Polynomial-Size Formulation for ATSP

This research paper discusses the polynomial-size formulation for the Asymmetric Traveling Salesman Problem (ATSP) and its optimal solution using mathematical models and exact algorithms.

Computer Science

Table of Contents
1. References of the Polynomial-Size Formulation for ATSP...........................................................2
2. Polynomial-Size Formulation for ATSP........................................................................................2
3. Constraints and the Variables........................................................................................................4
4. CPLEX OPL Modelling Language.................................................................................................6
5. Solution.............................................................................................................................................7

1. References of the Polynomial-Size Formulation for ATSP
To develop the new flow based formulations for solving some scheduling problem for
the asymmetric salesman issue is the aim of this research paper (Pdfs.semanticscholar.org,
2019). This is utilised for developing new ATSP formulations that shall solve the large size
problem by taking advantage of their relaxations. Based on the exponential number of Danzig
Fulkerson Johnson sub tour elimination constraints, for which we shall be implementing the
formulas. For solving the linear problem relaxations, the Lagrangian dual formulations and
subgradinate methods shall be used for optimum results. Alternate approach that we utilize
for deriving a set of strong valid inequalities based on our tighter formulations should be by a
suitable subrogation process for including within the more compact manageable
formulations. By getting the solution of the LP relaxations for all variations of our JSCD
formulation equal to the maximum total processing time among the jobs in the issue which
we have observed that the lower bound values obtained formulation which includes
precedence constraints in order to enforce a partial order on the sequence.
URL: https://pdfs.semanticscholar.org/abf8/e0f348148ed9a002f2eff848f1e6d3eec7a9.pdf
URL: https://arxiv.org/ftp/cs/papers/0609/0609005.pdf
2. Polynomial-Size Formulation for ATSP
For the very popular Asymmetric Traveling Salesman Problem (ATSP), this project
paper suggests the optimal solution by making use of the most effective mathematical
models and exact algorithms. The derivation of the classical (assignment, shortest
spanning r-arborescence, linear programming) relaxations which shall be carried out by
presenting of the fundamental Integer Linear Programming (ILP) model as was proposed by
Danzig, Fulkerson and Johnson, and also we will discuss the most effective branch-and-
bound and branch-and-cut algorithms. The next step in the process Presentation of the
polynomial ILP formulations proposed for the ATSP shall be presented and the
experimentally comparison value for the considered algorithms and formulations, will be
the last step, as a set of benchmark instances (Kucharzak, Zydek and Poźniak-Koszałka,
2012).

Let G = (V, A) be a given complete digraph, where V={1,…,n}V={1,…,n} is the
vertex set and A={(i,j):i,j∈V}A={(i,j):i,j∈V} the arc set, and let c ij be the cost associated
with arc (i,j)∈A(i,j)∈A (with cii=+∞,cii=+∞, for i∈Vi∈V). A Hamiltonian circuit (tour)
of G is a circuit visiting each vertex of Vexactly once. The Asymmetric Traveling Salesman
Problem (ATSP) is to find a Hamiltonian circuit G∗=(V,A∗)G∗=(V,A∗) of G whose
cost ∑(i,j)∈A∗cij∑(i,j)∈A∗cij is minimum. For the case where the considered graph G is
undirected, it is denoted as the Symmetric Traveling Salesman Problem (STSP).
ATSP issue has been researched very widely specially in the last 5 years by many
researchers as it is well known to be the NP-hard in the strong sense.
We shall study the exact algorithms proposed and then conduct the experiments for
comparing the most effective Integer Linear Programming (ILP) models, for calculating the
optimal solution of the ATSP.
If the considered graph G is undirected graph, than the Symmetric Traveling
Salesman Problem (STSP) is the designated problem as denoted. It proposed the following
ILP model (hereafter DFJ), utilizing n 2 binary variables x ij , for the ATSP:
(DFJ)min∑i=1n∑j=1ncijxij(DFJ)min∑i=1n∑j=1ncijxij (1)
s.t.∑i=1nxij=1,j=1,…,n,s.t.∑i=1nxij=1,j=1,…,n, (2)
∑j=1nxij=1,i=1,…,n,∑j=1nxij=1,i=1,…,n, (3)
∑i∈S∑j∈Sxij≤|S|−1,S⊂V:S≠∅,∑i∈S∑j∈Sxij≤|S|−1,S⊂V:S≠∅, (4)
xij∈{0,1},i,j=1,…,n,xij∈{0,1},i,j=1,…,n, (5)
where x ij is equal to 1 if and only if arc (i, j) (i=1,…,n;j=1,…,ni=1,…,n;j=1,…,n) is in the
optimal tour. Constraints (2) and (3) impose that the in-degree and out-degree of each
vertex, respectively, is equal to one, while constraints ( 4) are Subtour Elimination
Constraints (SECs) and impose that no partial circuit exists.
Moreover, it is well known that one can halve the number of SECs ( 4) by replacing them
with
∑i∈S∑j∈Sxij≤|S|−1,S⊂V∖{r}:S≠∅,∑i∈S∑j∈Sxij≤|S|−1,S⊂V∖{r}:S≠∅,
Where is any vertex of vertex set V.
Equivalently rewritten as Connectivity constraints will be the constraints (2) and (3) with
Constraint (4):
∑i∈S∑j∈V∖Sxij≥1,S⊂V:S≠∅.∑i∈S∑j∈V∖Sxij≥1,S⊂V:S≠∅. (6)

Also in this case, by replacing (6) with the below equation, one can halve the number of
connectivity constraints,
∑i∈S∑j∈V∖Sxij≥1,S⊂V:r∈S∑i∈S∑j∈V∖Sxij≥1,S⊂V:r∈S (7)
or with
∑i∈S∑j∈V∖Sxij≥1,S⊂V:S≠∅,r∉S∑i∈S∑j∈V∖Sxij≥1,S⊂V:S≠∅,r∉S (8)
Where r, is any fixed vertex.
A valid lower bound on the optimal solution value of the ATSP can be obtained ( 1)–(5) or
(1)–(3), (5) and (7), by optimally solving the Linear Programming (LP) relaxation of the
previous models, and obtained by replacing constraints (5) with constraints
xij≥0,i,j=1,…,n.xij≥0,i,j=1,…,n. (9)
The effective polynomial separation procedure proposed by Padberg and Rinaldi
for the STSP, where LP relaxations can be efficiently solved in polynomial time using this
and although the considered ILP models require an exponential number of Sub tour
Elimination or Connectivity constraints. Additional lower bounds can be obtained by
considering the different substructures of the ATSP, each of the associated with a subset of
constraints defining a well-structured relaxation. Making use of the constraints ( 2), (3) and
(9), with objective function (1), we shall define the well-known min-sum Assignment
Problem (AP). Visiting all the vertices of G, which requires finding the minimum-cost
collection of vertex-disjoint sub tours and the issues always, has an integer optimal
solution. In O(n 3) time, we can solve the relaxation AP. The constraints (2), (7) and (9),
with objective function (1) are used for defining the well-known shortest Spanning r-
Arborescence Problem (r-SAP).
(1) The in-degree of each vertex is exactly one, and (2) each vertex can be reached
from the root vertex r, which if for finding a minimum-cost spanning subdigraph G¯=
(V,A¯)G¯=(V,A¯) of G such that a problem always has an integer optimal solution.
Relaxation r-SAP can be solved in O(n 2) time, By finding the shortest spanning
arborescence rooted at vertex r and adding the minimum-cost arc entering vertex r.
3. Constraints and the Variables
One of the most prominent combinatorial optimization problems is the Traveling
Salesman Problem (TSP). For the shortest tour through all vertices with respect to the

distances d, the given a complete graph G = (V, E) and non-negative distances d for every
edge TSP. A branch and cut approach method of choice (Soltani and Shafiei, 2011) is for the
solution of the TSP to optimality. To identify violated inequalities by usually using the
integrality constraints relaxing them first, Fractional solutions will be the method for all such
separation processes. Working with only integer solutions without ever interfering with
fractional solutions, our approach for which we try to exploit the impressive performance of
current ILP-solvers by only easing the sub tour elimination constraints to stick to the very
simple ILP-model. The resulting problem is solved to integer optimality, when the violated
constraints (which are trivial to find) are added and this process keeps on repeating, until the
time a feasible solution is not searched and found. The speed of the algorithm can be
increased while pursuing many attempts for finding as many relevant sub tours as possible.
Additional insights gained from empirical observations and random graph theory, have
mentioned attempts are based on the clustering of vertices. Extracted from the TSPLIB95 and
on random Euclidean graphs respectively, the computational results are carried out on test
instances. With four widely famous books entirely written on its research, Traveling
Salesman/Salesperson Problem TSP is one of the most well-known and widely investigated
combinatorial optimization problems.
We shall calculate and then calculate the minimum, shortest tour with respect to the
distances de containing each vertex exactly once, given the complete graph G= (V, E) with |V|
= n and |E| = m = n(n - 1)/2, and nonnegative distances de for each e ∈ E, the TSP. Let δ(v):
= {e = (v, u) ∈ E∣u ∈ V} denote the set of all edges adjacent to v ∈ V. Introducing binary
variables xe for the possible inclusion of any edge e ∈ E in the tour we get the following
classical ILP formulation:
minimize∑e∈Edexe 1
s.t.∑e∈δ(v)xe=2∀v∈V, 2
∑e=(u,v)∈Eu,v∈Sxe≤|S|−1∀S⊂V,S≠∅, 3
xe∈{0,1}∀e∈E 4
Equation (1) defines the objective function.
Equation (2) is the degree equation for each vertex.
Equation (3) are the sub tour elimination constraints (SEC), which forbid solutions consisting
of several disconnected tours.

Equation (4) defines the integrality constraints.
We get pairs of SEC by enforcing the connection of S and S′, for the vertex sets S ⊂ V, S ≠ ∅,
and S′ = V\S, although there are no signs of SEC redundancy. Also, we can see that
depending crucially on the performance of the used LP-solver, the performance of the above
branch-and-cut approach, ILP-Solvers along with the highly efficient LP-solvers has been
available for quite some time now, by reaching an impressive performance after improving
rapidly for the last few decades. Without using the LP-relaxations explicitly, this motivated
the idea of a very simple approach for solving TSP.
After solving the remaining ILP model (corresponding to a weighted 2-matching
problem), relax the entire SEC (3) from the model. The obtained integer solution contains sub
tours is not checked, it will be an optimal TSP tour. All the subtours in the integral solution
(which can be done by a simple scan) have to be identified, or else this would not have been
the situation. Represented by the subset of vertices in the corresponding subtour SEC of the
corresponding model, it has to be added where it has been found each of these. By optimally
solving again this resultant enlarged ILP model which clearly leads to the optimal TSP tour
for carrying out the iterating process.
4. CPLEX OPL Modelling Language
The Cplex program allows one to deal with mixed integer linear problem and real
linear programs. The real linear program declare all variable from equation that denoted as R
and mixed integer linear program declare some variable that variable denoted as Z. We take
the house allocation variable and that declare the all variable. A Cplex environment
constructs by capex class. It available of all other concert technology classes and a pointer to
an implementation object (Sun, Liu and Chai, 2011). First step to create an environment and
next step to creating a model. The Cplex number variable representing modelling variables.
To take the all variable as linear expression which used to form the linear programming. In
this model contains integer variable, Boolean variable and continuous variable. Consider the
lower bound value and upper bound value for create the modelling variable. The float
numbers are continues variable, numeric value are called as integer variable and Boolean
value such as AND, OR, NAND etc. are called as Boolean variable. The given values related
to house allocation that contains dish, car, vacuum and lawn.so that are replaced as x

variable. The value are formed as the linear regression equation. Consider int i value is 10
and the equation are,
12X1+10X2+3X3+14X4 <=100
9X1+5X2+10X3+20X4 <=200
7X1+12X2+11X3+12X4<=120
16X1+11X2+9X3+15X4<=500
The objective function and decision variable are create for implemented the linear
equation. The Cplex solve the given problem and that return the value. If the graph id
directed graph that the issue is identified as asymmetric travelling salesman problem.
Symmetric Travelling Salesman issue is the graph which is undirected graph. To compare the
most effective integer linear programming model to exact algorithms, we have to find the
optimal answer for the Asymmetric Travelling Salesman Problem. The optimization problem
has been stored in model object and creates a Cplex object. If the solution is feasible then it
return the Boolean variable (Wood and Siebenmorgen, 2013). If the solution is not optimal
solution then it does not return the variable.
Linear program has product mix problem that has maximum value, subject to values
and constraint value. The value are used to implement the linear programming problem. We
take the given value for constraint value that denoted as x variable. The program represent the
minimum and maximum value based on objective function value. Blank spaces are ignored
and using symbols and alphanumeric for names. Objective function section has minimum or
maximum value and label with colon. The constraints section has keyword subject to and list
of constraints. Bounds section has lower bound value, upper value, and fixed value and
specifies a free variables. General section has list of integer variable and binary section has
list of binary variable. Finally that equation ended with end keyword.
5. Solution
Formulate the linear equation using given data value and that value are contains
objective function, subjective value and constraints. The constraints value are based on given
house allocation value which used to formulate x variable.
Declare the variable using given data and that contains the integer and string variable.
{string} Children = ...;

{string} Chores = ...;
int Cost[Children][Chores] = ...;
dvar int whoDoesWhat[Children][Chores] in 0..1;
Declare the maximum or minimum value for objective function
minimize
sum(i in Children, j in Chores)
Cost[i][j]*whoDoesWhat[i][j];
The code run by constraints so formulate the constraints variable.
subject to {
ct1: //Each house chore must be covered
forall( j in Chores )
sum(i in Children) whoDoesWhat[i][j] == 1;
ct2: //Each children gets one chore
forall(i in Children)
sum(j in Chores ) whoDoesWhat[i][j] == 1;
}
Finally write the cplex code and run the code on cplex optimization studio.
References
Kucharzak, M., Zydek, D. and Poźniak-Koszałka, I. (2012). Overlay Multicast Optimization:
IBM ILOG CPLEX. International Journal of Electronics and Telecommunications,
58(4), pp.381-388.
Soltani, H. and Shafiei, S. (2011). Heat exchanger networks retrofit with considering pressure
drop by coupling genetic algorithm with LP (linear programming) and ILP (integer
linear programming) methods. Energy, 36(5), pp.2381-2391.

Sun, L., Liu, W. and Chai, T. (2011). Tool sequence optimization based on three logical
modes of robot control via CPLEX optimization. Journal of Shanghai Jiaotong
University (Science), 16(4), pp.436-440.
Wood, D. and Siebenmorgen, T. (2013). Spotlight on Rice. CFW Plexus.