Presentation MCIP CIAC

8/7/2019 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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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



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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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Outline

Introduction

Approximation of 2-MCIP

Approximation of k-MCIP



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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


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!

Presentation MCIP CIAC

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