The Hamiltonian Strictly Alternating Cycle Problem

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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)
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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.
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Received: July, 2012
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