

Ford-Fulkerson: Java implementation

public class FordFulkerson{ private boolean[] marked; // true if s->v path in residual network private FlowEdge[] edgeTo; // last edge on s->v path private double value; // value of flow

public FordFulkerson(FlowNetwork G, int s, int t) { value = 0.0; while (hasAugmentingPath(G, s, t)) { double bottle = Double.POSITIVE_INFINITY; for (int v = t; v != s; v = edgeTo[v].other(v)) bottle = Math.min(bottle, edgeTo[v].residualCapacityTo(v));

for (int v = t; v != s; v = edgeTo[v].other(v)) edgeTo[v].addResidualFlowTo(v, bottle);

value += bottle; } }

private boolean hasAugmentingPath(FlowNetwork G, int s, int t) { /* See previous slide. */ }

public double value() { return value; }

public boolean inCut(int v) { return marked[v]; }}

compute bottleneck capacity

augment flow

is v reachable from s in residual network?

compute edgeTo[]



Algorithms

‣ introduction





‣ applications

6.4 MAXIMUM FLOW





Maxflow/mincut is a widely applicable problem-solving model.

・Data mining.

・Open-pit mining.

・Bipartite matching.

・Network reliability.

・Baseball elimination.

・Image segmentation.

・Network connectivity.

・Distributed computing.

・Security of statistical data.

・Egalitarian stable matching.

・Multi-camera scene reconstruction.

・Sensor placement for homeland security.

・Many, many, more.

Maxflow and mincut applications

60

liver and hepatic vascularization segmentation

Bipartite matching problem

61

N students apply for N jobs.

Each gets several offers.

Is there a way to match all students to jobs?

1

2

3

4

5

Alice

Bob

Carol

Dave

Eliza

AdobeAmazonGoogle

AdobeAmazon

AdobeFacebook Google

AmazonYahoo

AmazonYahoo

6

7

8

9

10

Adobe

Amazon

Facebook

Google

Yahoo

AliceBobCarol

AliceBobDaveEliza

Carol

AliceCarol

DaveEliza

bipartite matching problem

Given a bipartite graph, find a perfect matching.

Bipartite matching problem

62

bipartite graph

N students N companies

1

2

3

4

5

Alice

Bob

Carol

Dave

Eliza

AdobeAmazonGoogle

AdobeAmazon


AmazonYahoo

AmazonYahoo

6

7

8

9

10

Adobe

Amazon

Facebook

Google

Yahoo

AliceBobCarol

AliceBobDaveEliza

Carol

AliceCarol

DaveEliza

bipartite matching problemperfect matching (solution)

Alice

Bob

Carol

Dave

Eliza

—— Google

—— Adobe

—— Facebook

—— Yahoo

—— Amazon 3

1

5

2

4

6

8

9

7

10

Network flow formulation of bipartite matching

・Create s, t, one vertex for each student, and one vertex for each job.

・Add edge from s to each student (capacity 1).

・Add edge from each job to t (capacity 1).

・Add edge from student to each job offered (infinite capacity).

63

1

2

3

4

5

Alice

Bob

Carol

Dave

Eliza

AdobeAmazonGoogle

AdobeAmazon


AmazonYahoo

AmazonYahoo

6

7

8

9

10

Adobe

Amazon

Facebook

Google

Yahoo

AliceBobCarol

AliceBobDaveEliza

Carol

AliceCarol

DaveEliza

3

1

t

6

8

9

55

2

4

7

10

s

bipartite matching problem


flow network

1-1 correspondence between perfect matchings in bipartite graph and

integer-valued maxflows of value N.

Network flow formulation of bipartite matching

64

3

1

t

6

8

9

55

2

4

7

10

s

1

2

3

4

5

Alice

Bob

Carol

Dave

Eliza

AdobeAmazonGoogle

AdobeAmazon


AmazonYahoo

AmazonYahoo

6

7

8

9

10

Adobe

Amazon

Facebook

Google

Yahoo

AliceBobCarol

AliceBobDaveEliza

Carol

AliceCarol

DaveEliza

bipartite matching problemflow network


Goal. When no perfect matching, explain why.

What the mincut tells us

65

5

3

2

4

10

6

8

7

9

5

2

4

10

7

1

S = { 2, 4, 5 }T = { 7, 10 }

student in Scan be matched

only tocompanies in T

| S | > | T |

no perfect matching exists

Mincut. Consider mincut (A, B).

・Let S = students on s side of cut.

・Let T = companies on s side of cut.

・Fact: | S | > | T |; students in S can be matched only to companies in T.

Bottom line. When no perfect matching, mincut explains why.

What the mincut tells us

66

3

1

t

6

8

9

55

2

4

7

10

s

2

4

7

10

s

5

no perfect matching exists

S = { 2, 4, 5 }T = { 7, 10 }

student in Scan be matched

only tocompanies in T

| S | > | T |

Q. Which teams have a chance of finishing the season with the most wins?

Montreal is mathematically eliminated.

・Montreal finishes with ≤ 80 wins.

・Atlanta already has 83 wins.

Baseball elimination problem

67

i teamteam wins losses to play ATL PHI NYM MON

0 Atlanta 83 71 8 – 1 6 1

1 Philly 80 79 3 1 – 0 2

2 New York 78 78 6 6 0 – 0

3 Montreal 77 82 3 1 2 0 –


Philadelphia is mathematically eliminated.

・Philadelphia finishes with ≤ 83 wins.

・Either New York or Atlanta will finish with ≥ 84 wins.

Observation. Answer depends not only on how many games already won

and left to play, but on whom they're against.


68

i teamteam wins losses to play ATL PHI NYM MON

0 Atlanta 83 71 8 – 1 6 1

1 Philly 80 79 3 1 – 0 2

2 New York 78 78 6 6 0 – 0

3 Montreal 77 82 3 1 2 0 –


Detroit is mathematically eliminated.

・Detroit finishes with ≤ 76 wins.

・Wins for R = { NYY, BAL, BOS, TOR } = 278.

・Remaining games among { NYY, BAL, BOS, TOR } = 3 + 8 + 7 + 2 + 7 = 27.

・Average team in R wins 305/4 = 76.25 games.


69

i teamteam wins losses to play NYY BAL BOS TOR DET

0 New York 75 59 28 – 3 8 7 3

1 Baltimore 71 63 28 3 – 2 7 4

2 Boston 69 66 27 8 2 – 0 0

3 Toronto 63 72 27 7 7 0 – 0

4 Detroit 49 86 27 3 4 0 0 –

AL East (August 30, 1996)

Intuition. Remaining games flow from s to t.

Fact. Team 4 not eliminated iff all edges pointing from s are full in maxflow.

Baseball elimination problem: maxflow formulation

70

s g12 t

game vertices(each pair of teams other than 4)

team vertices(each team other than 4)

w4 + r4 – w2

1

0

3

2

0–2

0–3

1–3

0–1

2–3

∞

∞

1–2

games leftbetween 1 and 2

team 2 can still winthis many more games

Maximum flow algorithms: theory

(Yet another) holy grail for theoretical computer scientists.

71

year method worst case discovered by

1951 simplex E3 U Dantzig

1955 augmenting path E2 U Ford-Fulkerson

1970 shortest augmenting path E3 Dinitz, Edmonds-Karp

1970 fattest augmenting path E2 log E log( E U ) Dinitz, Edmonds-Karp

1977 blocking flow E 5/2 Cherkasky

1978 blocking flow E 7/3 Galil

1983 dynamic trees E2 log E Sleator-Tarjan

1985 capacity scaling E2 log U Gabow

1997 length function E3/2 log E log U Goldberg-Rao

2012 compact network E2 / log E Orlin

? ? E ?

maxflow algorithms for sparse digraphs with E edges, integer capacities between 1 and U

Maximum flow algorithms: practice

Warning. Worst-case order-of-growth is generally not useful for predicting

or comparing maxflow algorithm performance in practice.

Best in practice. Push-relabel method with gap relabeling: E 3/2.

72

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

E L S E V I E R European Journal of Operational Research 97 (1997) 509-542

T h e o r y a n d M e t h o d o l o g y

Computational investigations of maximum flow algorithms R a v i n d r a K . A h u j a a, M u r a l i K o d i a l a m b, A j a y K . M i s h r a c, J a m e s B . O r l i n d, .

a Department t~'lndustrial and Management Engineering. Indian Institute of Technology. Kanpur, 208 016, India b AT& T Bell Laboratories, Holmdel, NJ 07733, USA

c KA'F-Z Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260, USA d Sloun School of Management, Massachusetts Institute of Technology. Cambridge. MA 02139. USA

Received 30 August 1995; accepted 27 June 1996

A b s t r a c t

The maximum flow algorithm is distinguished by the long line of successive contributions researchers have made in obtaining algorithms with incrementally better worst-case complexity. Some, but not all, of these theoretical improvements have produced improvements in practice. The purpose of this paper is to test some of the major algorithmic ideas developed in the recent years and to assess their utility on the empirical front. However, our study differs from previous studies in several ways. Whereas previous studies focus primarily on CPU time analysis, our analysis goes further and provides detailed insight into algorithmic behavior. It not only observes how algorithms behave but also tries to explain why algorithms behave that way. We have limited our study to the best previous maximum flow algorithms and some of the recent algorithms that are likely to be efficient in practice. Our study encompasses ten maximum flow algorithms and five classes of networks. The augmenting path algorithms tested by us include Dinic's algorithm, the shortest augmenting path algorithm, and the capacity-scaling algorithm. The preflow-push algorithms tested by us include Karzanov's algorithm, three implementations of Goldberg-Tarjan's algorithm, and three versions of Ahuja-Orlin-Tarjan's excess-scaling algorithms. Among many findings, our study concludes that the preflow-push algorithms are substantially faster than other classes of algorithms, and the highest-label preflow-push algorithm is the fastest maximum flow algorithm for which the growth rate in the computational time is O(n LS) on four out of five of our problem classes. Further, in contrast to the results of the worst-case analysis of maximum flow algorithms, our study finds that the time to perform relabel operations (or constructing the layered networks) takes at least as much computation time as that taken by augmentations and/or pushes. © 1997 Published by Elsevier Science B.V.

1. I n t r o d u c t i o n

The maximum flow problem is one of the most fundamental problems in network optimization. Its intuitive appeal, mathematical simplicity, and wide applicabil i ty has made it a popular research topic

* Corresponding author.

0377-2217/97/$17.00 © 1997 Published by Elsevier Science B.V. All PII S0377-2217(96)00269-X

among mathematicians, operations researchers and computer scientists.

The maximum flow problem arises in a wide variety of situations. It occurs directly in problems as diverse as the flow of commodit ies in pipeline networks, parallel machine scheduling, distributed computing on multi-processor computers, matrix round- ing problems, the baseball el imination problem, and the statistical security of data. The maximum flow

rights reserved.

On Implement ing Push-Re labe l M e t h o d for the M a x i m u m Flow Problem

Boris V. Cherkassky 1 and Andrew V. Goldberg 2

1 Central Institute for Economics and Mathematics, Krasikova St. 32, 117418, Moscow, Russia

[email protected] 2 Computer Science Department, Stanford University

Stanford, CA 94305, USA goldberg ~cs. stanford, edu

Abst rac t . We study efficient implementations of the push-relabel method for the maximum flow problem. The resulting codes are faster than the previous codes, and much faster on some problem families. The speedup is due to the combination of heuristics used in our implementations. We also exhibit a family of problems for which the running time of all known methods seem to have a roughly quadratic growth rate.

1 I n t r o d u c t i o n

The rnaximum flow problem is a classical combinatorial problem that comes up in a wide variety of applications. In this paper we study implementations of the push-rdabel [13, 17] method for the problem.

The basic methods for the maximum flow problem include the network simplex method of Dantzig [6, 7], the augmenting path method of Ford and F~lker- son [12], the blocking flow method of Dinitz [10], and the push-relabel method of Goldberg and Tarjan [14, 17]. (An earlier algorithm of Cherkassky [5] has many features of the push-relabel method.) The best theoretical time bounds for the maximum flow problem, based on the latter method, are as follows. An algorithm of Goldberg and Tarjan [17] runs in O(nm log(n2/m)) time, an algo- r i thm of King et. al. [21] runs in O(nm + n TM) time for any constant e > 0, an algorithm of Cheriyan et. al. [3] runs in O(nm + (n logn) 2) time with high probability, and an algorithm of Ahuja et. al. [1] runs in O ( a m log (~ - -~ + 2 ) ) time.

Prior to the push-relabel method, several studies have shown that Dinitz' algorithm [10] is in practice superior to other methods, including the network simplex method [6, 7], Ford-giflkerson algorithm [11, 12], Karzanov's algorithm [20], and Tarjan's algorithm [23]. See e.g. [18]. Several recent studies (e.g. [2,

* Andrew V. Goldberg was supported in part by NSF Grant CCR-9307045 and a grant from Powell Foundation. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the above-mentioned NSF and Powell Foundation grants.

Summary

Mincut problem. Find an st-cut of minimum capacity.

Maxflow problem. Find an st-flow of maximum value.

Duality. Value of the maxflow = capacity of mincut.

Proven successful approaches.

・Ford-Fulkerson (various augmenting-path strategies).

・Preflow-push (various versions).

Open research challenges.

・Practice: solve real-world maxflow/mincut problems in linear time.

・Theory: prove it for worst-case inputs.

・Still much to be learned!

73

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