Presentation MCIP CIAC
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On the Minimum CommonInteger Partition Problem
Author: Xin Chen, Lan Liu,
Zheng Liu, Tao Jiang
Presenter: Lan Liu
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Outline Introduction
Problem definitions
Biological applications
Approximation of 2-MCIP
Approximation of k-MCIP
Conclusion and future work
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Problem Definitions P(n): given an integern, a partition is a set of
integers, say {n1,n2,, nr}, s.t.i=1rni=n.
Example: given n=4, {2,2} is a P(4);given n=3, {3} is a P(3).
Observation: S= IP(S)
Example: given S= {3, 3, 4}, {2,2,3,3} is anIP({3,3,4}).
IP(S): given a multiset S= {x1, 0, xm}, an integerpartition is a disjoint union
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MCIP is NP-hard Subset sum P MCIP
Subset sum problemGiven a set of integer x1, x2,, xn, s.t. X=ixi, askif
there is a subset with the sum X/2.
Reduction to MCIP problem
- Let S={X/2, X/2}, T={x1, x2,, xn}, find MCIP(S,T).- If {x1, x2,, xn} is a MCIP(S,T), the answer is yes to
Subset sum problem; otherwise, the answer is no.
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Biological Applications(1) The distance between
two strings
a b c d e f g h i j k hh i j k h e f g a b c d
Genetic distance between
two genomes
a b c d e f g h i j k h
h i j k h e f g a b c d
Minimum Common
Substring Partition
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Biological Applications(2) MCIP is a special case of Minimum Common
Substring Partition(MCSP)
MCIP(S',T')
S'= {x1, x2, 0, xm}
T'= {y1, y2, 0, yn}
aa...a |- aa...a |- aa...ax
1 x2 xn
aa...a -| aa...a -| aa...ay
1 y2 y
MCSP(S,T)
S=
T=
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Outline
Introduction
Approximation of 2-MCIP Positive results
Negative results
Approximation of k-MCIP Conclusion and future work
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Some basic facts |MCIP(S1,S2,,Sk)|
max(|S1|,|S2|,,|Sk|)
|MCIP(S,T)| m+n-1.
|S|=m,|T|=n
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Algorithm Analysis
|MCIP(S,T)| m+n-1
|MCIP(S,T)| max(m,n)
Approximation ratio is 2
An example: S= {3, 3, 4},T={2,2,6}
S T CIPRound
{1,3,4} {2,6} {2}1{3,4} {1,6} {2,1}2{2,4} {6} {2,1,1}3
{4} {4} {2,1,1,2}4
; ; {2,1,1,2,4}5
{3,3,4} {2,2,6} ;0
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Definitions for MRSP(1) Related multisets: ifS=Tand S,T{;, Sand T
are a pair of related multisets.
Example:{3
3
4
5
10}{2 2 687}
{334510}
{22687}
Basic related multisets: if there are no S' Sand T' T, s.t. S'and T'are related.
Example:
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Definitions for MRSP(2) Maximum Related Multiset Partition problem(MRSP)
Given S and T, partition them into related submultisets
with the maximum cardinality.
(2){334 10}
{22687}
(1){334 10}
{22687}
(3){334 10}
{22687}
Observation: IfS, Tare
a pair of basic related
multisets, |MRSP|=1.
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MRSP $ 2-MCIP CIP! RSP
S: {4 3 35 10}
T: {2 2 687}
CIP: {2 2 3 353 7}
For each component,
#edges #vertices 1
Each component is
related.
S: {4 3 35 10}
T: {22 687}
|CIP| m+n-|RSP|
|MCIP| m+n-|RSP|
m+n-|MRSP|
S: {4 33510}
T: {22 687}
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MRSP $ 2-MCIP |MRSP| = m+n|MCIP|
IfS, Tare a pair of basic related multisets,
|MCIP|= m+n-1, because |MRSP|=1. When m+n 5, |MCIP| =m+n-1 4/5(m+n).
A new way to solve MCIP
Step1. find MRSP;
Step2. for each basic related submultiset, run
Greedy_CIP(S', T').
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Approximate 2-MCIP Algorithm intuition:
Step 1. find related submulitsets
Step 2. set packing
Step 3. Greedy-CIP
mimic MRSP
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Set Packing Problem(1) Set Packing
Given a set of subsets S, find the largest number of
mutually disjoint subsets from S?
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Set Packing Problem(2) Bad news
- It is NP-hard to find related submultisets oflarge size.
- Set packing itself is NP-hard.
Good news
We can find the small related submultisets andapproximate set packing efficiently.
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Approximate 2-MCIP Main idea: use different strategies for the
submultisets with different sizes.
The approximation ratio is 5/4.
If there are no basic related submultisets with sizesmaller than 5, 4/5 (m+n) |MCIP| m+n-1.
Str t i sR v _c _i t r
r xi t _s t_ cki
Submultis t siz
,r mor r y_ I
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Outline
Introduction
Approximation of 2-MCIP Positive results
Negative results
Approximation ofk-
MCIP
Conclusion and future work
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General frameworkx f(x )
g(x ,y) y
IP
IP
SOLP
(x ) SOLP
(f(x ))
LinearReduction L
OPTP2(f(x)) E OPTP1(x)
|OPTP1(x)- g(x,y)| F|OPTP2(f(x))-y|
IfP1 cannot be approximated
within some constant ratio c,
P2 cannot be approximated by
some constant ratio c'.
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Maximum 3DM-3 Problem Definition
Given a set D XYZ, where X, Yand Z are disjoint sets,
and each element occurs in at most three triples, find amatching with the maximum cardinality.
Known fact
Maximum 3DM-3 cannotbe approximated within some
constant ratio. [Kann91]
X:
Y:
Z:
X:
Y:
Z:
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L-reduction(1) f: S={4i|i2X[Y[Z }
T={4i1+4i2+4i3|(i1,i2,i3)2D}
D
X Y ZS :
T:
OPTMCIP
70*OPT3DM
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g: - CIP ! RSP|OPTRSPSOLRSP| |OPTMCIPSOLMCIP|
- RSP ! 3DM
X
S1:
Y Z
T1:
10 1... ...
D
*
1 1
* *
di
i1 i3i2
OPT3DMOPTRSP
Each related submultiset
includes at least one triple
L-reduction(2)
|OPT3DMSOL3DM| |OPTRSPSOLRSP|
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Outline
Introduction
Approximation of 2-MCIP
Approximation of k-MCIP
Conclusion and future work
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Approximate k-MCIP Run Greedy_CIP(S,T) sequentially on S1,S2, ,
Sk.
|MCIP(S1,S2,,Sk)| |S1|+|S2|++|Sk|
|MCIP(S1,S2,,Sk)| max(|S1|,|S2|,,|Sk|)
Approximation ratio is k
We can get a {3k(k-1)}/(3k-2)- approximation
by removing the common elements.
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Outline
Introduction
Approximation of 2-MCIP Approximation of k-MCIP
Conclusion and future work
Upper bound5/4
{3k(k-1)}/(3k-2)2-MCIP
k-MCIP (k>2)
Lower boundAPX-hardAPX-hardAPX-hard
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Thanks for your time and
attention!