ADVANCED ALGORITHMS Lecture 21: LPs for matching, covering 1 - - PowerPoint PPT Presentation

ADVANCED ALGORITHMS

Lecture 21: LPs for matching, covering

ANNOUNCEMENTS

➤ HW 5 will be out tonight — due in two weeks ➤ Project “mid-way” report due next Friday

LAST CLASS

➤ Linear programming

N

k

i

Nn

ain

a b

AIN E be

aina'sbm

LAST CLASS

➤ Geometry of linear programs ➤ what is a “corner point”? (intersection of n of the planes) ➤ neighboring corners for u (directions given by A-1, where A defines u) ➤ no neighbor gives improvement => optimum! ➤ main idea behind simplex algorithm ➤ Can solve in polynomial time via Ellipsoid, interior point methods a localopt

globof

USING LP FOR COMBINATORIAL PROBLEMS

➤ Continuous formulations … matching?

discrete optimization

nu

g

variables

ng for edges

ij3

i

j

Cfoe

Constraints

• T.fi

eneigiiimcn lrL.repiAa

B

v j

nij

Voenij

Vi

1

max

Wijnij 4 0

USING LP FOR COMBINATORIAL PROBLEMS

➤ Continuous formulations … matching? ➤ Main problem: what if the solutions are fractions?


Main question: are the “corner points” integral?

MATCHING

➤ Theorem: all the corner points of the “matching polytope” are

integral.

➤ Thus, solving the LP gives a 0/1 solution!

a

setoffeasible solutions

a

to the linear

m

variables

m

• f edges

System

qq.gs

constraints

q

irregs

If

we take any m of the constraints

solve

we

get

an

integer point

PROOF 1 (OUTLINE) : TOTAL UNIMODULARITY

Claim

Take any

m of the

2mi 2n constraints

The intersection point hasonly

0 1 values

1 1 values

N

N12 I 7h22

Z

f

• k

i i

i

EE

ni El

m

1,1 1 1

1

PROOF 1 (OUTLINE) : TOTAL UNIMODULARITY

Em

Any

corner pt canbe obtained by solving

a

term

B n

z

where

B

comprises

linear Sys

precisely

m

rows of M

and z

is the RHS of

the corresponding constraint

n n a

Nu

Fit

B EH

• ITI

dit 113

Nij

PROOF 2: UNDERSTANDING CORNER POINTS …

z

Alternate characterization

f

z

z

u is

a

corner ptiff

u

For any perturbation direction Z

fl

z

p

Z

at most

• ne of

Utz

and u

z

is in the feasible set

Given

u if we need to show that

u w Not a corner

then

ive it suffices to inhibit

2 sit

utz and

u

2

are both feasible

PROOF 2

Claim

Let

u

lae any feasible

non integral point

satisfying Then F zto.it

utz and

u

z bothSati

E

ijs.t.ocuig.cl

• U
• i

i

i

a

8

i

e

i

Observation

Entry vertex has

degree

rO

O

O

I

g

Exercised

Any such graph has a cycle I

e

there

is

a cycle

using only

the edges of E

• no fading

Ptu

if

8

be

a

• o.us

can

pt

For small enough 8

this will still be

a

feasible solution

This gives the perturbation

z that

we want

FLOWS IN NETWORKS

➤ Theorem: all the corner points of the “flow polytope” are integral.

for

finding max flow

we

can simply

use

linear

programming

WARNINGS

➤ This is a very special phenomenon! ➤ Polytopes usually have fractional corners …

MONITORING EDGES (A.K.A. VERTEX COVER)

➤ Problem. given undirected graph G = (V

, E), find a small set of nodes S such that every edge has at least one of its neighbors chosen.

find the smallest

Lo

• f nodes so that

all

edges are

monitored

Vats

p

NuC

• I

7 01411

relaxation

LP FOR VERTEX COVER

Constants Hedges

i j

ni taj 31

min

ni

7

312

O E ni I l

U

Of Nu

Nv Nw E 1

Now

nut nu 31

v

O

Nu Nu Nw 42

Nr t Kw71

Nwt Nu

I

BAD CORNERS

“ROUNDING” SOLUTION

APPROXIMATION ALGORITHM

OTHER PROBLEMS — INDEPENDENT SET

➤ Problem. given undirected graph G = (V

, E), find the largest possible set of nodes S such that has no edges within.

LP AND ITS LIMITATIONS

IN SUMMARY

➤ LP (“continuous”) formulations for discrete problems ➤ can get lucky — all corners are integral (i.e., discrete) ➤ corners can be “somewhat integral” — approximation algorithms ➤ corners can be totally useless — means better LP is needed! ➤ Next time … randomized rounding