Framework for Small Traveling Salesman Problems
RICHARD H. WARREN
Retired: Lockheed Martin Corporation, King of Prussia, PA 19406, USA
Address: 403 Bluebird Crossing, Glen Mills, PA 19342-3362, USA
rhw3@psu.edu
Abstract: We study small traveling salesman problems (TSPs) because current quantum computers
can find optional solutions for TSPs with up to 14 cities. Also, we study small TSPs because TSPs
have been recommended to be benchmarks to measure quantum optimization on all types of
quantum hardware. This means comparisons of quantum data about small TSPs. We extent
previous numerical results that were reported in “Small Traveling Salesman Problems” for 6, 8
and 10 cities. The new results in this paper are for 10 – 14 cities in symmetric TSPs. The data for
this new range of cities is consistent with the previous data and can be the basis for estimates of
results from quantum computers that are upgraded to handle more than 14 cities. The work and
analysis suggest two conjectures that we discuss. The paper also contains an annotated survey of
recent publications about TSPs.
Keywords: Traveling salesman problem, optimal tour, combinatorial analysis, discrete
optimization, quantum annealing, quantum computer
Received: March 24, 2023. Revised: September 13, 2024. Accepted: October 9, 2024. Published: November 12, 2024.
1 Introduction
Research interest in the traveling salesman
problem (TSP) has been stimulated by
demonstrated ability to find optimal solutions
on quantum processors [28] and by questions
such as: What are the characteristics that
distinguish easy to solve TSPs from those
that are difficult to find an optimal solution
on a quantum computer? Additional interest
in TSPs has come from proposals to use TSPs
as benchmarks for quantum optimization on
various hardware [9, 32, 33].
Due to its many applications and
computational complexity, the worldwide
opinion of the TSP has escalated from an
obscure novelty in the 1960’s to a leading
example of combinatorial optimization
problems. Reference [1], which has
prominent contributors and editors, was
instrumental in this transition. Textbooks [2
- 5] contributed to the rise. of the TSP.
The current paper is a continuation of [6]
where the structure of TSPs for 6, 8 and 10
cities is examined. We observe similar
structure in the current study for 10 – 14 cities
and show the supporting data in Tables 3 and
4. Interestingly, Table 3 shows without
exception that as the number of cities
increases, the number of optimal tours
increases.
The effort for [6] was motivated by the
need to have reference TSPs for comparison
to the results of the D-Wave quantum solver
that has about 2,000 qubits and can optimally
solve TSPs with up to about 8 cities. An
upgraded D-Wave quantum solver with about
5,600 qubits can optimally solve TSPs with
up to 14 cities. This improvement stimulated
the current study to understand the nature of
optimal solutions. To obtain an optimal
solution on D-Wave’s upgraded quantum
processor, the size of the TSP is limited to about
14 cities due to difficulties embedding all-to-all
connections of cities. Therefore, in this study we
report about optimal solutions of TSPs with at
most 14 cities. The technique in [32] to solve the
TSP is limited to about 8 cities.
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DOI: 10.37394/232030.2024.3.7
Richard H. Warren
E-ISSN: 2945-0489
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We define the TSP. Given a set of cities
and the distance between each pair of cities,
the TSP asks for a shortest route that visits
each city once and returns to the starting city.
A shortest route is called an optimal tour. If
for each pair of cities (A, B), the distance
from city A to city B is the same as the
distance from city B to city A, then the TSP
is called symmetric. The term TSP includes
those that are symmetric and those that are
not symmetric. Initial work defining the TSP
on a quantum annealing computer was
published in [7, 8, 30].
We studied symmetric and non-symmetric
TSPs in [6] without distinction. References
[9] and [10] recommend symmetric TSPs as
a benchmark for quantum optimization
problems and disqualify non-symmetric
TSPs as benchmarks. Therefore, the current
study is only about symmetric TSPs. This is
very significant because there are extremely
few random TSPs that are symmetric
compared to the number that are non-
symmetric.
Since we are interested primarily in an
optimal answer, our work does not consider
hybrid solvers (quantum and digital
combined, each doing part of the solution)
because their analog nature usually produces
answers that are not optimal. We are
interested in the total number of optimal tours
and the frequency of occurrence because
quantum results can contain several
minimum energy solutions that may be near-
optimal or optimal tours. This gives the
salesman more than one option for a tour.
We outline the contents of this paper. The
next Section contains insights and two
conjectures. Section 3 contains the settings
for the parameters. Section 4 has the data
generated in the study. Future studies of the
TSP are recommended in Section 5. Section
6 is an annotated survey of recent, relevant
articles, mostly about the TSP. Conclusion
and results are in Section 7.
2 Open Questions and Conjectures
Symmetric TSPs are beginning to be used to
benchmark quantum algorithms and quantum
hardware for quantitative optimization
problems [27]. This effort is generating new
questions about TSPs. Some of them are: (i)
How many shortest routes does a TSP have?
(ii) What is the gap between the length of a
shortest route and the length of a next-to-
shortest route for TSPs?
A conjecture related to Question (i): If a
TSP has few shortest routes, then adverse
quantum effects are likely to occur that
curtail solving the TSP on a quantum
machine. A conjecture related to Question
(ii): If the gap for a TSP is small, then adverse
quantum effects are likely to occur. The
adverse quantum effects for the TSP are
excessive time to solve, failure to return a
route, and inability to find a shortest route.
We expect that answers to the questions for
TSPs can help predict outcomes for
quantitative optimization problems.
A study has begun to quantify TSP
characteristics by examining TSPs with 6
cities [11]. The small number of qubits at that
time limited the size of the TSPs that could
be examined. Now an increase in the number
of qubits requires larger TSPs. In the current
paper we show results for TSPs with 10 – 14
cities with data collected from classical
processors.
It is widely recognized that quantum
computers are analog devices that may
deviate from the theory of an algorithm [12].
This difficulty, coupled with numeric
imprecision, caused some quantum
calculations to miss optimal solutions for
TSPs. For some TSPs the D-Wave quantum
computer could not distinguish between an
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Richard H. Warren
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optimal solution and a close-to-optimal
solution.
3 Methodology and Parameter
Settings
Table 1 lists the parameters and their
settings in [6] and the current paper.
Table 1. Settings for the parameters in two studies
Parameter
Reference [6] Setting
Current Paper Setting
Number of Cities
6, 8, 10
10, 11, 12, 13, 14
Number of TSPs studied
5,000 for each number of cities
200 for 10 cities
100 for 11 and 12 cities
51 for 13 cities
5 for 14 cities
Distances between cities
Random integers ∈ {1, 2, …,
21}
Random integers ∈ {1, 2, …, 21}
Type of TSP
Did not distinguish between
symmetric and non-symmetric
Symmetric
Solution algorithm
Python exact, examine all tours
Python exact, examine all tours
Next, we show two examples of TSPs.
Example 1 is a symmetric TSP on 6 cities.
Let the cities be designated A, B, C, D, E, F.
Let a distance matrix X be given that contains
the distance between each pair of cities.
Since the TSP is symmetric, X is a symmetric
matrix, i.e., upper triangular. The diagonal
elements of X have no role in the TSP.
Example 2 is a symmetric TSP on 8 cities.
We can describe Example 2 as an expansion
of Example 1. Two additional designations
are needed for cities and additional distances
are needed to expand X.
We comment about the distances restricted
to 1, 2, …, 21 in Table 1. A D-Wave
implementation transcribes coefficients to
the interval [-10, 10]. When the original
coefficients are integers 1, 2, …, 21, then the
D-Wave mapping to the [-10, 10] has the
greatest accuracy because it is a 1 to 1
mapping of integers onto integers [13]. TSPs
with all distances 1 and 2 are NP-complete
and have been studied in [14].
Since it takes significantly longer to test
TSPs with a larger number of cities, we
scaled down the total number tested as the
number of cities increased.
4 Data Analysis and Findings
Table 2 shows the distribution of random,
symmetric TSPs according to the number of
optimal tours. Since the TSPs are symmetric,
a tour and its inverse have the same length,
which means the number of optimal tours is
an even integer. The data for 14 cities is
weak since the sample size is very small.
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Richard H. Warren
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Table 2. Distribution of random, symmetric TSPs for 10 to 14 cities
10 Cities
11 Cities
12 Cities
13 Cities
14 Cities
159
75
67
32
3
32
16
21
12
8
9
8
5
1
1
1
1
3
1
1
200
100
100
51
5
502
268
304
180
20
Table 3 is Table 2 normalized to 100 TSPs for each number of cities.
Table 3. Normalized distribution of random, symmetric TSPs for 10 to 14 cities
10 Cities
11 Cities
12 Cities
13 Cities
14 Cities
79.5
75
67
64
60
16
16
21
24
4
9
8
10
20
0.5
1
20
3
2
2
251
268
304
360
400
Table 4 shows data for the distribution of the average length of optimal tours. We did
not compute this for TSPs with 14 cities and for TSPs with 13 cities that have 18 and
20 optimal tours due to a small number of TSPs.
Table 4. Distribution of average length of optimal tours for 10 to 13 cities
10 Cities
11 Cities
12 Cities
13 Cities
44.2
44.8
48.4
45.8
48.2
47.3
51.4
46.8
47.0
49.9
50.4
44.0
53.0
52.0
60.0
200
100
100
49
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Results that are like those in Table 4 are
shown for TSPs with 6, 8 and 10 cities in
Figures 1 – 3 of [6].
Tests have been run on the D-Wave
quantum machine that solve small,
symmetric TSPs exactly. In addition to our
work, reference [28] reports several TSPs
solved exactly.
5 Research Directions in the
Future
When D-Wave upgrades its array of qubits in
its quantum processor, then the size of TSPs
that can be processed without heuristics or
hybrid methods is expected to increase. This
number of cities will need to be determined.
Then a study like the current one should be
undertaken for the new numbers of cities
beyond 14 and for 13 & 14 cities for overlap.
The classical software Concorde [15] is
recommended to establish a baseline, since it
is the gold standard for solving symmetric
TSPs.
The average gap between the length of an
optimal tour and the length of the next
shortest tour can be determined for various
categories of TSPs. Is there a correlation
between the size of the gap and easy or
difficult to solve with a quantum algorithm,
i.e., do large gaps correspond to ‘easy to
solve’ on a quantum processor? ‘Easy to
solve’ can be described numerically by the
time to solve, accuracy of the solution, and/or
a percentage of the optimal tours found. This
leads to generating and assembling TSP data
about the two conjectures in the Preface.
Based on future data, we can begin to decide
the validity of the conjectures and how to
quantify them.
A major challenge is the need to deal with
more sophisticated, real-world problems.
Let the number of cities be fixed. What
sample size is needed to have X% confidence
that all symmetric TSPs have a property of
the samples? Are the sample sizes in this
paper adequate?
Lastly, we comment that large companies
including Google, Microsoft, IBM, D-Wave
and [34] are working to improve their
quantum hardware and algorithms in order to
secure their marketplace for this new
technology in the business world.
6 Recent Publications Related to
Experimental Results about TSPs
In this section we provide fresh insights,
breakthroughs and collaborations from the
literature.
Reference [16] categorizes an extensive list
of references in their Tables 9 - 11. The TSP
is included in the analysis and comparisons.
The authors of [17] introduce an improved
version of quantum annealing to handle local
optimal results when solving the TSP on a D-
Wave processor. Since there are difficulties
in the theory, the experimental results are
remarkable.
The quantum modeling techniques in [18]
are designed for variants of the TSP. It may
be interesting to recast them for basic TSPs
and test them on a D-Wave quantum
processor for 6-city problems. This is the
smallest number of cities that can have two
distinct loops; in this case each loop has three
cities.
Paper [19] uses a logical embedding of a
TSP in the qubit array of D-Wave’s 2,000 and
5,000 quantum processors. The conclusions
agree with what is known previously. In
general, a TSP with at most 8 cities can be
embedded in D-Wave’s 2,000 qubit machine
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[20 Section 5.1] and a TSP with at most 13
cities can usually be embedded in D-Wave’s
5,000 qubit machine [10 Section 2].
The remarkable TSP results that are
claimed in [20] need an independent
investigation that repeats the experiments.
The authors of paper [21] investigate the
performance of two classical and two
quantum optimization algorithms to solve the
TSP. The overall conclusion of the authors is
that current classical devices significantly
outperform the IBM quantum devices.
Publication [22] has the same authors and
topic as [21]. In [22] results on the TSP for
two classical optimization algorithms are
compared with one quantum algorithm on a
range of IBM quantum devices. There is
insufficient information about the attributes
of the TSPs in the experiments. The overall
conclusion of the authors is that the classical
optimization techniques outperform the
quantum methods in both computational time
and solution quality.
We call attention to paper [23] because it
has similarities to the current paper. In [23]
pairs of integers from a 100 x 100 square are
candidates for cities. Randomly N pairs are
chosen from the square for the cities of a TSP.
The distance between cities is the Euclidean
distance rounded down to an integer.
According to [23 page 3], the plan is to
address four questions with empirical data. 1.
What is the distribution of the distances? 2.
What is the distribution of the lengths of tours
for instances of the same size? 3. What is the
distribution of lengths of optimal tours for
instances of the same size? 4. How difficult
is it to solve an instance? Results for 1 – 3
are in the paper but apparently not for 4.
Articles [24] and [25] claim to have a technique
to reduce the number of variables in a D-Wave
QUBO for the TSP. The result is expected to be
larger TSPs processed by a quantum annealer
without heuristics, resulting in better quality
solutions at the expense of longer quantum
execution time [25 Table 1]. The papers lack
comparisons of experimental data for solving
TSPs with and without this enhancement.
We recommend that the data be collected for
random, symmetric TSPs with various
numbers of cities on both the D-Wave 2000
and 5000 processors. It is recommended that
the data be presented in a table that identifies
the quantum processor, the TSP, the number
of cities, the length of an optimal tour, length
of the shortest tour (found by the quantum
processor) with and without the enhancement
and number of qubits used with and without
the enhancement. Also, time for the quantum
processing unit with and without the
enhancement is a useful comparison.
The authors of [26] have developed a
computational method for generating metric
TSPs that are hard for Concordia [15] to
solve, i.e., a long runtime is needed by
Concorde to find an optimal tour.
Using simulated Ising quantum software
with all-to-all connections, the eleven authors
of [31] show details to find an optimal
solution for a 9-city, symmetric TSP. Based
on that technique they experimentally solve a
70-city TSP. The simulation operates with
high energy efficiency compared to several
quantum computers, including D-Wave’s
2,000 qubit Ising machine.
The researchers of paper [33] use the TSP
on gate-based NISQ hardware to show that
their quantum solution algorithm is superior
to digitized quantum annealing and the
quantum approximate optimization
algorithm. It appears that [33] reports
success for a 3-city TSP solved on an IBM
superconducting machine and for a 3-city
TSP and a 4-city TSP solved on an IonQ
trapped-ion machine. These very small TSPs
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are trivial to solve. Results for 100 random
6-city TSPs would be much more useful.
7 Conclusions
The current study examines symmetric TSPs
that have 10 – 14 cities with distances
between cities restricted to random integers
in the interval 1 to 21. The conclusions in a
previous study [6] are essentially the same in
the current study. We describe them. Let n
be an integer and 6 ≤ n ≤ 13. Then according
to [6] and Tables 3 and 4 for large collections
of n-city TSPs, most likely the number of
optimal tours per TSP and the average length
of an optimal tour increase together. This
data is the first connection between the
number of optimal tours per TPS and the
average length of an optimal tour.
The data in the last line of Table 3 shows
for 10 – 14 cities that the number of optimal
tours increases as the number of cities
increases. This result has no exceptions.
Recalling that each column of Table 3 is
normalized to 100 TSPs. The rate of increase
of optimal tours across Table 3 is very slow
compared to the factorial rate of increase of
tours.
Looking down the columns of Table 3, the
conclusion is that as the number of optimal
tours increases the frequency of optimal row
tours decreases, except for two anomalies,
one for 12 cities and the other for 14 cities.
Table 4 indicates that for 2 – 6 optimal tours
as the number of cities increases from 10 to
11 and from 11 to 12, the average length of
an optimal tour increases. The only
exception occurs for 4 optimal tours
transitioning from 10 cities to 11 cities.
Looking down the columns of Table 4 we
observe that in most cases (8 of 11) as the
number of optimal tours increases, the length
of an optimal tour increases. The three
exceptions occur between 4 and 6 optimal
tours.
Similar results shown are anticipated for
quantum annealing solutions of large
numbers of TSPs.
In conclusion, we point to an outstanding
lecture about the TSP [29].
Acknowledgement All glory, praise and
honor to Jesus Christ who is my Lord and
Savior.
Disclosure The author reports there are no
competing interests to declare. This research
did not receive funding from an agency in the
public, commercial, or not-for-profit sectors.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.7
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E-ISSN: 2945-0489
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