On the Minimum Common Integer Partition Problem Author: Xin Chen, Lan Liu, Zheng Liu, Tao Jiang Presenter: Lan Liu

Outline  Introduction Problem definitions Biological applications Approximation of 2-MCIP Approximation of k-MCIP Conclusion and future work

Problem Definitions P(n): given an integer n, a partition is a set of integers, say {n 1,n 2,…, n r }, s.t.  i=1 r n i =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 an IP({3,3,4}).  IP(S): given a multiset S= {x 1, , x m }, an integer partition is a disjoint union

Examples CIP(S 1, S 2, …, S k ): given multisets S 1, S 2, …, S k, a common integer partition of all multisets. Example: given S= {3, 3, 4}, T={2,2,6}, {2,2,3,3} is a CIP(S,T); {1,1,2,2,4} is also a CIP(S,T). # P(100)= Observation: (1) 9 CIP(S 1,…, S k ) $  S 1 =…=  S k (2) |CIP(S 1,…, S k )| ¸ |S i |, i 2 {1,..,k}  MCIP(S 1, S 2, , S k ): a common integer partition with the minimum cardinality. Example: {2,2,3,3} is a MCIP(S,T).

MCIP is NP-hard Subset sum · P MCIP Subset sum problem Given a set of integer x 1, x 2, …, x n, s.t. X=  i x i, ask if there is a subset with the sum X/2. Reduction to MCIP problem - Let S={X/2, X/2}, T={x 1, x 2, …, x n }, find MCIP(S,T). - If {x 1, x 2, …, x n } is a MCIP(S,T), the answer is “ yes ” to Subset sum problem; otherwise, the answer is “ no ”.

Biological Applications(1) The distance between two strings a b c d e f g h i j k h h 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

Biological Applications(2) MCIP is a special case of Minimum Common Substring Partition(MCSP) MCIP(S',T') S'= {x 1, x 2, , x m } T'= {y 1, y 2, , y n } MCSP(S,T) S= T=

Outline  Introduction  Approximation of 2-MCIP Positive results Negative results Approximation of k-MCIP Conclusion and future work

Some basic facts |MCIP(S 1,S 2,…,S k )| ¸ max(|S 1 |,|S 2 |,…,|S k |) |MCIP(S,T)| · m+n-1. |S|=m,|T|=n

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}

Definitions for MRSP(1) Related multisets: if  S=  T and S,T  ;, S and T are a pair of related multisets. Example:  Basic related multisets: if there are no S' ½ S and T' ½ T, s.t. S' and T' are related. Example:

Definitions for MRSP(2) Maximum Related Multiset Partition problem(MRSP) Given S and T, partition them into related submultisets with the maximum cardinality. Observation: If S, T are a pair of basic related multisets, |MRSP|=1.

MRSP $ 2-MCIP CIP ! RSP For each component, #edges ¸ #vertices –1 Each component is related. |CIP| ¸ m+n-|RSP| |MCIP| ¸ m+n-|RSP| ¸ m+n-|MRSP|

MRSP $ 2-MCIP CIP Ã RSP For each related submultisets (S', T'), we run Greedy_CIP(S', T'), |CIP (S', T')| · |S'|+ |T'|-1 |CIP| · m+ n- |RSP| |MCIP| · |CIP| · m+ n-| MRSP|

MRSP $ 2-MCIP |MRSP| = m+n –|MCIP| If S, T are 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').

Approximate 2-MCIP Algorithm intuition: Step 1. find related submulitsets Step 2. set packing Step 3. Greedy-CIP mimic MRSP

Set Packing Problem(1)  Set Packing Given a set of subsets S, find the largest number of mutually disjoint subsets from S?

Set Packing Problem(2) Bad news - It is NP-hard to find related submultisets of large size. - Set packing itself is NP-hard. Good news We can find the small related submultisets and approximate set packing efficiently.

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 size smaller than 5, 4/5 (m+n) · |MCIP| · m+n-1.

Outline  Introduction  Approximation of 2-MCIP –Positive results –Negative results  Approximation of k-MCIP  Conclusion and future work

General framework  Linear Reduction · L  OPT P2 (f(x)) ·  OPT P1 (x)  | OPT P1 (x)- g(x,y)| ·  |OPT P2 (f(x))-y| If P1 cannot be approximated within some constant ratio c, P2 cannot be approximated by some constant ratio c'.

Maximum 3DM-3 Problem Definition Given a set D µ X £ Y £ Z, where X, Y and Z are disjoint sets, and each element occurs in at most three triples, find a matching with the maximum cardinality. Known fact Maximum 3DM-3 cannot be approximated within some constant ratio. [Kann91]

L-reduction(1) f: S={4 i | i 2 X [ Y [ Z } T={4 i1 +4 i2 +4 i3 | (i 1,i 2,i 3 ) 2 D} OPT MCIP · 70*OPT 3DM

 g: - CIP ! RSP |OPT RSP –SOL RSP | · |OPT MCIP – SOL MCIP | - RSP ! 3DM OPT 3DM ¼ OPT RSP Each related submultiset includes at least one triple L-reduction(2) |OPT 3DM –SOL 3DM | · |OPT RSP – SOL RSP |

L-reduction(3)  There is a constant c s.t. Maximum 3DM-3 cannot be approximated within c.  There is a L-reduction s.t.  OPT MCIP · 70*OPT 3DM  |OPT 3DM –SOL 3DM | · |OPT MCIP – SOL MCIP |  There is a constant c' s.t. 2-MCIP cannot be approximated within c'. c'<5/4

Outline  Introduction  Approximation of 2-MCIP  Approximation of k-MCIP  Conclusion and future work

Approximate k-MCIP Run Greedy_CIP(S,T) sequentially on S 1,S 2, …, S k. |MCIP(S 1,S 2,…,S k )| · |S 1 |+|S 2 |+…+|S k | |MCIP(S 1,S 2,…,S k )| ¸ max(|S 1 |,|S 2 |,…,|S k |) Approximation ratio is k We can get a {3k(k-1)}/(3k-2)- approximation by removing the common elements.

Outline  Introduction  Approximation of 2-MCIP  Approximation of k-MCIP  Conclusion and future work

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