Problem Solving 5
Applications of NP-hardness
Knapsack
Knapsack
max c1 x1  c 2 x 2    cnxn
s. t. s1 x1  s 2 x 2    snxn  S
xi {0, 1}.
Assume ci  0 and si  S .
Knapsack is NP-hard
max c1 x1  c 2 x 2    cnxn
s. t. s1 x1  s 2 x 2    snxn  S
xi {0, 1}.
Assume ci  0 and si  S .
Decision Version of Knapsack
Given c1 , c2 ,..., cn , s1 , s2 ,..., sn , S and k  0, determine
whether th ere exist x1 , x2 ,..., xn such that
c1 x1  c2 x2    cn xn  k
s1 x1  s2 x2    sn xn  S .
Partition  mp Knapsack
Partition
Given a set of n positive integers, A  {a1 , a2 ,..., an }, is there a partition
A  A1  A2 such that
 a   a.
a A1
a A2
Partition  mp Decision v ersion of Knapsack
1 n
Set ci  si  ai and k  S  i 1 ai .
2
Then Yes for Partition  Yes for Knapsack.
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 setcover.)
Vertex - Cover  mp Set - Cover
Vertex-Cover
Given a graph G  (V , E ), find a minimum vertex cover
where a vertex cover is a subset of vertices such that
every edge has an endpoint in the subset.
Decision Version of Vertex-Cover
Given a graph G  (V , E ) and k ,1  k | V |, determine
whether th ere exists a vertex cover with at most k vertices.
Decision Version of Set-Cover
Given a collection C of subsets of a finite set X and an integer
k  0, determine whether t here exists a set cover with at most
k subsets.
Decision Version of Vertex - Cover  mp Decision Version of Set - Cover
Let S (v) be the set of edges covered by v V . Set C  {S (v) | v V }
and X  E. Then G has a vertex cover of size at most k  C contains
a set cover of size at most k .
Broadcast in Multi-Channel
Wireless Networks
Problem
Consider a network G  (V , E ) that at each node v  V ,
there exists a set of channels (postive integers), A(v) that
v can use to send/recei ve messages.
Given a source node s, find F (v)  A(v) for every v  V
such that (1) {( u,v) | F (u )  A(u )  } contains a broadcast
tree; (2) vV | F (v) | is minimized.
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 DNA
Disadvantages
• 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 Matrix
viruses
probes
p1
p2
p3
pi
pt
c1
0
0
1
0
.
.
0
.
.
0
c2
0
1
0
0
c3
0
0
0
1
…
…
…
…
0
0
0
0
…
…
…
…
0
0
0
0
cj
… 0
… 0
… 0
… 0
…
…
…
…
0
0
0
0
…
…
…
…
0
0
0
0
…
…
…
…
0
0
0
0
…
…
…
…
cn
0
0
0
0
0
0 … 0 … 0 … 1 … 0 … 0 … 0 … 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
probes
p1
p2
p3
c1
1
0
0
1
0
0
c2
1
0
0
0
1
0
c3
1
0
0
0
0
1
0
1
0
1
0
0
0
1
0
0
1
0
0
1
0
0
0
1
cj
0
0
1
1
0
0
0
0
1
0
1
0
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 Matrix
viruses
p1
p2
p3
probes
pi
pt
c1
0
0
1
0
.
.
0
.
.
0
c2 c3
cj
0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0
1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0
0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0
0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0
0
0 … 0 … 0 … 1 … 0 … 0 … 0 … 0
0
0 … 0 … 0 … 0 … 0 … 0 … 0 … 0
All unions of up to d columns are distinct.
Decoding: O(nd)
cn
d-Disjunct Matrix
c1 c2
p1 0 0
p2 0 1
p3 1 0
probes
0 0
.
.
pi 0 0
.
.
pt 0 0
0 … 0 … 0
c3
0
0
0
1
…
…
…
…
0
0
0
0
viruses
cj
… 0 … 0
… 0 … 0
… 0 … 0
… 0 … 0
…
…
…
…
0
0
0
0
…
…
…
…
0
0
0
0
…
…
…
…
0
0
0
0
…
…
…
…
cn
0
0
0
0
0 … 0 … 0 … 1 … 0 … 0 … 0 … 0
0 … 0 … 0 … 0 … 0 …
Each column is different from the union of every d other columns
Decoding: 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
What complexity class do Nonunique Probe
Selection problems belong to?
Do they belong to NP?
Complexity?
• All three problems may not be in NP, why?
• Guess a t x n matrix M, verify if M is dseparable (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 vertexcover
• Given a graph G and h > 0, does G have a
vertex cover of size at most h?
VC  mp Complement of d - Separabili ty Test
Suppose G  (V , E ).
Choose v*  V . Design the following pools :
{v} for v  V ;
{v*}  e for e  E ,
and set d  h  1.
-
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 NPhard.
• In general, the problem is conjectured to
p
be Σ2 –complete.
What is  2p ?
What is Σ - complete?
p
2