ADVANCED ALGORITHMS Lecture 22: Rounding solutions, duality overview - - PowerPoint PPT Presentation

ADVANCED ALGORITHMS

Lecture 22: “Rounding” solutions, duality overview

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ANNOUNCEMENTS

➤ HW 5 is out — best 5 out of 6 ➤ Project “mid-way” report due this Friday (or weekend)

2

A linear programs

Hw

next week

I

submit

• n

canvas

• nly
• ne submission

per group

LAST WEEK

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➤ Linear programming

LAST WEEK

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➤ Geometry of linear programs ➤ what is a “corner point”? (intersection of n of the planes) ➤ neighboring corners ➤ simplex algorithm ➤ alternate definition of corner point ➤ LP for combinatorial problems ➤ Matching linear program — all corner points are integral! (proved

by two methods) (determinants; characterization of corners)

IO

N

Nn

m

k

it

ii

ft H

1

weighted matching

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

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➤ 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. ∀edges {i, j}, xi + xj ≥ 1 Variables: xi minimize ∑

i

xi 0 ≤ xi ≤ 1 1/2 1/2 1/2

Optimum value of the LP can be smaller than “real” opt value…

Note that it can’t be larger!

end points

a

na my 2 1

I

e

Because any integer solution is also feasiblefor

the LP

“ROUNDING” SOLUTION

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Theorem

If

OPTy

denotes the optimum valueofthe

LP

and

PTuc

denotes the value ofthe optimum 0 1

Solution

then

PT

3

OPTu Z L

• pTu

En

OPT

PT

I

Proof

By

rounding

a

fractional sdn

ni

to

• ne

0 1 solution

sothat the objective

values

are not too different

satish g

IV edges

n

Start with

some

ni

i come up with

Yi C

• I

fifnido

5

sety.ir

if

ni

3

0.5

set Yi

L

If

nitaj

31

then is yityj 71

what happens to objective value

Can wereeak Eti

R

E Yi

For energi Yi E 2mi

yi

E

2

xi

to

APPROXIMATION ALGORITHM

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whated

F

an

efficient poly time algorithm

that

produces

a

solution

yi

whose cost is

I

OPT

up

E

2

OP

Tue

Relax

and

dound

paradigm

fractional integral

writidg a discreteOPI mone lo continuous opt

OTHER PROBLEMS — INDEPENDENT SET

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➤ Problem. given undirected graph G = (V

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

There is no

kntown

poly time

n

algorithm that has an

approx ratio

even n 7

Ni

i 0 1 variables

H edges

i j

ni

nj El

LIMITATIONS

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max

Xi me 2

b t

H edges

i j

ai

nj

El

OPT

12

SUMMARY SO FAR

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➤ 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 — better LP?

TFheduling clustering

PROBABILISTIC ROUNDING

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➤ Example. consider “hiring problem”; given n people, each having

expertise on a subset of m skills, hire k people in order to maximize total number of distinct skills.

iii

Ein

h

• 0bs

skill j

is

covered if

at least one of the i's that

j

are enperts in skill

jinosen

O

FORMULATING THE LP

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Zg

supposed to be

a boolean van that is I

if

skill j

ai covered and 0 otherwise

Skill

j

is

covered

E

ni

31

I

ienbrslj

Zj l

I

tinnnijit

0EZj El

and Zg E

E Ni

k

ienbrs j

Objedinfn

maximize E Zz

relaxing

variables

Ki C

• i

OEnitt

constraint

ni

K

• bjective

maximize

midlife.sij

O

j

BAD SOLUTIONS?

13

j

people

skills

2 1 2 a

l

k

X

1 b o

• integer OPT

usohy

2

X

2

5

ta

ET

g

Id

• 4

but

OPTLP

6

2

e

5

zfo

PROBABILISTIC ROUNDING

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Turns out we can compute … Using this, can show that expected # of topics “covered” > 0.63 * OPT

I

APPROXIMATION FOR MAX-K-COVER

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SUMMARY SO FAR

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➤ LP (“continuous”) formulations for discrete problems ➤ can get lucky — all corners are integral (i.e., discrete) ➤ corners can be “somewhat integral” — approximation algorithms ➤ Randomized rounding — natural idea, reasonably simple to analyze