Problem Solving 5

Applications of NP-hardness

Knapsack

Knapsack

. and 0 Assume

}.1 ,0{

t.s.

max

2211

2211

Ssc

x

Sxsxsxs

xcxcxc

ii

i

nn

nn

Knapsack is NP-hard

. and 0 Assume

}.1 ,0{

t.s.

max

2211

2211

Ssc

x

Sxsxsxs

xcxcxc

ii

i

nn

nn

Decision Version of Knapsack

Knapsack Partition

.

such that ,...,,exist erewhether th

determine ,0 and ,,...,,,,...,,Given

2211

221

21

2121

1

pm

nn

nn

n

nn

Sxsxsxs

kxcxcxc

xxx

kSsssccc

Partition

1 2

. such that

partition a thereis },,...,,{ integers, positive ofset aGiven

21

21

Aa Aa

n

aaAAA

aaaAn

Knapsack ofersion Decision v Partition pm

Knapsack.for Partition for Then

.2

1 and Set

1

YesYes

aSkascn

i iiii

Set-Cover

Set-Cover

Given a collection C of subsets of a set X, find a minimum subcollection C’ of C such that every element of X appears in a subset in C’ . (Such a C’ is called a set-cover.)

Cover-Set Cover -Vertex pm

Vertex-Cover

subset. in theendpoint an has edgeevery

such that verticesofsubset a iscover vertex a where

cover vertex minimum a find ),,(graph aGiven EVG

Decision Version of Vertex-Cover

vertices.most at cover with vertex a exists erewhether th

determine |,|1, and ),(graph aGiven

k

VkkEVG

Decision Version of Set-Cover

subsets.

most at cover withset a exists here whether tdetermine ,0

integer an and set finite a of subsets of collection aGiven

k

k

XC

Cover-Set ofVersion Decision Cover -Vertex ofVersion Decision pm

.most at size ofcover set a

contains most at size ofcover vertex a has Then . and

}|)({Set .by covered edges ofset thebe )(Let

k

CkGEX

VvvSCVvvS

Broadcast in Multi-Channel Wireless Networks

Problem

minimized. is |)(| (2) tree;

broadcast a contains })()( | ){( (1)such that

every for )( )( find , node source aGiven

messages. vesend/recei tousecan

that )( integers), (postive channels ofset a exists there

, nodeeach at that ),(network aConsider

vF

uAuFu,v

VvvAvFs

v

vA

VvEVG

Vv

Nonunique Probe Selection and Group Testing

DNA Hybridization

Polymerase Chain Reaction

(PCR)• cell-free method of

DNA cloning

Advantages• much faster than cell

based method• need very small

amount of target DNADisadvantages• need to synthesize

primers• applies only to short

DNA fragments(<5kb)

Preparation of a DNA Library• DNA library: a collection of cloned DNA fragments• usually from a specific organism

DNA Library Screening

Problem

• If a probe doesn’t uniquely determine a virus, i.e., a probe determine a group of viruses, how to select a subset of probes from a given set of probes, in order to be able to find up to d viruses in a blood sample.

Binary Matrixviruses

c1 c2 c3 cj cn

p1 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p2 0 1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p3 1 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0

probes 0 0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0 ..

pi 0 0 0 … 0 … 0 … 1 … 0 … 0 … 0 … 0 ..

pt 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0

The cell (i, j) contains 1 iff the ith probe hybridizes the jth virus.

Binary Matrix of Example

virus

c1 c2 c3 cj p1 1 1 1 0 0 0 0 0 0 p2 0 0 0 1 1 1 0 0 0 p3 0 0 0 0 0 0 1 1 1

probes 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 00 0 1 0 0 1 0 0 1

Observation: All columns are distinct.

To identify up to d viruses, all unions of up to d columns should be distinct!

d-Separable Matrixviruses

c1 c2 c3 cj cn

p1 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p2 0 1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0

p3 1 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 probes 0 0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0

.

. pi 0 0 0 … 0 … 0 … 1 … 0 … 0 … 0 … 0

.

. pt 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0

All unions of up to d columns are distinct.Decoding: O(nd)

_

d-Disjunct Matrix viruses

c1 c2 c3 cj cn

p1 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p2 0 1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p3 1 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0

probes 0 0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0 ..

pi 0 0 0 … 0 … 0 … 1 … 0 … 0 … 0 … 0 ..

pt 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0

Each column is different from the union of every d other columnsDecoding: O(n)Remove all clones in negative pools. Remaining clones are all positive.

Nonunique Probe Selection• Given a binary matrix, find a d-separable

submatrix with the same number of columns and the minimum number of rows.

• Given a binary matrix, find a d-disjunct submatrix with the same number of columns and the minimum number of rows.

• Given a binary matrix, find a d-separable submatrix with the same number of columns and the minimum number of rows

_

Problem

NP? tobelong they Do

to?belong problemsSelection

Probe Nonunique do class complexityWhat

Complexity?

• All three problems may not be in NP, why?

• Guess a t x n matrix M, verify if M is d-separable (d-separable, d-disjunct).

-

d-Separability Test

• Given a matrix M and d, is M d-separable?

• It is co-NP-complete.

d-Separability Test

• Given a matrix M and d, is M d-separable?

• This is co-NP-complete.

(a) It is in co-NP.

Guess two samples from space S(n,d). Check if M gives the same test outcome on the two samples.

-

-

-

(b) Reduction from vertex-cover

• Given a graph G and h > 0, does G have a vertex cover of size at most h?

.1set and

,for *}{

;for }{

:pools following Design the .* Choose

).,( Suppose

hd

Eeev

Vvv

Vv

EVG

Testty Separabili- of Complement dVC pm

d-Separability Test Reduces to d-Separability Test

• Put a zero column to M to form a new matrix M*

• Then M is d-separable if and only if M* is d-separable.

-

-

d-Disjunct Test

• Given a matrix M and d, is M d-disjunct?

• This is co-NP-complete.

Minimum d-Separable Submatrix

• Given a binary matrix, find a d-separable submatrix with minimum number of rows and the same number of columns.

• For any fixed d >0, the problem is NP-hard.

• In general, the problem is conjectured to be Σ2 –complete.

p

complete?- isWhat

? isWhat

2

2

p

p

Σ