Cycle Mutation: Evolving Permutations via Cycle Induction
Abstract
:1. Introduction
2. Background
2.1. Permutation Mutation Operators
2.2. Permutation Cycles
2.3. Cycle Crossover (CX)
2.4. Test Problems
2.4.1. TSP
2.4.2. QAP
2.4.3. LCS
3. Methods
3.1. Cycle Mutation
3.1.1. Shared Notation and Operations
Algorithm 1 $\mathrm{CreateCycle}(p,\mathrm{indexes})$ 

Algorithm 2 $\mathrm{Sample}(n,k)$ 

Algorithm 3 $\mathrm{InsertionSample}(n,k)$ 

3.1.2. $\mathrm{Cycle}\left(\mathit{kmax}\right)$
Algorithm 4 $\mathrm{CycleMutation}(p,\mathit{kmax})$ 

3.1.3. $\mathrm{Cycle}\left(\alpha \right)$
Algorithm 5 $\mathrm{CycleMutation}(p,\alpha )$ 

3.1.4. Asymptotic Runtime Summary
3.2. New Measures of Permutation Distance
3.2.1. Cycle Distance
3.2.2. Cycle Edit Distance
3.2.3. KCycle Distance
3.3. Fitness Landscape Analysis
3.3.1. Fitness Landscape Diameter
3.3.2. Fitness Distance Correlation
3.3.3. Search Landscape Calculus
3.3.4. Summary of Fitness Landscape Analysis Findings
4. Results
4.1. TSP Results
4.2. QAP Results
4.3. LCS Results
5. Discussion and Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
CX  Cycle crossover 
EA  Evolutionary algorithm 
ES  Evolution strategies 
FDC  Fitness distance correlation 
GA  Genetic algorithm 
LCS  Largest common subgraph 
QAP  Quadratic assignment problem 
SA  Simulated annealing 
TSP  Traveling salesperson problem 
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Mutation Operator  Worst Case  Average Case 

$\mathrm{Cycle}\left(\mathit{kmax}\right)$  $O(min(n,{\mathit{kmax}}^{2}))$  $O(min(n,{\mathit{kmax}}^{2}))$ 
$\mathrm{Cycle}\left(\alpha \right)$  $O\left(n\right)$  $O(min(n,{(\frac{2\alpha}{1\alpha})}^{2}))$ 
$\mathrm{Swap}$  $O\left(1\right)$  $O\left(1\right)$ 
$\mathrm{Insertion}$  $O\left(n\right)$  $O\left(n\right)$ 
$\mathrm{Reversal}$  $O\left(n\right)$  $O\left(n\right)$ 
$\mathrm{Scramble}$  $O\left(n\right)$  $O\left(n\right)$ 
Mutation Operator  Diameter 

$\mathrm{Cycle}\left(\mathit{kmax}\right),\phantom{\rule{0.166667em}{0ex}}\mathit{kmax}\ge 5$  $\approx 2n/\mathit{kmax}$ 
$\mathrm{Cycle}\left(\mathit{kmax}\right),\phantom{\rule{0.166667em}{0ex}}\mathit{kmax}\le 4$  $max\left(\right)open="\{"\; close="\}">[n/2],\phantom{\rule{0.166667em}{0ex}}\left[\right(n1)/(\mathit{kmax}1\left)\right]$ 
$\mathrm{Cycle}\left(\alpha \right)$  2 
$\mathrm{Swap}$  $n1$ 
$\mathrm{Insertion}$  $n1$ 
$\mathrm{Reversal}$  $n1$ 
$\mathrm{Scramble}$  1 
Mutation Operator  TSP  QAP  LCS 

$\mathrm{Cycle}\left(\alpha \right)$  −0.0569  0.0213  −0.0278 
$\mathrm{Cycle}\left(5\right)$  0.1801  0.1339  −0.5342 
$\mathrm{Cycle}\left(4\right)$  0.1667  0.1737  −0.3984 
$\mathrm{Cycle}\left(3\right)$  0.2482  0.2210  −0.6180 
$\mathrm{Swap}$  0.3318  0.2245  −0.6355 
$\mathrm{Insertion}$  0.5277  0.0305  −0.3547 
$\mathrm{Reversal}$  0.8459  0.0189  −0.0350 
$\mathrm{Scramble}$  0.0117  0.0048  −0.0340 
Mutation Operator  $\mathsf{\Delta}\left[{\mathit{\delta}}_{\mathit{em}}\right]$  $\mathsf{\Delta}\left[{\mathit{\delta}}_{\mathit{ce}}\right]$ 

$\mathrm{Cycle}\left(\alpha \right)$  $(2\alpha )/(1\alpha )$  $(42\alpha )/(1\alpha )$ 
$\mathrm{Cycle}\left(\mathit{kmax}\right)$  $(\mathit{kmax}+2)/2$  $\mathit{kmax}+2$ 
$\mathrm{Swap}$  2  4 
$\mathrm{Insertion}$  $(n+4)/3$  3 
$\mathrm{Reversal}$  $\left[\right(n+1)/3,(n+4)/3]$  2 
$\mathrm{Scramble}$  $(n+1)/3$  $(n+1)/3$ 
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Cicirello, V.A. Cycle Mutation: Evolving Permutations via Cycle Induction. Appl. Sci. 2022, 12, 5506. https://doi.org/10.3390/app12115506
Cicirello VA. Cycle Mutation: Evolving Permutations via Cycle Induction. Applied Sciences. 2022; 12(11):5506. https://doi.org/10.3390/app12115506
Chicago/Turabian StyleCicirello, Vincent A. 2022. "Cycle Mutation: Evolving Permutations via Cycle Induction" Applied Sciences 12, no. 11: 5506. https://doi.org/10.3390/app12115506