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Advanced Studies in Biology, Vol. 4, 2012, no. 10, 491 - 495 The Hamiltonian Strictly Alternating Cycle Problem Anna Gorbenko Department of Intelligent Systems and Robotics Ural Federal University 620083 Ekaterinburg, Russia [email protected] Vladimir Popov Department of Intelligent Systems and Robotics Ural Federal University 620083 Ekaterinburg, Russia [email protected] Abstract In this paper we consider an approach to solve the Hamiltonian strictly alternating cycle problem. This approach is based on constructing logical models for the problem. Keywords: Hamiltonian strictly alternating cycle, edge-colored graphs, logical model, satisﬁability problem, NP-complete Diﬀerent regularities extensively investigated in bioinformatics (see e.g. [1], [2]). For instance, the study of genome rearrangements has drawn a lot of attention (see e.g. [3]). In particular, according to Bennett’s model of cytogenetics the order of the chromosomes is determined by an alternating Hamiltonian path of some graph G (see e.g. [4]). Therefore, it is natural to investigate different problems related to alternating Hamiltonian paths and cycles. In this paper we consider the Hamiltonian strictly alternating cycle problem. We assume that each edge of a graph has a color. If the number of colors is restricted by an integer c, we speak about c-edge-colored graphs. Let Knc be a complete c-edge-colored graph with the set of vertices V = {v1 , v2 , . . . , vn }. Let {0, 1, . . . , c − 1} be the set of colors of Knc . Let Et is the set of edges of tth color where 0 ≤ t ≤ c − 1. A strictly alternating cycle in Knc is a cycle of length pc, where p is an integer, such that the sequence of colors (01 . . . c − 1) 492 A. Gorbenko and V. Popov is repeated p times. A strictly alternating cycle in Knc is called Hamiltonian if it contains all the vertices of Knc . The Hamiltonian strictly alternating cycle problem (HSAC): Instance: Knc , Et , 0 ≤ t ≤ c − 1. Question: Is there a Hamiltonian strictly alternating cycle in Knc ? Note that HSAC, c ≥ 3, is NP-complete [5]. Recently, encoding hard problems as Boolean satisﬁability and solving them with very eﬃcient satisﬁability algorithms has caused considerable interest (see e.g. [6] – [10]). In this paper, we consider an approach to solve HSAC. This approach is based on constructing logical models for the problem. Let ϕ[1] = ∧1≤i≤n ∨1≤j≤n x[i, j], ϕ[2] = ∧1≤i≤n ∧1≤j[1]<j[2]≤n (¬x[i, j[1]] ∨ ¬x[i, j[2]]), ϕ[3] = ∧1≤j≤n ∧1≤i[1]<i[2]≤n (¬x[i[1], j] ∨ ¬x[i[2], j]), δ[1] = ∧1≤i<n ∧1≤j[1]≤n (¬x[i, j[1]] ∨ ¬x[i + 1, j[2]]), 1≤j[2]≤n (vj[1] ,vj[2] )∈E / t t=i mod c δ[2] = ∧1≤j[1]≤n (¬x[n, j[1]] ∨ ¬x[1, j[2]]), 1≤j[2]≤n (vj[1] ,vj[2] )∈E / t t=i mod c ξ = (∧3i=1 ϕ[i]) ∧ (∧2p=1 δ[p]). It is clear that ξ is a CNF. It is easy to check that ξ gives us an explicit reduction from HSAC to SAT. By direct veriﬁcation we can check that α ⇔ (α ∨ β1 ∨ β2 ) ∧ (α ∨ ¬β1 ∨ β2 ) ∧ (α ∨ β1 ∨ ¬β2 ) ∧ (α ∨ ¬β1 ∨ ¬β2 ), l ∨j=1 αj ⇔ (α1 ∨ α2 ∨ β1 ) ∧ α1 ∨ α2 ∨4j=1 αj l−4 (∧i=1 (¬βi ∨ αi+2 ∨ βi+1 )) ∧ (¬βl−3 ∨ αl−1 ∨ αl ), ⇔ (α1 ∨ α2 ∨ β) ∧ (α1 ∨ α2 ∨ ¬β), ⇔ (α1 ∨ α2 ∨ β1 ) ∧ (¬β1 ∨ α3 ∨ α4 ) (1) (2) (3) (4) 493 The Hamiltonian strictly alternating cycle problem time OA1 (N) OA1 (G) OA1 (A) OA2 (N) OA2 (G) OA2 (A) average max best 2.6 h 11.2 h 9.3 min 2.4 h 9.8 h 8.1 min 2.7 h 10.9 h 27.6 min 2.63 h 10.1 h 14.2 min 2.1 h 8.6 h 11.4 min 2.28 h 9.7 h 37 sec Table 1: Experimental results for reduction to 3SAT where OA1 (N) is OA from [11] for natural instances, OA1 (G) is OA from [11] for automatic generation of recognition modules, OA1 (A) is OA from [11] for algebraic instances, OA2 (N) is OA from [12] for natural instances, OA2 (G) is OA from [12] for automatic generation of recognition modules, OA2 (A) is OA from [12] for algebraic instances. where l > 4. Using relations (1) – (4) we can easily obtain an explicit transformation ξ into ζ such that ξ ⇔ ζ and ζ is a 3-CNF. It is clear that ζ gives us an explicit reduction from HSAC to 3SAT. We consider 3SAT solvers from [11], [12]. We have created a generator of natural instances for HSAC. In many self-learning systems, to solve the problem of automatic generation of recognition modules we need a training set which represents a set of conﬁgurations of the system and a set of conﬁrmations and contradictions for the object (see e.g. [13]). It is easy to see that such training set can be represented by some edge-colored graph G. In this case, to setup a self-learning process we need to solve HSAC for G. We have designed a generator of instances for automatic generation of recognition modules. Algorithmic problems of algebra have been studied intensively in the last decades (see e.g. [14] – [23]). Many algorithmic problems of algebra can be solved by an interpretation of Minsky machines and Turing machines (see e.g. [24] – [32]). Such interpretations can be simulated by edge-colored graphs. Using this idea we have created a generator of algebraic instances for HSAC. We have used heterogeneous cluster for our computational experiments. Each test was runned on a cluster of at least 100 nodes. Selected experimental results are given in Table 1. References [1] V. Yu. Popov, Computational complexity of problems related to DNA sequencing by hybridization, Doklady Mathematics, 72 (2005), 642-644. [2] V. Popov, The approximate period problem for DNA alphabet, Theoretical Computer Science, 304 (2003), 443-447. [3] V. Popov, Multiple genome rearrangement by swaps and by element duplications, Theoretical Computer Science, 385 (2007), 115-126. 494 A. Gorbenko and V. Popov [4] D. Dorninger, Hamiltonian circuits determining the order of chromosomes, Discrete Applied Mathematics, 50 (1994), 159-168. [5] A. Benkouar, Y. Manoussakis, V. Paschos and R.Saad, On the complexity of Hamiltonian and Eulerian problems in edge-colored complete graphs, Lecture Notes in Computer Sciences, 557 (1991), 190-198. [6] A. Gorbenko and V. Popov, The Longest Common Subsequence Problem, Advanced Studies in Biology, 4 (2012), 373-380. [7] A. Gorbenko and V. Popov, The set of parameterized k-covers problem, Theoretical Computer Science, 423 (2012), 19-24. [8] A. Gorbenko and V. Popov, Element Duplication Centre Problem and Railroad Tracks Recognition, Advanced Studies in Biology, 4 (2012), 381384. [9] A. Gorbenko and V. Popov, Programming for Modular Reconﬁgurable Robots, Programming and Computer Software, 38 (2012), 13-23. [10] A. Gorbenko, V. Popov, and A. Sheka, Localization on Discrete Grid Graphs, Proceedings of the CICA 2011, (2012), 971-978. [11] A. Gorbenko and V. Popov, Task-resource Scheduling Problem, International Journal of Automation and Computing, 9 (2012), 429-441. [12] A. Gorbenko and V. Popov, On the Optimal Reconﬁguration Planning for Modular Self-Reconﬁgurable DNA Nanomechanical Robots, Advanced Studies in Biology, 4 (2012), 95-101. [13] A. Gorbenko and V. Popov, Self-Learning Algorithm for Visual Recognition and Object Categorization for Autonomous Mobile Robots, Proceedings of the CICA 2011, (2012), 1289-1295. [14] V. Yu. Popov, Critical theories of supervarieties of the variety of commutative associative rings, Siberian Mathematical Journal, 36 (1995), 11841193. [15] V. Yu. Popov, Critical theories of varieties of nilpotent rings, Siberian Mathematical Journal, 38 (1997), 155-164. [16] V. Yu. Popov, A ring variety without an independent basis, Mathematical Notes, 69 (2001), 657-673. [17] Yu. M. Vazhenin and V. Yu. Popov, On positive and critical theories of some classes of rings, Siberian Mathematical Journal, 41 (2000), 224-228. 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Popov, On complexity of the word problem for ﬁnitely presented commutative semigroups, Siberian Mathematical Journal, 43 (2002), 10861093. [25] V. Yu. Popov, On the decidability of equational theories of varieties of rings, Mathematical Notes, 63 (1998), 770-776. [26] V. Yu. Popov, On equational theories of varieties of anticommutative rings, Mathematical Notes, 65 (1999), 188-201. [27] V. Yu. Popov, On the derivable identity problem in varieties of rings, Siberian Mathematical Journal, 40 (1999), 948-953. [28] V. Yu. Popov, On equational theories of classes of ﬁnite rings, Siberian Mathematical Journal, 40 (1999), 556-564. [29] V. Yu. Popov, Undecidability of the word problem in relatively free rings, Mathematical Notes, 67 (2000), 495-504. [30] V. Yu. Popov, Equational theories for classes of ﬁnite semigroups, Algebra and Logic, 40 (2001), 55-66. [31] V. Yu. Popov, Decidability of equational theories of coverings of semigroup varieties, Siberian Mathematical Journal, 42 (2001), 1132-1141. [32] V. Popov, The word problem for relatively free semigroups, Semigroup Forum, 72 (2006), 1-14. Received: July, 2012