Connective Probe Machine Based Algorithm for Solving
3- Vertex-Coloring Problem
Jianzhong Cui1,2, Houjia Fang1, Jing Yang2, Zhixiang Yin2
1
Department of Computer
HuaiNan Union University. Huainan Anhui China
2
School of Mathematics and Physics
Anhui University of Science and Technology. Huainan Anhui China
[email protected]
Abstract：
The graph coloring problem has been the subject of extensive research for many years. The
problem of deciding whether a given graph is 3-colorable is NP-complete. Motivated by recently
reported computational model, named probe machine, we demonstrated that 3-vertex-coloring
problem could be decided by only one step of probe operation regardless of the size of the graph.
The proposed algorithm shows the massive parallelism inherited in probe machine as well as
computational capacity that may surpass Turing machine when tackling NP-complete problem.
Keywords:
probe machine, NP-complete, 3-vertex-coloring
1. Introduction:
An important concept in computer science is Turing machine, which is a theoretical abstraction of
general purpose computer. In 1936, Turing [1] proposed the computational model from what he
conceived that a mathematician handled with calculating. To do a calculation, the mathematician
access to an unlimited number of pieces of paper, it can manipulate the pages by going from one
to another, it can manipulate the symbols on the papers by erasing and writing a new symbol, the
“eye” allows the mathematician to see what symbols are written on a page, the “memory”
allow the mathematician to remember intermediate results (like a carry in addition) as it goes
from page to page, the “rules” are the rules of arithmetic or other mathematical system that the
mathematician must follow to get a correct result[2]. While, in Turing machine, the data is stored
on a tape that is indefinitely long in both directions, so that a calculation cannot fail because the
machine runs out of calculating space. The “rules” and “memory” are placed in a box, called
finite controller. The controller accesses the tape by means of a read/write head and the controller
can move across the tape from one position to another.
On the basis of Turing machine, in 1945, Von Neumann built up the architecture of modern
computer using electronic component. Then the following year saw the birth of the first
electronic programmable computer- ENIAC. Ever since then, computer has undergone
tremendous development and computational power has been dramatically increased. To these
days, computer has become a dispensable tool in nearly every academic field, such as, formal
verification, coding theory, protein secondary structure prediction, etc. As we solve larger and
more complex tasks with greater computational power and more optimized algorithms, the
problems that we cannot tackle begin to stand out, the P versus NP problem. It is the most
fundamental question that remains unsolved in computer science and is named as one of the
seven Millennium Prize Problems by the Clay Math Institute in 2000.
In 1965, Edmonds [3] gave a formal dentition of efficient computation (runs in time a fixed
polynomial of the input size). The collection of problems with efficient solutions became known
as P for Polynomial Time. Meanwhile, there is also a sizable class of very applicable and
significant problems that does not seem to have efficient algorithm, whereas given a potential
solution it is quiet easy to verified efficiently. The collection of problems with efficiently
verifiable solutions was categorized into NP for Nondeterministic Polynomial Time. And this led
to complexity's most important concept, NP-completeness. A problem p in NP is NP-complete if
any other problem in NP can be reduced into p in polynomial time. In 1971, Cook [4] proved that
the satisfiability problem is NP-complete. In the following years, Karp’s work [5] showed that
eight central combinatorial problems are all NP-complete. In 1973, Levin [6] proved that a
variant of the tiling problem is also NP-complete. NP-completeness has considered to be one of
the most insightful and fundamental theories in the mathematics of the last half century, since an
efficient solution to any NP-complete problem would imply an efficient solution to every
NP-complete problem and, thus P = NP.
The past decades saw much effort had being devoted to seeking efficient solutions to numerous
NP-complete problems. And a number of novel computing paradigms had been proposed, such as
bionic computing (artificial neural network, evolutionary computing), optical computing,
biological computing [7-10], quantum computing [11-14], etc. To date, bionic computing
paradigms rely thoroughly on electronic computer to perform computation. The computation
model of optical computing is in essence Turing machine and differs from Turing machine in
implementation materials, optical components and electronic components respectively. As for
quantum computing which is on the basis of quantum Turing machine or simply, quantum
computer, there are currently disputes over whether P versus NP problem can be solved or not.
From the computational complexity theory point of view, the collection of problems that can be
efficiently solved by quantum computers is called BQP, for bounded error, quantum, and
polynomial time. BQP is classified in the complexity class #p, which is a subclass of PSPACE
[15]. BQP is suspected to be disjoint from NP-complete, but that is not yet known. Figure 1
illustrates the suspected relationship of BQP problems to PSPACE, NP, NP-complete and P
problem.
PSPACE
Problem
NP Problem
NP-complete
BQP
P problem
Figure 1. The suspected relationship of BQP to other problems
In biological computing, it seems that P versus NP problem may be tackled under certain given
size. Lipton’s work [16] showed satisfiability problem could be efficiently solved, but the number
of variables in given satisfiability problem was limited to 60-70 if brute force strategy was
adopted. Yoshida [17] proposed that, with the breadth-first search algorithm, the number of
variables could be theoretically increased to about 120 variables. At present, biological
computing is still in its infancy, due to error-prone nature of bio-chemical reaction, difficulty of
final solution detection, etc. The answer to P versus NP is unknown. It seems that we are quiet far
from achieving this goal. Nevertheless, the massive parallelism inherited in hybrid reaction
makes it possible that biological-based computing model surpass Turing machine in terms of
computational capacity when tackling NP-complete problem. In 2011, on the 100th anniversary
of Turing’s birth, an open solicitation for computing model that surpassed Turing machine was
made worldwide. In 2016, probe machine [18] is proposed as a theoretical computing model. In
the paper, Turing machine was proved to be a special case of probe machine. Motivated by the
paper, we proposed the algorithm for solving 3- Vertex-Coloring Problem.
The rest of paper is organized as follows. The following section will give a brief outline of probe
machine, followed by, the definition of 3-vertex-coloring problem and our algorithm for it. We
end with the discussion and our future work in section 4.
2. Probe machine
Probe machine (PM) is defined as the following nine-turple,
PM  ( X , Y , 1 , 2 , , ,, Q, C)
where each element responds to data library, probe library, data controller, probe controller, probe
operation, computing platform, detector, true solution storage and residue collector, respectively.
The data library X consists of n data sub-libraries which encode the given problem, denoted
by, X 1 , X 2 , , X n . In sub-library X i (i  1,2,, n) , vast copies of data, denoted by xi , are
stored. Each data xi comprises of a data body and data fibers. Data fibers, denoted by xil , are
attached to data body. Figure 2 gives schematic diagram of data library, data sub-library X i , data
and data body.
(a) data library
(b) sub-library
(c) data fiber
(d) data body
Figure 2. Schematic diagram of data library, data sub-library, data and data body
Note that, in Figure 2.(c), different data fibers are attached to different regions of data body. They
differ in distinct colors. Let S ( xi )  S i denotes the set of p i types of data fibers of xi , namely,
S ( xi )  {xi1 , xi2 ,, xipi } .Then data xi is defined as the following,
xi  (i; xi1 , xi2 ,, xipi )  (i; Si )
For i  1,2,, n ,where i is the data body of xi .
Probe library Y consists of probe sub-libraries Yit . In sub-libraries Yit , vast copies of probes,
denoted by xil xtm , are stored. Note here, xil xtm represents the complement of xil xtm . A Probe may
be seen as a gluer that can find a pair of data and connect them together by means of data fibers.
Figure 3 illustrates schematically data fiber xil and xtm are glued together, since data fibers are
attached to data body, data xi and xt are thus glued together under the action of probe xil xtm . The
process is also called basic probe operation.
Figure 3. Schematic diagram of basic probe operation
According to whether Yit  Yti or not, probe sub-library can be divided into connective and
transitive probe sub-library. Similarly, connective or transitive probe machines are defined. The
probe sub-library is called complete if all data fibers of xi and x j can be connected together.
Let S ( xi )  pi and S ( xt )  pt . For complete connective probe sub-library, there are a total of
Yit  pi  pt types of probes, where
Yit  {xi1 xt1 , xi1 xt2 ,, xi1 xtpt ; xi2 xt1 , xi2 xt2 ,, xi2 xtpt ; xipi xt1 , xipi xt2 ,, xipi xtpt } .
For each type of probes xia xtb , we construct a probe sub-library Yitab , which contains vast copies
of probes xia xtb , for a  1,2,, pi and b  1,2,, pt , as shown in Figure 4.
Figure 4. Schematic diagram of connective probe library
Date controller  1 and probe controller  2 are controllers that can take data or probes and
place them into computing platform  to perform probe operation. The number of date
controllers and probe controllers correspond to the number of data sub-libraries and probe
sub-libraries, respectively.
Probe operation  is a process of performing basic probe operations concurrently. Let X ' and
Y ' be the subset of data and probe libraries, respectively. The result of a probe operation  on
X ' and Y ' , denoted by  ,
 ( X ' ,Y ' )   .
Computing platform  is the circumstance under which probes can smoothly find data fibers
and perform probe operations. We refer to 2-data-polymer as result of basic probe operation,
3-data-polymer that contains 3 data together with 2 probes. M-data-polymer is similarly denfied,
and denote the order of M-data-polymer by M . In special case, a data is called 1-data-polymer.
Detector  determines whether the given problem has solutions, and separates true solutions
from candidate solutions. A probe operation graph of data subset X ' and probe subset Y ' ,
denoted by,
'
'
G ( X ,Y ) , is the topological structure of the polymer as result of probe operation. The
vertex set V (G( X
'
,Y ' )
'
'
) is the set of data in a polymer, and the edge set E(G( X ,Y ) ) is the set of
'
'
probes. A candidate solution is true solution if and only if its topology is isomorphic to G ( X ,Y ) .
The functions of detector are list as follows.
(1) For a M-data-polymer, if M  V (G ( X ,Y ) ) or the number of probes in M-data-polymer do
'
'
'
'
not equal to E (G ( X ,Y ) ) , then the detector separates it into residue collector.
(2) For a M-data-polymer, if M  V (G ( X ,Y ) ) and the number of probes in M-data-polymer
'
'
'
'
equal to E (G ( X ,Y ) ) , then the detector separates it into true solution storage.
True solution storage Q is used to store true solutions and output them correctly.
Residue collector C is mainly used to recycle residues, decompose them back into data, and
return data to original data sub-library. Figure 5 shows computing platform, detector, true
solution storage and residue collector.
Figure 5. Schematic diagram of computing platform, detector, true solution storage and residue
collector
3. Algorithm for solving 3-vertex-coloring problem
The graph G considered in this paper is a simple, undirected graph with the set of vertices and
edges, denoted by, V (G ) , E (G ) respectively. Let V (G)  n and E(G)  m .
The graph G is 3-vertex-colorable if there is a assignment from vertex set V (G ) to color set
{1,2,3} , f : V (G )  {1,2,3} such that uv  E (G ) , f (u )  f (v) . The K - vertex-coloring is
defined similarly.
The problem is NP-hard. In particular, it is NP-hard to compute the chromatic number  (G ) [19].
It is NP-complete to decide whether a given graph is K -coloring for K  3 . The 3-coloring
problem remains NP-complete even on planar graphs of degree 4[20].
The graph coloring problem has wide applications in scheduling and storage retrieve problem. It
has been addressed in the biological computing as well [21-23].
For solving 3-coloring problem, we begin by constructing data library .
Step 1. Construction of data library
Each vertex vi  V (G) is represented by a data xi ( i  1,2,  , n ). Each possible color
assignment to vertex vi is represented by one type of data fibers attached to a data body. If
vertex vi has degree k , then k data fibers of the same color are attached to the data body.
Denote vertex vi with degree k as xik . Note here, the second subscript specifies the degree of
the vertex. Since degree sequece of the given graph specifies degrees of corresponding vertices,
the number of data fibers attached to corresponding vertices is thus chosed. Figure 6 illustrates
the representation of vertex vi with degree K  4 and possible color assignment.
xi14
xi24
xi34
Figure 6. Schematic diagram of data xi 4
Therefore, the data library is constructed on the basis of degree sequence as follows:
X   ni1{xik1 , xik2 , xik3 } .
Step 2.Construction of probe library
Since adjacent vertices in the graph can not be assigned to the same color and every vertex has
three possible color assignments, for each edge vi v j  E (G )(i  j ) , let deg( vi )  k and
deg( v j )  l ,we construct six types of probes as follows:
{xik1 x 2jl , xik1 x 3jl , xik2 x1jl , xik2 x 3jl , xik3 x1jl , xik3 x 2jl } .
Therefore, the probe sub-libraries is denoted as follows:
Yij  {xik1 x 2jl , xik1 x 3jl , xik2 x1jl , xik2 x 3jl , xik3 x1jl , xik3 x 2jl } , Yij  6 E (G )  6m .
Step 3. Probe operation
The data controllers  1 take an appropriate amount of data from data library X, and add them
into the computing platform λ. Meanwhile, the probe controllers  2 take an appropriate amount
of probes from probe library, and add them into  , and then the probe operation  is
performed.
Step 4. True solution detection
If the given graph G is 3-colorable, there must exist n - data-polymers with m probes that is
isomorphic to G . Such polymers are separated into true solution storage Q by the detector  ,
and others are separated into the residue collector C .
Next we solve an instance of 3-coloring problem. In figure 7, an example of
seven- edge graph with a possible color assignment is given.
five-vertex and
v1
v5
v2
v3
v4
Figure 7. A five-vertex simple, undirected graph with a possible color assignment. Color 1, 2 and 3 is depicted in circle, square
and triangle, respectively.
Degree sequence of the given graph is (4,3,3,2,2) . We construct data library corresponding to
degree sequence as follows:
1
3
1
2
3
2
3
2
3
1
2
3
X  {x14
, x142 , x14
; x33
, x33
, x33
; x143 , x 43
, x 43
; x122 , x 22
, x 22
; x52
, x52
, x52
}
For each edge in the given graph, we construct probe library as follows:
1 2
1 3
3
3 1
3 2
Y12  {x14
x22 , x14
x22 , x142 x122 , x142 x22
, x14
x22 , x14
x22 }
1 2
1 3
1
3
3 1
3 2
Y13  {x14
x33 , x14
x33 , x142 x33
, x142 x33
, x14
x33 , x14
x33}
1 2
1 3
3
3 1
3 2
Y14  {x14
x43 , x14
x43 , x142 x143 , x142 x43
, x14
x43 , x14
x43}
1 2
1 3
1
3
3 1
3 2
Y15  {x14
x52 , x14
x52 , x142 x52
, x142 x52
, x14
x52 , x14
x52}
2
3
2 1
2 3
3 1
3 2
Y23  {x122 x33
, x122 x33
, x22
x33 , x22
x33 , x22
x33 , x22
x33}
1 2
1 3
2 1
2 3
3 1
3 2
Y34  {x33
x43 , x33
x43 , x33
x43 , x33
x43 , x33
x43 , x33
x43}
2
3
2 1
2 3
3 1
3 2
Y45  {x143 x52
, x143 x52
, x43
x52 , x43
x52 , x43
x52 , x43
x52}
Y  Y12  Y13  Y14  Y15  Y23  Y34  Y45 .
Probe operation and detect whether there exists 5 - data-polymers with 7 probes that is
isomorphic to G . If such polymers exist then the given graph is 3-colorable else is not
3-colorable. The problem of 3-coloring is thus decided.
4. Conclusion and discussion
In this paper, we proposed probe machine based algorithm for solving 3-vertex-coloring problem.
For n -vertex and m -edge graph, we constructed 3n different types of data and 6m different
types of probes, performed one step of probe operation to decide whether the given graph is
3-colorable regardless of the size of the graph. As a computing model, probe machine is
theoretical. But the prototype of probe machine does exist in the real world. For instance, a
biological nervous system is a typical connective probe machine [18].
Compared with Turing machine, probe machine alters the mode of data storage. Data
participating in the computing is not necessarily required to be adjacent on one hand. Vast copies
of probe can “process” concurrently a pair of data on the other hand. These appealing
characteristics endows probe machine with more powerful computing capability. Although probe
machine has difficulty in implementation technology currently, the research of this bio-inspired
computing model will surely lead to aboundant brand-new algorithms for various problems and
more insightful understandings of P versus NP problem. That will be our future work.
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