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Optimization Problem Relating to Sting Case Solution & Answer

Optimization Problem Relating to Sting Case Study Solution

Background and Objective:

The entire article is dealing with the work of the Minimum common string partition (MCSP) problem. The problem has been described through 2 related input strings S1 and S2; they both have same length which is described through variable n over the finite variable ∑. Given that, DNA sequences S1 = AGACTG, S2 = ACTAGG (Both have same input strings with just change in order.) Therefore, the valid solution for MCSP obtained through partitioning the S1 into set P1 and, S2 into set P2 of non-overlapping sub strings like P1 = P2 = {A, A, C, T, G, G} (Objective function), but the optimal function has half value then objective is 3, P1 = P2 = {ACT, AG, G}. However, we want to find the valid solution that will be an absolute set of P1 = P2. Basically, the MCSP problem has been introduced in 2005 by the Chen et al, due to the relation of the genome rearrangement for obtaining the answer of the question that, given DNA string is possible to generate through rearrangement of another DNA string. Another study has been obtained, called upgradation of MCSP is K-MCSP in 2005 by Goldstein, Kolman & Zheng in which each letter occurs at most K times in each input string. It is the modified version of the simple greedy algorithm with approximation ratio of O (K^2). The 2-MCSP problem was showed to be APX-hard in 2005 by Goldstein et al; where, when the input strings are over the alphabet C (C ≥ 2), problem shown as MCSPc, after that, it is proven by Jiang, Zhu, Zhu & Zhu in 2012. Considerably many researchers have been dealing with the approximability of the MCSP like Cormode in 2002 by O(lognlog*n), approximate for the edit distance with moves problem. Sapira and Storer (2002) had extended their work on it, other approaches were obtained by Kolman and Walen in 2007. In 2004, Kolman and Sgall showed the simple greedy approach concerning the 2-MCSP problem ratio is 3 and 4-MCSP problem ratio is Ω(log(n)). In article there are 2 contributions; first one relates to the IBM ILOG CAPLEX and second one consists of 2-phase heuristic which is based on integer liner program model. First heuristic outperforms competitor algorithms from literature, second relates to large problem instances.

Methods (notation, formulation & solution approach):

It is the first article who represent the Integer Linear Program Model to solve MCSP. Formula states that:

m

min ∑ xi

i=1

subject to               m

∑ M1ij . xi  = 1   for j = 1,……n

i=1

m

∑ M2ij . xi  = 1   for j = 1,……n

i=1

i   {0,1}    for i = 1,……….m

Here is common block bi of S1 and S2, which is denoted by the triple (ti, k1i, k2i), where ti is a string at the position of 1 ≤ k1i ≤ n in string S1 and 1 ≤ k2i ≤ n in string S2. Moreover B = {b1,………bm}is the set of all possible common blocks of S1 and S2. Moreover, any valid solution of S of MCSP problem is subset of B (S ∁ B). To solve two binary m X n matrices M1 M2 defined 1 ≤ i ≤ m corresponds to bi  B and 1 ≤ j ≤ n corresponds position j in input string S1 and S2. Complete set of common blocks B consists 30 artificial instances and 15 real life instances. The benchmark consists of 4 subsets, group 1 contains the 10 artificial instances with 200 maximum length, group 2 and group 3 contains 10 artificial instances each with 201 to 400 and 401 to 600 length respectively. Finally, the 4 set contains the 15 real life instances with 200 to 600 multiple lengths………….

 

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