The Eleventh International Symposium on Artificial Life and Robotics 2006 (AROB 11th'06), B-con Plaza, Beppu, Oita, Japan, January 23-25, 2006
Molecular Computing Approach for Constraint Assignment Problem Zuwairie Ibrahim1&2, Yusei Tsuboi2, and Osamu Ono2 1
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Department of Mechatronics and Robotics Faculty of Electrical Engineering Universiti Teknologi Malaysia 81310 UTM Skudai, Johor Darul Takzim Malaysia
[email protected]
Institute of Applied DNA Computing Meiji University 1-1-1 Higashi-mita, Tama-ku Kawasaki-shi, Kanagawa-ken 214-8571 Japan (zuwairie, tsuboi, ono)@isc.meiji.ac.jp
Abstract
computing called direct-proportional length-based DNA computing is chosen and extended for the development of a DNA algorithm for the computation of a constraint assignment problem.
The concept of direct-proportional length-based DNA computing is considered in order to solve a constraint assignment problem on an unconventional molecular computer. Constraint assignment problem is an extended version of unconstraint assignment problem, where a constraint that is the cost of an assignment is taken into account during the optimization. This paper shows that the complexity of constraint assignment problem could be reduced to a path-finding problem and an algorithm based on direct-proportional length-based DNA computing is design for searching the optimal assignment.
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Modified Graph
A simple input of a constraint assignment problem is depicted in Figure 1. This input consists of a set employees, E = {a, b, c} and a set jobs, J = {d, e, f}. The cost of an assignment is also shown. Based on the input, the optimal assignment is a-d, b-e, and c-f, where the assignment cost is 5.
Keywords: Constraint Assignment Problem, Optimization, Direct-Proportional Length-Based DNA Computing 1
Introduction
In the previous paper [1-2], we proposed a DNAbased computing algorithm for solving unconstraint assignment problem. The objective of the unconstraint assignment problem is to establish a full one-to-one correspondence between two set E (employees) and J (jobs), both of which have N elements. An assignment is a one-to-one mapping, α: E → J [3]. The assignment problem is so fundamental in operations research as well as in engineering field. It is very useful because assigning n tasks to n people is a basic primitive in many applications [4]. Based on the constraint assignment problem, a constraint is considered that is the cost of assigning an element in E to an element in J, and thus, the objective of the assignment now is to find one-to-one correspondence between two set E and J, both of which have N elements with the lowest cost. Basically, from the DNA computing point of view, there are several characteristic of DNA that could be manipulated in order to solve a weighted graph problem. Those are length [5], concentration [6], and melting temperature [7]. However, a length-based DNA
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Figure 1: Input of a constraint assignment problem. In order to reduce the constraint assignment problem to a path-finding problem, the input graph in Figure 1 is visualized slightly different into a modified graph as shown in Figure 2. According to the input, it is clear that the nodes a, b, and c are connected to nodes d, e, and f. The modified graph is constructed in such as way that the nodes d, e, and f that belong to node a are connected by connective edges to node b. Similarly, nodes d, e, and f that belong to node b are connected by connective edges to node c. Lastly, nodes d, e, and f that belong to node c are connected by connective edges to node end. We introduce a node end as a stopper. The costs of connective edges are set to 0. According to the modified graph, the constraint assignment problem can be solved in vitro on a molecular computer by computing the shortest path problem employing direct-proportional length-based DNA computing as shown in Figure 3.
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The Eleventh International Symposium on Artificial Life and Robotics 2006 (AROB 11th'06), B-con Plaza, Beppu, Oita, Japan, January 23-25, 2006
amplify exponentially generated double stranded DNAs (dsDNAs) that begin from node a and end at node end.
Figure 2: A modified graph. Figure 4: All possible solutions of the constraint assignment problem. In constraint assignment problem, an employee should be assigned only to one machine, either d, e, or f. In order to select only possible solutions that satisfy this rule, magnetic bead separation can be applied three times, particularly, for extraction of subsequences d, e, and f. Magnetic bead separation is a biotechnology tool to extract dsDNAs that contain certain subsequences from a solution. Possible solutions that survived after the magnetic bead separation are shown in Figure 5.
Figure 3: The shortest path as solution to a constraint assignment problem. 3
DNA sequence design
The sequence design is exactly the same as directproportional length-based DNA computing as follows: a. If there is a connection between V1 to Vj, where j ≠ n, synthesize the oligo for edge as V1 (20) + W1j (ω- 30) + Vj (20) b. If there is a connection between Vi to Vj, where i ≠ 1, j ≠ n, synthesize the oligo for edge as Vi (20) + Wij (ω- 20) + Vj (20) c. If there is a connection between Vi to Vn, where i ≠ 1, synthesize the oligo for edge as Vi (20) + Win (ω- 30) + Vn (20) where V, W, and ‘+’ denote the DNA sequences for nodes, DNA sequences for weight, and ‘join’ respectively. The synthesized oligos consist of two node segments and an edge segment. ‘ω’ denotes the weight value for corresponding DNA sequences for weight Wij, where Wij denotes the DNA sequences representing a cost between node Vi and Vj. The value in parenthesis indicates the number of DNA bases or nucleotides for each segment. Table 1 lists all the oligos designed using DNASequenceGenerator [8] for the in vitro computation. 4
Figure 5: Possible solutions survived after magnetic bead separation of subsequence d, e, and f. All possible solutions survived after the previous reactions contain one-to-one assignment of employees and jobs with various costs. The solution of the constraint assignment problem is the one-to-one assignment with the lowest cost. The selection of that kind of assignment is possible by performing a method called gel electrophoresis. Gel electrophoresis, in fact, is a method to separate and list dsDNAs according to their lengths, in an agarose or polyacrylamide gel. The separation by gel electrophoresis is shown conceptually in Figure 6. After the gel electrophoresis is successfully done, the shortest band appears in the gel represents the solution of the constraint assignment problem. However, a method to extract the information represented by dsDNAs in the shortest band is required. Fortunately, it is possible to extract the information by using a method called graduated PCR. Graduated PCR can be performed by extracting the dsDNAs of interest from the gel and applying three different PCRs with three different set of
In vitro computation
In this research, POA is performed for initial pool generation. After parallel overlap assembly is done, all possible solutions of the constraint assignment problem as shown in Figure 4 could be generated. The values in parenthesis indicate the cost of each assignment. Polymerase chain reaction (PCR) is performed to
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The Eleventh International Symposium on Artificial Life and Robotics 2006 (AROB 11th'06), B-con Plaza, Beppu, Oita, Japan, January 23-25, 2006
Table 1: DNA sequences required for the in vitro computation Name a b c d e f end Name a b c d e f end Name a-d a-e a-f b-d b-e b-f c-d c-e c-f d-b e-b f-b d-c e-c f-c d-end e-end f-end
Nodes DNA sequences (5’-3’) CGTCTGTCTAGCGGACCTTA TACACCGACTACCCATGTGC GATCCCTGAGAGGTGAATGG GTTCAACTGACAGGTGTGCC CTAAGTTGTGGTGAGTGGGC GAATATCCCGTCCTCTACGC TGGTCCCAGTGATACCAGTC Complements of nodes DNA sequences (5’-3’) TAAGGTCCGCTAGACAGACG GCACATGGGTAGTCGGTGTA CCATTCACCTCTCAGGGATC GGCACACCTGTCAGTTGAAC GCCCACTCACCACAACTTAG GCGTAGAGGACGGGATATTC GACTGGTATCACTGGGACCA Edges DNA sequences (5’-3’) CGTCTGTCTAGCGGACCTTAGTTCAACTGACAGGTCTCGG CGTCTGTCTAGCGGACCTTAACGTGCTAAGTTGTGGTGAGTGGGC CGTCTGTCTAGCGGACCTTACCGTCTTTTAGAATATCCCGTCCTCTACGC TACACCGACTACCCATGTGCAGGTCGTTCAACTGACAGGTGTGCC TACACCGACTACCCATGTGCCTAAGTTGTGGTGAGTGGGC TACACCGACTACCCATGTGCGCGTGCTCTCGAATATCCCGTCCTCTACGC GATCCCTGAGAGGTGAATGGACGTGTTTTGGTTCAACTGACAGGTGTGCC GATCCCTGAGAGGTGAATGGCTAAGTTGTGGTGAGTGGGC GATCCCTGAGAGGTGAATGGGCGTGGAATATCCCGTCCTCTACGA GTTCAACTGACAGGTGTGCCTACACCGACTACCCATGTGC CTAAGTTGTGGTGAGTGGGCTACACCGACTACCCATGTGC GAATATCCCGTCCTCTACGCTACACCGACTACCCATGTGC GTTCAACTGACAGGTGTGCCGATCCCTGAGAGGTGAATGG CTAAGTTGTGGTGAGTGGGCGATCCCTGAGAGGTGAATGG GAATATCCCGTCCTCTACGCGATCCCTGAGAGGTGAATGG GTTCAACTGACAGGTGTGCCTGGTCCCAGTGATACCAGTC CTAAGTTGTGGTGAGTGGGCTGGTCCCAGTGATACCAGTC GAATATCCCGTCCTCTACGCTGGTCCCAGTGATACCAGTC
Concluding Remarks
This research concerning with an extended version of DNA-based algorithm based on direct-proportional length-based DNA computing for solving an example of
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engineering problem, namely, constraint assignment problem. A constraint that is the cost for each assignment is taken into account in this paper. It has been shown that the complexity of the constraint assignment problem could be reduced to path-finding problem and the directproportional length-based DNA computing could be performed for the in vitro computation. For this purpose, direct-proportional length-based DNA computing is improved by additional reactions called magnetic bead separation and graduated PCR. For the future works, the proposed algorithm will be implemented in laboratory experiments.
forward and reverse primers. After that, gel electrophoresis is applied again to the three different products of PCRs and the resultant lengths are observed. The conceptual description of graduated PCR is shown in Figure 7. If the expected product of gel electrophoresis shown in Figure 7 is observed, one can conclude that the best assignment is a-d, b-e, and c-f. 6
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The Eleventh International Symposium on Artificial Life and Robotics 2006 (AROB 11th'06), B-con Plaza, Beppu, Oita, Japan, January 23-25, 2006
Acknowledgments The first author is very thankful to Universiti Teknologi Malaysia for granting a study leave in Meiji University under JPA-SLAB scholarship. References [1] Z. Ibrahim, Y. Tsuboi, and O. Ono, “Solving unconstraint assignment problem by a molecularbased computing algorithm,” IEEE International Symposium on Industrial Electronics 2004, pp. 1473-1478, 2004. [2] Z. Ibrahim, Y. Tsuboi, O. Ono, and M. Khalid, “Unconstraint assignment problem: a molecular computing approach,” International Arab Journal of Information Technology, Zarqa Private University, Vol. 3, No. 2, pp. 183-188, 2006. [3] G. Liu and R.M. Haralick, “Assignment problem in edge detection performance evaluation,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Vol. 1, pp. 26-31, 2000. [4] A. Arora, A. Frieze, and H. Kaplan, “A new rounding procedure for the assignment problem with applications to dense graph arrangement problems,” Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pp. 21-30, 1996. [4] D. M. Norton, “Polymerase chain reaction-based methods for detection of Listeria monocytogenes: toward real-time screening for food and environmental samples,” Journal of AOAC International, Vol. 85, pp. 505-515, 2002. [5] Z. Ibrahim, Y. Tsuboi, O. Ono, and M. Khalid, “Experimental implementation of directproportional length-based DNA computing for numerical optimization of the shortest path problem”, European Conference on Artificial Life 2005: Workshop on Unconventional Computing, pp. 35-46, 2005. [6] M. Yamamoto, A. Kameda, N. Matsuura, T. Shiba, Y. Kawazoe, and A. Ohuchi, “Local search by concentration-controlled DNA computing,” International Journal of Computational Intelligence and Applications, Vol. 2, No. 4, pp. 447-455, 2002. [7] J.Y. Lee, S.Y. Shin, S.J. Augh, T.H. Park, B.T. Zhang, “Temperature gradient-based DNA computing for graph problems with weighted edges,” Lecture Notes in Computer Science, Vol. 2568, pp. 73-84, 2003. [8] F. Udo, S. Sam, B. Wolfgang, and R. Hilmar, “DNA sequence generator: a program for the construction of DNA sequences,” Proceedings of the 7th International Workshop on DNA Based Computers, pp. 23-32, 2001.
Figure 6: Separation by gel electrophoresis of possible solutions after magnetic bead separation of subsequence d, e, and f.
Figure 7: Expected output of graduated PCR for extraction of molecular information of the constraint assignment problem.
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