My presentation on special issue of Genedis 2016, called Nubacom.

1. A Quantum Inspired GVNS: Some preliminary results
*Christos Papalitsas1 Panayiotis Karakostas2 Kalliopi
Kastampolidou1
1Department of Informatics
Ionian University
2Department of Applied Informatics
University of Macedonia
Genedis, 2016
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 1 / 18

2. Preview of our study
Unconventional Computing
Meta-heuristic procedures
Variable Neighborhood Search
qGVNS
TSP
Experimental results
Conclusion
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 2 / 18

3. Introduction
Unconventional Computing
Unconventional Computing.
New or unusual proposed computational models.
Natural Computing.
1 A part of unconventional computing.
2 A procedure that makes us think naturally about the computational
processes.
Quantum inspired methods.
1 A subsection of natural computing processes.
2 Inspired by quantum physics.
Our work is inspired by quantum computing principles.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 3 / 18

4. Introduction
Meta-heuristic methods An overview
Meta-heuristics: optimization frameworks which can be modified
properly in order to produce heuristic approaches for hard
optimization problems.
Ideal for problems with missing or incomplete information, or with
limited computing capacity.
They do not guarantee a global optimal solution for each category of
problems.
By searching into a large set of feasible solutions, the meta-heuristic
procedures can often find good solutions with less computational
effort.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 4 / 18

5. Introduction
VNS Variable Neighborhood Search(1/2)
VNS is a meta-heuristic method for solving a set of combinatorial
problems and global optimization problems.
This strategy is driven by three principles:
1 a local minimum for a neighborhood structure cannot be a local
minimum to a different neighborhood structure
2 a global minimum is a local minimum for all possible structures of the
neighborhood, and
3 a local minimum is closely related to the total minimum for many
classes of problems.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 5 / 18

6. Introduction
VNS Variable Neighborhood Search(2/2)
The systematic neighborhood change try to find an optimal (or a
close-to-optimal) solution.
VNS heuristic consists of three parts:
1 A shaking procedure, used to escape local optimal solutions.
2 Improvement phase: exploration of neighborhood structures through
different local search moves. systematic neighborhood change to reach
an optimal (or a close-to-optimal) solution.
3 The neighborhood change ,where the following neighborhood structure
is determined.
(an approval or rejection criterion is also applied on the last solution
found.)
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 6 / 18

7. Introduction
qGVNS (1/3) An overview
The quantum-inspired GVNS (qGVNS) consists of:
A VND local search.
1 the relocate local search operator: solutions obtained by nodes
relocation in tour.
2 2-opt: solutions obtained by break and different reconnect of two tour
arcs.
A quantum inspired shaking procedure.
Neighborhood change step.
The main difference between qGVNS and classic GVNS is in the
diversification(shaking) phase.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 7 / 18

8. Introduction
qGVNS (2/3) Quantum inspired perturbation
Perturbation phase adopts quantum computation principles.
Step 1: shaking begins: a quantum register generate necessary qubits.
Step 2: generated qubits produce the corresponding components.
(These components will be equal or higher than the number of the
nodes in the tour)
Step 3: algorithm serially choose the components, and put them in a
1 x n vector.
Step 4: each one will be matched to each node of our current
solution.(flag) Note, that components can be 0 ≤ C ≤ 1
Step 5: sorting the first vector will affect the node’s order in the
solution vector.
This is the route occurring after the shaking move, driving our exploration
effort in another search space.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 8 / 18

9. Introduction
qGVNS (3/3) qGVNS Pseudocode
Data: an initial solution
Result: an optimized solution
Initialization of the feasibility distance matrix
begin
X < − Nearest Neighbor heuristic;
repeat
X’ < − Quantum-Perturbation(X)
X” < − VND(X’)
if X” is better than X’ then
X < − X”
end
until optimal solution is found or time limit is met;
end
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 9 / 18

10. Introduction
TSP
Numerous applications in many different areas (Logistics, Artificial
Intelligence, Machine Learning, Software Technology etc).
Symmetric TSP, Asymmetric TSP, Multiple TSP.
TSP is the problem of finding the shortest Hamiltonian cycle.
NP-hard.
Significant in various fields, such as operational research and
theoretical computer science.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 10 / 18

11. Introduction
TSP Formulation
An undirected graph G = (V, E), if symmetrical.
A directed graph G = (V, A), if asymmetric.
The set V = {1,2,3, ..., n} is the set of vertices.
A = {(i, j): i, j ∈V, i < j} is the set of the edges.
By n we denote a number of cities (nodes).
A cost matrix C = (ci,j ) defined on E.
The total number of possible paths is equal to (n-1)!/2.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 11 / 18

12. Introduction
Experimental results (1/5)
Proposed qGVNS was implemented in Fortran.
Applied on 16 benchmark TSPLib instances (9 sTSPs and 7 aTSPs).
Stopping criteria:
1 an optimal solution
2 60-second limit
Each instance was solved 5 times.
a comparison between first and best improvement search strategies of
qGVNS and GVNS.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 12 / 18

13. Introduction
Experimental results (2/5)
Table: qGVNS on sTSP instances
Best Improvement First Improvement
Problem Av. Value Av. CPU Time Best Value Av. Value Av. CPU Time Best Value
bayg29 1610 18.84s 1610 1610 20.27s 1610
bays29 2020 1.23s 2020 2020 26.47s 2020
fri26 937 0.27s 937 937 0.59s 937
gr17 2085 0.12s 2085 2085 0.03s 2085
gr24 1272 5.52s 1272 1272 2.29s 1272
ulysses16 6859 0.01s 6859 6859 0.05s 6859
ulysses22 7013 0.01s 7013 7013 0.02s 7013
gr48 5049.4 37.67s 5046 5056.8 53s 5046
hk48 11508.6 43.3s 11461 11519 60s 11470
The best improvement search strategy of qGVNS has been the most
efficient.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 13 / 18

14. Introduction
Experimental results (3/5)
Table: qGVNS on aTSP instances
Best Improvement First Improvement
Problem Av. Value Av. CPU Time Best Value Av. Value Av. CPU Time Best Value
br17 39 0 39 39 0 39
ftv33 1357.8 60s 1345 1444 60s 1420
ftv35 1544.6 60s 1516 1639.4 60s 1615
ftv38 1616.6 60s 1593 1749.4 60s 1635
ftv44 1763.8 60s 1739 1937 60s 1895
p43 5654 60s 5625 5661.4 60s 5636
ry48p 14698.2 60s 14651 14991 60s 14824
The best improvement search strategy of qGVNS has been the most
efficient.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 14 / 18

15. Introduction
Experimental results (4/5)
Table: Optimal Values of sTSP instances
Problem Optimal qGVNS
bayg29 1610 1610
bays29 2020 2020
fri26 937 937
gr17 2085 2085
gr24 1272 1272
ulysses16 6859 6859
ulysses22 7013 7013
gr48 5048 5049.4
hk48 11461 11508.6
qGVNS detects the optimal solution at seven out of nine in total
benchmarks.
Table: Optimal Values of aTSP instances
Problem Optimal qGVNS
br17 39 39
ftv33 1286 1357.8
ftv35 1473 1544.6
ftv38 1530 1616.6
ftv44 1613 1763.8
p43 5620 5654
ry48p 14422 14698.2
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 15 / 18

16. Introduction
Experimental results (5/5)
Table: Comparison between qGVNS and GVNS
Problem Av. Value(qGVNS) Best(GVNS) Problem Av. Value(qGVNS) Best(GVNS)
bayg29 1610 1653 br17 39 39
bays29 2020 2069 ftv33 1357.8 1489
fri26 937 969 ftv35 1544.6 1791
gr17 2085 2085 ftv38 1616.6 1778
gr24 1272 1278 ftv44 1763.8 2014
ulysses16 6859 6859 p43 5554 5629
ulysses22 7013 7013 ry48p 14698.2 15134
gr48 5049.4 5325
hk48 11508.6 11884
qGVNS has achieved better results than the equivalent GVNS, for small
TSP instances. The best value achieved between first and best
improvement strategies is selected as the best result of each instance
(sTSP and aTSP) with GVNS.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 16 / 18

17. Introduction
Conclusion
qGVNS for solving the TSP has been introduced.
We presented our methodology through experimental results for small
dimension benchmark problems.
Our method seems promising and it was shown to outperform the
equivalent classic GVNS.
Future work:
1 Extended comparative study between classic GVNS and qGVNS.
2 Apply different neighborhood structures and neighborhood change
moves.
3 The current qGVNS scheme may be applied on many TSP variants.
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 17 / 18

18. Appendix References
For Further Reading I
Mladenovic, N., Hansen, P.: Variable neighborhood search.
Computers Operations Research 24(11), 10971100 (1997). DOI
10.1016/s0305-0548(97)00031-2
Narayanan, A., Moore, M.: Quantum-inspired genetic algorithms.
Proceedings of IEEE International Conference on Evolutionary
Computation DOI 10.1109/icec.1996.542334
Papalitsas, Ch., Giannakis, K., Andronikos, Th., Theotokis, D.,
Sifaleras, A.: Initialization methods for the tsp with time windows
using variable neighborhood search. In:IEEE Proc. of the 6th
International Conference on Information, Intelligence, Systems and
Applications (IISA 2015), 6-8 July, Corfu, Greece (2015)
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 18 / 18

19. Appendix References
Any Questions?
Ch. Papalitsas et al. qGVNS: Some preliminary results Genedis, 2016 18 / 18