The Hamiltonian Strictly Alternating Cycle Problem

Transcription

1 Advanced Studies in Biology, Vol. 4, 2012, no. 10, The Hamiltonian Strictly Alternating Cycle Problem Anna Gorbenko Department of Intelligent Systems and Robotics Ural Federal University Ekaterinburg, Russia Vladimir Popov Department of Intelligent Systems and Robotics Ural Federal University Ekaterinburg, Russia 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 Kn c be a complete c-edge-colored graph with the set of vertices V = {v 1,v 2,...,v n }. Let {0, 1,...,c 1} be the set of colors of Kn c. Let E t is the set of edges of tth color where 0 t c 1. A strictly alternating cycle in Kn c is a cycle of length pc, where p is an integer, such that the sequence of colors (01...c 1)

2 492 A. Gorbenko and V. Popov is repeated p times. A strictly alternating cycle in Kn c is called Hamiltonian if it contains all the vertices of Kn c. The Hamiltonian strictly alternating cycle problem (HSAC): Instance: Kn c, E t, 0 t c 1. Question: Is there a Hamiltonian strictly alternating cycle in Kn? c 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 1 j[2] n (v j[1],v j[2] )/ E t t=i mod c δ[2] = 1 j[1] n 1 j[2] n (v j[1],v j[2] )/ E t t=i mod c ( x[i, j[1]] x[i +1,j[2]]), ( x[n, j[1]] x[1,j[2]]), ξ =( 3 i=1 ϕ[i]) ( 2 p=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 β 1 β 2 ) (α β 1 β 2 ) (α β 1 β 2 ) (α β 1 β 2 ), (1) l j=1 α j (α 1 α 2 β 1 ) ( i=1 l 4 i α i+2 β i+1 )) ( β l 3 α l 1 α l ), (2) α 1 α 2 (α 1 α 2 β) (α 1 α 2 β), (3) 4 j=1 α j (α 1 α 2 β 1 ) ( β 1 α 3 α 4 ) (4)

3 The Hamiltonian strictly alternating cycle problem 493 time OA1 (N) OA1 (G) OA1 (A) OA2 (N) OA2 (G) OA2 (A) average 2.6 h 2.4 h 2.1 h 2.7 h 2.63 h 2.28 h max 11.2 h 9.8 h 8.6 h 10.9 h 10.1 h 9.7 h best 9.3 min 8.1 min 11.4 min 27.6 min 14.2 min 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), [2] V. Popov, The approximate period problem for DNA alphabet, Theoretical Computer Science, 304 (2003), [3] V. Popov, Multiple genome rearrangement by swaps and by element duplications, Theoretical Computer Science, 385 (2007),

4 494 A. Gorbenko and V. Popov [4] D. Dorninger, Hamiltonian circuits determining the order of chromosomes, Discrete Applied Mathematics, 50 (1994), [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), [6] A. Gorbenko and V. Popov, The Longest Common Subsequence Problem, Advanced Studies in Biology, 4 (2012), [7] A. Gorbenko and V. Popov, The set of parameterized k-covers problem, Theoretical Computer Science, 423 (2012), [8] A. Gorbenko and V. Popov, Element Duplication Centre Problem and Railroad Tracks Recognition, Advanced Studies in Biology, 4 (2012), [9] A. Gorbenko and V. Popov, Programming for Modular Reconfigurable Robots, Programming and Computer Software, 38 (2012), [10] A. Gorbenko, V. Popov, and A. Sheka, Localization on Discrete Grid Graphs, Proceedings of the CICA 2011, (2012), [11] A. Gorbenko and V. Popov, Task-resource Scheduling Problem, International Journal of Automation and Computing, 9 (2012), [12] A. Gorbenko and V. Popov, On the Optimal Reconfiguration Planning for Modular Self-Reconfigurable DNA Nanomechanical Robots, Advanced Studies in Biology, 4 (2012), [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), [14] V. Yu. Popov, Critical theories of supervarieties of the variety of commutative associative rings, Siberian Mathematical Journal, 36 (1995), [15] V. Yu. Popov, Critical theories of varieties of nilpotent rings, Siberian Mathematical Journal, 38 (1997), [16] V. Yu. Popov, A ring variety without an independent basis, Mathematical Notes, 69 (2001), [17] Yu. M. Vazhenin and V. Yu. Popov, On positive and critical theories of some classes of rings, Siberian Mathematical Journal, 41 (2000),

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