graph partitioning problems

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Graph Partitioning Problems s1 s3 s4 s2 T1 T4 T2 T3 s1 s4 s2 s3 t3 t1 t2 t4 A region R1 R2 C1 C2

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Graph Partitioning Problems. T2. s1. s2. T1. T3. T4. s3. R1. R2. s1. s4. s4. t1. t3. t2. C1. C2. A region. s3. s2. t4. Graph Partitioning Problems. General setting: to remove a minimum (weight) set of edges to cut the graph into pieces. Examples: - PowerPoint PPT Presentation

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Page 1: Graph Partitioning Problems

Graph Partitioning Problems

s1

s3

s4

s2

T1

T4

T2

T3

s1s4

s2 s3

t3

t1

t2

t4

A region

R1 R2

C1

C2

Page 2: Graph Partitioning Problems

Graph Partitioning Problems

General setting: to remove a minimum (weight) set of edges

to cut the graph into pieces.

Examples:

Minimum (s-t) cut

Multiway cut

Multicut

Sparsest cut

Minimum bisection

Page 3: Graph Partitioning Problems

Minimum s-t Cut

s

Mininium s-t cut = Max s-t flow

Minimum s-t cut = minimum (weighted) set of edges to disconnect s and t

t

Page 4: Graph Partitioning Problems

Multiway Cut

Given a set of terminals S = {s1, s2, …, sk},

a multiway cut is a set of edges whose removal

disconnects the terminals from each other.

The multiway cut problem asks for the minimum weight multiway cut.

s1

s3

s4

s2

Page 5: Graph Partitioning Problems

Multicut

Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)},

a multicut is a set of edges whose removal disconnects each source-sink pair.

The multicut problem asks for the minimum weight multicut.

s1s4

s2 s3

t3

t1

t2

t4

Page 6: Graph Partitioning Problems

Multicut vs Multiway cut

Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)},

a multicut is a set of edges whose removal disconnects each source-sink pair.

Given a set of terminals S = {s1, s2, …, sk},

a multiway cut is a set of edges whose removal

disconnects the terminals from each other.

What is the relationship between these two problems?

Multicut is a generalization of multiway cut. Why?

Because we can set each (si,sj) as a source-sink pair.

Page 7: Graph Partitioning Problems

Sparsest Cut

Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)}.

For a set of edges U, let c(U) denote the total weight.

Let dem(U) denote the number of pairs that U disconnects.

The sparsest cut problem asks for a set U which minimizes c(U)/dem(U).

In other words, the sparsest cut problem asks for the

most cost effective way to disconnect source-sink pairs,

i.e. the average cost to disconnect a pair is minimized.

Page 8: Graph Partitioning Problems

Sparsest Cut

Suppose every pair is a source-sink pair.

For a set of edges U, let c(U) denote the total weight.

Let dem(U) denote the number of pairs that U disconnects.

The sparsest cut problem asks for a set U which minimizes c(U)/dem(U).

S V-S Minimize

Page 9: Graph Partitioning Problems

Minimum Bisection

The minimum bisection problem is to divide the vertex set into

two equal size parts and minimize the total weights of the edges in

between.

This problem is very useful in designing approximation algorithms

for other problems – to use it in a divide-and-conquer strategy.

Page 10: Graph Partitioning Problems

Relations

Minimum cut

Multiway cut

Multicut Sparsest cut

Minimum bisection

Page 11: Graph Partitioning Problems

Results

Minimum cut Polynomial time solvable.

Multiway cut a combintorial 2-approximation algorithm an elegant LP-based 1.34-approximation

Multicut an O(log n)-approximation algorithm. some “evidence” that no constant factor algorithm exists.

Sparsest cut an O(log n)-approximation algorithm based on multicut. an O(√log n)-approximation based on semidefinite programming. some “evidence” that no constant factor algorithm exists.

Min bisection an O(log n)-approximation algorithm based on sparsest cut. (this statement is not quite accurate but close enough).

Page 12: Graph Partitioning Problems

Relations

Minimum cut

Multiway cut

Multicut Sparsest cut

Minimum bisection

2-approx

O(log n)-approx

Region Growing

O(log n)-approx

O(log n)-approx

Page 13: Graph Partitioning Problems

Multiway Cut

Given a set of terminals S = {s1, s2, …, sk},

a multiway cut is a set of edges whose removal

disconnects the terminals from each other.

The multiway cut problem asks for the minimum weight multiway cut.

s1

s3

s4

s2

This picture leads to

a natural algorithm!

Page 14: Graph Partitioning Problems

Algorithm

(Multiway cut 2-approximation algorithm)

1. For each i, compute a minimum weight isolating cut for s(i), say C(i).

2. Output the union of C(i).

Define an isolating cut for s(i) to be a set of edges whose

removal disconnects s(i) from the rest of the terminals.

How to compute a minimum isolating cut?

Page 15: Graph Partitioning Problems

s1

s3

s4

s2

Analysis

Why is it a 2-approximation?

Imagine this is an optimal solution.

The (thick) red edges form an isolating cut for s1, call it T1.

Since we find a minimum isolating cut for s1, we have w(C1) <= w(T1).

Page 16: Graph Partitioning Problems

s1

s3

s4

s2

Analysis

Why is it a 2-approximation?

Imagine this is an optimal solution.

T1

T4

T2

T3

Key: w(Ci) <= w(Ti)

ALG = w(C1) + w(C2) + w(C3) + w(C4) OPT = (w(T1) + w(T2) + w(T3) + w(T4)) / 2

So, ALG <= 2OPT.

Page 17: Graph Partitioning Problems

Bad Example

2

22

2

1.0001

1.0001

1.0001

1.0001

OPT = 4.0004

ALG = 8

Page 18: Graph Partitioning Problems

Multicut

Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)},

a multicut is a set of edges whose removal disconnects each source-sink pair.

The multicut problem asks for the minimum weight multicut.

s1s4

s2 s3

t3

t1

t2

t4

Can we use the idea in the

isolating cut algorithm?

Page 19: Graph Partitioning Problems

Bad Example

……

.. ……

..

s1

s2

sk

t1

tk

t2

1 1

1

11

11.0001

OPT = 1.0001

ALG = k

Algorithm: take the union of minimum si-ti cut.

Page 20: Graph Partitioning Problems

Linear Program

for each path p connecting a source-sink pair

Separation oracle: given a fractional solution d, decide if d is feasible.

Shortest path computations between source-sink pairs.

Page 21: Graph Partitioning Problems

Multicut

Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)},

a multicut is a set of edges whose removal disconnects each source-sink pair.

The multicut problem asks for the minimum weight multicut.

Now we only consider a tree!

s1

s2

t2

t1

Page 22: Graph Partitioning Problems

Hardness

Is this problem hard?

Vertex cover.

Page 23: Graph Partitioning Problems

Primal Dual Programs

Primal complementary slackness condition:

Dual complementary slackness condition:

Page 24: Graph Partitioning Problems

Approximate Optimality Conditions

Primal complementary slackness condition:

Dual complementary slackness condition:

Only pick saturated edges.

At most one edge can be picked from a path carrying nonzero flow.

Relaxed dual complementary slackness condition:

two

Page 25: Graph Partitioning Problems

Algorithmic Idea

Only pick saturated edges.

At most two edges can be picked from a path carrying nonzero flow.

We want to construct flow and cut so that:

How to choose the flow?

We should avoid sending flow along a long path…

Page 26: Graph Partitioning Problems

Algorithm

Multicut in Tree (2-approximation algorithm)

Initially, f=0, D=0

Flow routing:

Find a pair (si,ti) with lowest lowest common ancestor,

Greedily send flow from si to ti.

Add all the saturated edges to D.

Let e1, e2,…, el be the ordered list of edges in D.

Reverse delete:

In reverse order, delete an edge e from D if D-e is still a multicut.

Output the flow and the cut.

Page 27: Graph Partitioning Problems

Reverse Delete

At most two edges can be picked from a path carrying nonzero flow.

Lemma. After reverse delete, the above condition is satisfied.

si ti

v

e’

e

sj

tj

uthere exists j wheree is the only edge. j is processed

earlier than i.

i can send a flow, so e is added after e*.

e*