BioSystems 150 (2016) 119–131 Contents lists available at ScienceDirect BioSystems jo ur nal home p age: www.elsevier.com/locate/biosystems A general resolution of intractable problems in polynomial time through DNA Computing C.A.A. Sanches ∗ , N.Y. Soma Instituto Tecnológico de Aeronáutica – CTA/ITA/IEC, Prac ¸ a Mal. Eduardo Gomes, 50, São José dos Campos, SP 12228-900, Brazil a r t i c l e i n f o Article history: Received 8 April 2016 Received in revised form 15 July 2016 Accepted 23 September 2016 Available online 28 September 2016 Keywords: DNA Computing NP-Hard problems Co-NP class a b s t r a c t Based on a set of known biological operations, a general resolution of intractable problems in polyno- mial time through DNA Computing is presented. This scheme has been applied to solve two NP-Hard problems (Minimization of Open Stacks Problem and Matrix Bandwidth Minimization Problem) and three co-NP-Complete problems (associated with Hamiltonian Path, Traveling Salesman and Hamiltonian Cir- cuit), which have not been solved with this model. Conclusions and open questions concerning the computational capacity of this model are presented, and research topics are suggested. © 2016 Elsevier Ireland Ltd. All rights reserved. 1. Introduction Just over two decades ago, Adleman (1994) used DNA sequences to solve an instance of the directed Hamiltonian Path Problem (dHPP) and ushered in a new area of research that has become known as DNA Computing. Among its main features are the large stor- age capacity and inherent parallelism of the biological operations involved. Since then, various algorithms with a polynomial num- ber of biological operations for presumably intractable problems 1 have appeared in the literature: 3-Satisfiability (3-SAT) Lipton (1995), Chang et al. (2008); 3-SAT, 3-Coloring and Independent-Set Fu (1997); Maximal Clique Quyang et al. (1997); Subgraph Isomorphism Hsieh and Chen (2008), Hsieh et al. (2008), 3-Coloring, Maximal Clique and Independent-Set Amos (1997); Dominating-Set Guo et al. (2004); 3-Dimensional Matching and Set-Packing Chang and Guo (2002); Set-Covering Chang and Guo (2003); Set-Partition Chang (2007); Subset-Sum Chang et al. (2004); Knapsack Darehmiraki and Nehi (2007, 2007), Henkela et al. (2007), Pérez-Jiménez and Sancho-Caparrini (2002); Bin-Packing Sanches and Soma (2009); Factoring Integers Chang et al. (2005); Quadratic Diophantine Equa- tion Sanches and Soma (2014); Social Networks Chen and Yang (2010); Minimum Edge Cover Wang et al. (2013); Assignment Wang and Tan (2014); among others. In general, these algorithms follow the strategy of brute force in which all combinations of data that can be matched to an answer ∗ Corresponding author. E-mail addresses: alonso@ita.br (C.A.A. Sanches), soma@ita.br (N.Y. Soma). 1 Throughout this text, we assume that P / = NP. to the problem are generated based on a given instance. Each com- bination is associated with a unique DNA sequence, and the result is calculated and attached to it. The sequence with the best evalu- ation is selected, and an optimal solution in polynomial time can be found because of the parallelism in the execution of biological operations. Because these algorithms are based on the same common prin- ciple, the inherent parallelism of this model may be exploited through a general scheme for solving intractable problems via DNA Computing, more specifically, problems that have search space 2 n , n ! or n n (or polynomial combinations of these values), where n is the size of the input. Tools for treating the DNA sequences as binary values have been developed based on the work by Roweis et al. (1998), who introduced the concept of stickers. Thus, the known data storage techniques and mathematical operations performed on binary machines can be simulated in DNA Computing. A set of biological operations and the basic principles of our general scheme are presented in the next section. A number of arithmetic calculations that can be used in various DNA Comput- ing algorithms are shown in Section 3. This scheme will be applied in Sections 4 and 5 to solve two NP-Hard problems (Minimization of Open Stacks and Matrix Bandwidth Minimization) and three co-NP- Complete problems (associated with Hamiltonian Path, Traveling Salesman and Hamiltonian Circuit) in polynomial time, which have never been approached in DNA Computing. We also present exam- ples of these resolutions and, to demonstrate the correctness of main algorithms, we included invariants and a higher level nota- tion. The final considerations, in which open issues are enumerated and research topics are suggested, will be addressed in the last section. http://dx.doi.org/10.1016/j.biosystems.2016.09.008 0303-2647/© 2016 Elsevier Ireland Ltd. All rights reserved.