# The Hamiltonian Strictly Alternating Cycle Problem

```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
gorbenko.ann@gmail.com
Department of Intelligent Systems and Robotics
Ural Federal University
620083 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 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]&lt;j[2]≤n (&not;x[i, j[1]] ∨ &not;x[i, j[2]]),
ϕ[3] = ∧1≤j≤n ∧1≤i[1]&lt;i[2]≤n (&not;x[i[1], j] ∨ &not;x[i[2], j]),
δ[1] = ∧1≤i&lt;n ∧1≤j[1]≤n
(&not;x[i, j[1]] ∨ &not;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
(&not;x[n, j[1]] ∨ &not;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 ∨ &not;β1 ∨ β2 ) ∧
(α ∨ β1 ∨ &not;β2 ) ∧
(α ∨ &not;β1 ∨ &not;β2 ),
l
∨j=1 αj ⇔ (α1 ∨ α2 ∨ β1 ) ∧
α1 ∨ α2
∨4j=1 αj
l−4
(∧i=1
(&not;βi ∨ αi+2 ∨ βi+1 )) ∧
(&not;βl−3 ∨ αl−1 ∨ αl ),
⇔ (α1 ∨ α2 ∨ β) ∧
(α1 ∨ α2 ∨ &not;β),
⇔ (α1 ∨ α2 ∨ β1 ) ∧
(&not;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 &gt; 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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