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22
Pareto-optimal Algorithms for Metric TSP: Experimental Research
Ekaterina N. Beresneva (Chirkova) and Prof. Sergey M. Avdoshin
Abstract—The Travelling Salesman Problem (TSP) is a fundamental task in combinatorial optimization. A special case of the TSP is Metric TSP, where the triangle inequality holds. Solutions of the TSP are generally used for costs minimization, such as finding the best tour for round-the-world trip or construction of very large-scale integration schemes. Since the TSP is NP-hard, heuristic algorithms providing near optimal solutions will be considered. The objective of this article is to find a group of Pareto optimal heuristic algorithms for Metric TSP under criteria of run time efficiency and qualitative performance as a part of the experimental study. Classification of algorithms for Metric TSP is presented. Feasible heuristic algorithms and their prior estimates are described. The data structure and the details of the research methodology are provided. Finally, results and prospective research are discussed.
Keywords—travelling salesman problem, resource-efficient algorithm, heuristic algorithm, posterior estimate, computational experiment.
I. Introduction
The Travelling Salesman Problem (TSP) is one of the most widely known questions in a class of combinatorial optimization problems. Essentially, to meet a challenge of the TSP is to find a Hamiltonian circuit of minimal length. A subcase of the TSP is Metric TSP where all of the edge costs are symmetric, and they satisfy the triangle inequality.
The methods for solving the TSP have been developed for many years, and since the problem is NP-hard, it continues to be topical. The TSP has seen applications in the areas of logistics, genetics, manufacturing, telecommunications and neuroscience [1]. The most common practical interpretation of the TSP relates to the movement of people and vehicles around tours, such as searching for the shortest tour through N cities, school bus route planning, and postal delivery. In addition, the TSP plays an important role in very large-scale integration (VLSI) [2] The purpose of this study is to determine the group of Pareto-optimal heuristic algorithms for Metric TSP by criteria of run time and qualitative performance as part of the experimental investigation.
Manuscript received April 18, 2017.
Ekaterina N. Beresneva is with the Department of Software Engineering, Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia (e-mail: enchirkova@edu.hse.ru).
Prof. Sergey M. Avdoshin is with the Department of Software Engineering, Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia (e-mail: savdoshin@hse.ru).
Clearly, a study of this type is inevitably restricted by various constraints, in this research only heuristic algorithms constructing near optimal solutions in polynomial time will be considered instead of the exact ones.
The paper is structured as follows. First, the theoretical basis is described. It presents the mathematical formulation of Metric TSP, the specification of metric types and, at last, definition of Pareto optimization. Next, a classification of algorithms for Metric TSP is given, including a literature review of popular heuristics. Then the description of methods to be used is provided with their prior estimates. After that the details of the research methodology and expected results are mentioned.
II. Theoretical basis A. Problem Formulation
In this paper, mathematical formulation of Metric TSP is adopted as follows.
Given a complete weighted undirected graph G = (V, V2) which contains N = |7| vertices. Graph vertices are indexed as index = V ^ I, I = {1,2,...,N}, and (Vvt £ V)(Vvj £ V) it is true that if Vi ± Vj then i ± j, where i = index (Vi). The distance between two vertices vt and Vj is calculated by distance function d(y>i, Vj). Here a real-valued function d(•,•) on V x V satisfies [3]:
1. d(vi, Vj) > 0 (non-negativity axiom)
2. d(vt, Vj) = 0 if and only if vt = Vj (identity axiom)
3. d(vt, Vj) = d(vj, Vj) (symmetry axiom)
4. d(vt, vk) < d(vi, Vj) + d(vj, vk) (triangle inequality axiom)
Let S be a set of all Hamiltonian cycles of G. It is defined as 5 = {p\ V ^ 7|(p(1) = 1&(Vi £ V)(Vj £ V) (p(i) = p(j) => i = j}. An example of a Hamiltonian circuit s £ S is fa, p2,..., pN), where pt is used as abbreviated notation of p(i).
Weight of a Hamiltonian cycle s £ S can be found according to the formula (1):
f(s) = d(plt pN) + lEfdfa, pi+1) (1)
The set of vertices V is determined in Euclidean space Rn by their coordinates. Under these circumstances, the distance between two vertices v = (x1, x2,..., xn) and w = (xlt x2,..., xn) is defined as (2):
d0 (v, w) = (4)
The formulation for the metric travelling salesman problem is to find such s0 that f(s0) = mins6S/(s) for the given metric d0.
B. Resourse-efficient parameters
Let M be a set of given heuristic algorithms for Metric TSP. There are two parameters of resource-efficiency for m E M for each N:
• fs(m, N) - qualitative performance;
• ft (m, N) - running time.
Qualitative performance can be calculated using (7):
fs(rn, N)= * 100%, (7)
where f(s) is the obtained tour length and f(s0) is the optimal tour length. The values of optimal tour lengths are taken from the open library TSPLIB as the lengths of the best reported solutions for each of the instances [4].
C. Pareto-optimality
Pareto optimization [5] or multi-objective optimization problem for the TSP can be formulated as (8):
minmEM [fs (rn), ft (m)} (8)
Pareto optimal solutions are solutions that cannot be improved in any of the objectives without degrading, at least, one of the other objectives. In mathematical terms, a feasible solution m1 EM is said to (Pareto) dominate another solution m2 E M, if
1. f(m1) < f(m2), for all indices i E [e, t}.
2. fj(m1) < fj(m2), for at least one index j E [e, t}.
III. Related works
Algorithms for solving the TSP may be divided into two classes:
• Exact algorithms, and
• Heuristic (or approximate) algorithms.
Exact algorithms are aimed at finding optimal solutions. Both widely known subtypes of exact methods - linear programming and branch-and-bound techniques - are described in details by Applegate [1]. However, a major drawback is connected with their time efficiency. It is a common knowledge that there are no exact algorithms running
in polynomial time. Thus, only small datasets can be solved in reasonable time. For example, the 4410-vertex problem is believed to be the largest Metric TSP ever solved with respect to optimality [6].
In this paper, only a class of heuristic search algorithms will be taken into account. They are designed to run quickly and to get an approximate solution to a given problem. Heuristic algorithms are subdivided into two groups.
The first group includes tour construction algorithms that have one feature in common - the tour is built by adding a new vertex at each step. The following are alternative ways of doing this [7]:
• Ordered sequence methods:
At the start of the heuristic, all the edges are ranked by some suitable criteria. At each iteration the best edge according to the criteria is added to the solution. The most common order sequence methods are Greedy [8] and Savings [9] [10].
• Increasing path methods:
One node or edge is selected as the starting point of a path. At each iteration one edge is selected according to some criterion, and added to one of the ends of the path. A disadvantage of this approach is that it is possible for the ends of the path to be 'far' from one another when the heuristic terminates, i.e. the last edge needed to change the path to a tour may be an expensive edge. There are two groups of increasing path methods - neighbour group [11] [12] and spacefilling curves group [13].
• Subtour insertion methods:
An initial subtour is selected, for example one edge is selected. In each iteration, a node is selected and inserted into the subtour according to some criterion. To insert a node an edge in the subtour is replaced by two edges not in the subtour. The insertion heuristics are described by Johnson and McGeoch [14].
• Combined methods:
These are well-known algorithms based on minimum spanning tree that were introduced by Christofides [15] [16].
The second group consists of tour-improving algorithms that have their roots in TSP papers from the 1950s [17] [18] According to Applegate, '... These heuristics take as input an approximate solution to a problem and attempt to iteratively improve it by moving to new solutions that are close to the original' [1]. Most of these algorithms are described by Aarts and Lenstra [19]. Currently, the most simple tour-improving heuristic used in practice is 2-Opt heuristic [20]. It was introduced and described by Flood [21], Croes [18] and Bock [17]. The later algorithm of Lin and Kernighan (LKH) [22] appeared on the basis of k-Opt tour-finding approach. Also this group consists of simulated annealing [23], evolutionary [24] and swarm intelligence methods [25] [26].
IV. Algorithms
In this paper, most of the methods mentioned in Part III are implemented and assessed through experiments.
It should be noted that Savings algorithm was not chosen because of its high computational complexity - 0(N3 log N) -in comparison with Greedy one - 0(N2 logN). In addition, it was mentioned in [1] that for most instances Greedy algorithm gives better results than Savings.
In order to restrict our investigation, it was decided to choose only two types of tour improving algorithms - the most simple one (2-Opt) and the most perspective one (LKH).
The list of used algorithms for Metric TSP is following:
1) Nearest Neighbour (NN).
The key to NN is to initially choose a random vertex and to add repeatedly the nearest vertex to the last appended, unless all vertices are used [21].
2) Double Ended Nearest Neighbour (DENN).
This algorithm is a modification of NN. Unlike NN, not only the last appended vertex is taken into consideration, so the closest vertex to both of endpoints in the tour is added [21].
3) Greedy (GRD).
The Greedy heuristic constructs a path by adding the shortest edge to the tour until a cycle with A edges, A < N, is created, or the degree of any vertex exceeds two [27].
4) Nearest Addition (NA), Simple Cheapest Insertion (SCI), Simple Nearest Segment Insertion (SNSI).
The fundamental idea of NA, SCI and SNSI is to start with an initial subtour made of the shortest edge and to add repeatedly other vertices using various rules. Depending on the algorithm the vertex not yet in the cycle should be inserted so that:
a) In NA it is the closest to any node in the tour;
b) In SCI its addition to the tour gives a minor increment of its length;
c) In SNSI distance between the node and any edge in the tour is minimal.
The previous step should be repeated until all vertices are added to the cycle [19].
5) Nearest Insertion (NI), Cheapest Insertion (CI), Nearest Segment Insertion (NSI).
Algorithms NI, CI and NSI are variations of NA, SCI and SNSI respectively. The feature of modified methods is additional computation that selects the best place for each inserting node.
6) Double Minimum Spanning Tree (DMST).
DMST method is based on the construction of a minimal spanning tree (MST) from the set of all vertices. After MST is built, the edges are doubled in order to obtain an Eulerian cycle, containing each vertex at least once. Finally, a Hamiltonian circuit is made from an Eulerian circuit by sequential (or greedy) removing occurrences of each node [16].
7) Double Minimum Spanning Tree Modified (DMST-M).
This algorithm is a modification of DMST. Unlike DMST, it is necessary to remove duplicate nodes from an Eulerian cycle using triangle inequality instead of greedy method.
8) Christofides (CHR).
This method is a modification of DMST that was proposed by Christofides [15]. The difference between CHR and DMST is addition of minimum weight matching calculation to the first algorithm.
9) Moore Curve (MC).
This is recursive geometric method. Vertices are sorted by the order they are located on the plane. Only the two-dimensional example of Moore curve is implemented. Figure 2 shows the order of the cells after one, two and three subdivision steps respectively [28] .
Fig. 1. The order for the Moore curve after 1, 2 and 3 subdivision steps.
10) 2-Opt.
The main idea behind 2-Opt is to take a tour that has one or more self-intersections and to remove them repeatedly. In mathematical terms, edges ab and cd should be deleted and new edges ac and bd should be inserted, if d(a, b) + d(c, d) > d(a, c) + d(b, d) (Fig. 1) [19].
a
a
2-Opt -►
'd wb
Fig. 2. 2-Opt modification.
11) Helsgaun's Lin and Kernighan Heuristic (LKH). LKH uses the principle of 2-Opt algorithm and generalizes it. In this heuristic, the fc-Opt, where k = 2..4n, is applied,
so the switches of two or more edges are made in order to improve the tour. This method is adaptive, so decision about how many edges should be replaced is taken at each step [22].
It should be noted that because of complexity of LKH algorithm, it was not implemented by the authors of research. The original open source code [29] was used to carry out experiments. All the parameters were not changed, so they were used by default.
Estimated upper bounds for the algorithms can be
f(s)
calculated as are the ratio of —— (see Table 1). According to
f(s o)
[30], for any k--Opt algorithm, where k < N/4, problems may be constructed such that the error is almost 100%. So 2-Opt and LKH algorithms have approximate upper bound 2. Running times of the algorithms are represented in Table 2.
TABLE I.
Upper-bound estimates of algorithms
# Algorithm Upper-bound estimate
1 NN DENN 0.5\\og2N + 11
2 GRD 0.5[log2 N + 11
3 NA, SCI, SNSI NI, CI, NSI 2 2 - N
4 DMST, DMST-M 2 2 - N
5 CHR 3 2
6 2-Opt ~ 2
7 LKH ~ 2
8 MC log N
TABLE II. Running time of algorithms
# Algorithm Running time
1 NN DENN 0(N2)
2 GRD 0(N2 logN)
3 NA, SCI, SNSI NI, CI, NSI 0(N2)
4 DMST, DMST-M 0(N2)
5 CHR 0(N3)
6 2-Opt 0(N2)
7 LKH 0(N2
8 MC O(NlogN)
V. Wave function as the data structure in use
The key aspect of realization of the algorithms is data structure they make use of. In the study the Hamiltonian circuits are presented as wave functions, which are fixed-length one-dimensional arrays [31]. Their sizes are equal to N.
The wave function data structure decreases the computational time of the algorithms in comparison with standard C++ template std::vector<T> (Fig. 3).
Number of vertices, N Fig. 3. The time-efficiency comparison of wave and vector data structures.
VI. Experimental research
This section documents details of the research methodology. The experiment is carried out on a 1.3 GHz Intel Core i5 MacBook Air. It includes the qualitative performance and the run time efficiency of the current implementations.
Heuristics are implemented in C++. VLSI data sets from an open library TSPLIB [2] are selected as input data for algorithms. There are 102 instances in the VLSI collection that range in size from 131 vertices up to 744,710 vertices. There is one data set for each number of vertices. The integer Euclidean metric distance is used, so coordinates of nodes and distances between them have integer values, thus without loss of generality (4) is transformed into:
di(v,w) = [J\x(v) - x(w)\2 + \y(v) - y(w)\2 + 0.5J
The computational experiment corresponds to the following scenario:
for algorithm m in range [1...9, 11] (see IV) for data set N in range [1.102] for i in range [1.11]
fet(m,N), ft.(m,N) are calculated if i > 1 then
fF . (m,N) is memorized
' t mtn v y
ft (m,N) is calculated
' t sum v y
ft-avg ^^
ftsum(m,N) 10
// 2-Opt stage
if m\ = 11 (m is not LKH)
f (m + 2-Opt,N), ft(m + 2-Opt, N) are calculated
E{fmn (m,N)) ,<r (^J™^) for an N - ?
max(fSmin(m,N)} ,min(femin(m,N)) for all N - ?
Metrics used in scenario have following meanings: f£min (m, N) - best qualitative performance of m, ftsum (m, ^) - accumulative running time of m (sec), ftavg (m, N) - average running time of m (sec), f£(m + 2-Opt,N) - qualitative performance of m + 2-Op t,
ft (m + 2-0pt,N) - running time of m + 2-0pt,
E^femtn (m, N)) - expected value of qualitative performance of m for all N,
o(fSmin(m, N)^ - standard deviation of qualitative performance of m for all N,
max (femin(m, N)^, min (femin(m, N)^ - maximum and
minimum values of qualitative performance of m for all N.
Qualitative performance metrics are represented in Table 3. Table color scheme varies from green (the best result in a column) to red (the worst value in a column).
The line graphs that illustrate f£ (m, N) and ft (m, N) of the algorithms are provided (see Fig. 4, Fig. 5). There are no MC algorithms as their key figures are outstanding (Table 3).
Fig. 4. Qualitative performance of the algorithms.
Fig. 5. Running time of the algorithms.
TABLE III.
Qualitative performance metrics of the algorithms
Algorithm E(£) max £ min £
LKH 5/10 0,07% 0,05% 0,23% 0,00%
CHR + 2-Opt 5,77% 0,68% 11,02% 3,47%
GRD + 2-opt 6,22% 0,70% 9,89% 4,69%
DENN + 2-opt 10,95% 4,42% 23,51% 3,82%
NN + 2-opt 11,44% 1,87% 24,77% 4,26%
CHR 12,61% 1,08% 17,79% 9,31%
CIM + 2-opt 13,05% 2,29% 21,86% 6,74%
NIM + 2-opt 14,60% 5,53% 29,66% 5,86%
DMST-M + 2-opt 16,08% 8,39% 40,61% 4,80%
NSIM + 2-opt 17,63% 5,82% 33,65% 8,92%
GRD 18,12% 2,91% 31,34% 10,30%
DMST + 2-opt 19,08% 9,54% 39,12% 6,91%
CIM 20,31% 1,44% 27,54% 12,46%
DENN 23,28% 1,62% 32,53% 13,88%
NN 23,94% 1,64% 30,97% 12,94%
CI 26,25% 2,70% 33,05% 17,94%
NIM 27,98% 1,89% 35,29% 14,89%
DMST-M 32,41% 3,15% 41,68% 18,55%
MC + 2-opt 32,41% 22,54% 177,83% 6,21%
NSIM 36,23% 4,23% 48,17% 19,15%
DMST 40,09% 2,34% 48,88% 33,16%
NSI 43,55% 5,64% 55,61% 25,89%
NI 52,46% 3,19% 60,94% 36,77%
MC 63,93% 25,58% 242,41% 33,07%
VII. PARETO-OPTIMAL ALGORITHMS
We decided to select six different data sets with N = {1083, 5087, 10150, 30440, 52057, 104815} to plot charts that illustrate Pareto-optimal algorithms.
The charts are shown below (see Fig. 6, Fig. 8, Fig. 10, Fig. 12, Fig. 14, Fig. 16). The name of TSPLIB instance is shown in chart title. The horizontal axis represents the time performance of methods in seconds. The vertical axis shows the gap between optimal and obtained solutions, expressed in percent. Pareto-optimal methods are highlighted in red. The points which are represented by Pareto solutions are bigger than non-Pareto-optimal solutions.
There are five charts (see Fig. 7, Fig. 9, Fig. 11, Fig. 13, Fig. 15), except one with N = 104814, where not all algorithms are compared. These auxiliary charts are enlarged copies of their originals. Their role is to graphically illustrate Pareto-optimal algorithms at scale-up.
30%
1 20% I
10%
0%
4 XIT1083
• •
• •
1 • *
• • • •
л • •
î 1
Time, яес
Fig. 6. Pareto-optimal algorithms for XIT1083.tsp, N = 1083.
60%
(]%
* ' XIT1083 i (not full)
• •
a •
• • щ
0
Time, sec
«NI
■К + 2hj|X
*SNSI
•NSI + ?-0[K
•mi
•a
•CI + •DUST • :>.1>; 1 ■DMST-M +
• DMST + 2-üfn •ffîfl
NN + 2-оцн
• DENN
• DENN + 3-ofrf »IRL»
•GRD + 1-apt •MC
• мг: ■• I
•cuk
•CHR + 2-itpt
• NA •NI
• SNS1
• NSI
•sei
• CI
• DMST
• DMST-M
• NN
■ DENN
• GRD
• MC
300%
250%
I 200%
I
f.150%
1 100%
&
FQM5087
9
•
f Я w Л. • •4 •
200 300
Time, sec
• NA •NI
N1 + 2-opi
• SNSI •NSI
•NSI - 2-Opl
#SCI
•ci
•CI+2-opt
• DM ST
• DMST-M
• DMST-M -2-opt « DMST-2-opt •NN
NN + 2-opt
• DENN
• DENN + 2-opl
• GRD
■GRD + 2-opt #MC
• MC ■+ 2-Opt
• CHR
•CHR - 2-opl •l.KH 5 1«
Fig. 8. Pareto-optimal algorithms for FQM5087.tsp, N = 5087.
g 200% J
£J50%
S
I 100%
I
50% 0%
FQM5087 (not full)
1
• • • \ 4 > Ш
Time, sec
■NA •N1
• SNS] •NSI •SCI •CI
• DM ST •DMST-M
• NN •DENN •GRD
• MC
Fig. 9. Pareto-optimal algorithms for FQM5087.tsp, N = 5087 (scale-up).
70%
50%
20% 10% 0%
У ÎMC1U15II
■ »
«
f » t в
• «
w • • 9
A 9 •
500
: ooo
Time, sec
I 500
NJ + ♦SNSI
■ NSI
•NSI + 2-Otrf •SCI •Ci
•CI + Jh^i
■ DM ST •DMST-M •DMST-M+2-ofJ
* DMST + 2-opl ♦NN
NN +2-^* •DENN •DENN + 2-Ofrf •CRD •GRD + 3-a|rf
•mc
• MI :>•I
•CHR
2 000 »CHR + 2-ofn ♦t.kiiF.m
Fig. 10. Pareto-optimal algorithms for XMC10150.tsp, N = 10150.
Fig. 7. Pareto-optimal algorithms, for XIT1083.tsp, N = 1083 (scale-up).
60%
s; 1 50%
I
| 40% t
¡S 30%
Ci
I 20% 10% 0%
XMC1015Ö (Hot full)
• »
4 i
___1 .j i •
■ • • • •
6 s
Time, sec
10
12
14
• NA ■NI •SNSI •NSI •SCI
• Cl
• DMST
• DMST-M
• NK
• DENN
• CRU
• MC
Fig. 11. Pareto-optimal algorithms for XMC10150.tsp, N = 10150 (scale-up). 70%
60%
10%
0%
» PBH30440
• f •
t
• •
■ • •
r • • •
m
2000
4000
6000
p me iscc
;j] + •SNSI •NSI
• v.: ■ : < -1
•SCI
'ma
•CI + 2-tifiï •DMST •DMST-M
• DMST-M - ï-.K|KI
• DMST+ 2-ïrfn ♦SIN
UN + ♦¡>1 NN
•ih N \ ■ 1-t^l •hc
■ MC + 2-Ofrt 12000 "c™
•lkhs/II
8 40%
I Ë
M
.1 | 20% 3
10%
■ i ■ • FNA52057
• • ■
» __ H • •
• •
0 2000 4000 6000
Time, sec
Fig. 14. Pareto-optimal algorithms for FNA52057.tsp, N = 52057.
60%
50%
Î 40% E
I
^30%
I -
£ 20%
CS
I
10%
¡00 3 50
Time, sec
•NT NI + 2-0)fa
• SNSI
• NSI
• NSI +2-OIII
• SCI
•ci
#CÏ ■ " .|H
• DMST •DMST-M
■ DMST-M 4-Î-ùïiI
• DMST + 2-Ofrf »SS
NN + 2-opi
• DENN + ,'h-III •MC
• MC + 2-Opi
f. t FNA52057 (not full)
• • • •
• • • •
•
•NA « NI •SNSI •NSI •SCI
• Cl •DMST
• DMST-M
• NN
• DENN
• MC
Fig. 15. Pareto-optimal algorithms for FNA52057.tsp, N = 52057 (scale-up).
Fig. 12. Pareto-optimal algorithms for PBH30440.tsp, N = 30440.
70% 60%
I 50%
1 20% Cr
10% 0%
1 PBH3044II (not full)
• •
• •
•
• • •
40 60
Time- sec
•NA ■ N1 •SNSI
• NS1 •SCI
• Cl •DMST
• DMST-M
• NN
• DENN
• MC
Fig. 13. Pareto-optimal algorithms for PBH30440.tsp, N = 30440 (scale-up).
60%
50%
: 30%
I 20% ô
0%
» SRAl 04815
h 9
• m • • ■
• *• • • . « . «
6 000 K 000 Time, sec
10 000 12 000 14 000
•NA
• NI
-Nl + 2-opl
♦SNSI
•NSI
•NSJ + 2-Opi •SCI
•ci
•ci + I-opl
•DMST
•DMST-M
• DMST-M + 2-oftf
• DMST + 2-oirt
•nw
NN +2-QJII
•denn •denn + 2-<ipl |MC
•MC + 2-Opl
Fig. 16. Pareto-optimal algorithms for SRA104815.tsp, N = 104815.
Full results are presented in Table 4. Dash sign '-' is used because some methods have not tested on large N yet. So we cannot claim whether these algorithms are Pareto-optimal or not.
TABLE IV. Pareto-optimal algorithms for 6 groups of N
XIT1083 FQM5087 XMC10150 PBH30440 FNA52057 SRA104815
SCI SCI
CI CI CI CI CI CI
CI+2-Opt CI+2-Opt
NN NN NN NN NN NN
DENN DENN DENN DENN
DENN+ 2-Opt DENN+ 2-Opt
GRD
MC MC MC MC MC MC
CHR - -
LKH LKH LKH LKH - -
The rough estimations can be suggested on the basis of VLSI data sets only:
1. SCI, and CHR methods are efficient for small N < 5000.
2. GRD is one of the bests for N ~ 10000.
3. DENN is high-performance for N > 10000.
4. CI + 2-Opt combination gives good results for large N > 50000.
5. DENN + 2-Opt is efficient on N ~ 30000 and N ~ 100000.
6. There are three heuristics included in each Pareto-optimal group for different N. They are CI, NN and MC.
Despite the lack of information on LKH algorithm for N = 52057 and N = 104814, this method continues to be the most perspective heuristic. As it can be seen from Table 3, expected value of qualitative performance of LKH is 0.07%. From there, we decided to add LKH to the generalized group of Pareto-optimal algorithms.
Overall, the generalized group of Pareto-optimal algorithms consists of CI, NN, MC and LKH heuristics.
VIII. Conclusion
The presented study is undertaken to determine what heuristics for Metric TSP should be used in specific circumstances with limited resources.
This paper provides an overview of eleven heuristic algorithms implemented in C++ and tested on the VLSI data set. In the course of computational experiments, the comparative figures are obtained and on their basis multi-objective optimization is provided. Overall, the generalized group of Pareto-optimal algorithms consists of CI, NN, MC and LKH heuristics.
In our future work, we are going to fine-tune parameters of LKH method using genetic algorithms of search optimization. Further, it is possible to increase the number of heuristic algorithms, to transit to other types of test data and to conduct experiments using different metrics in order to ensure that a Pareto optimal group is sustainable.
The practical applicability of our findings is to present Pareto optimal algorithms that lead to solutions with maximum accuracy under the given resource limitations. The results can be used for scientific purposes by other researchers and for cost minimization tasks.
References
[1] D. L. Applegate, The Traveling Salesman Problem, Princeton: Princeton University Press, 2006.
[2] Heidelberg University, "TSPLIB," [Online]. Available: https://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/. [Accessed 01 02 2017].
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