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Algorithm Design 6

Algorithm Design 6. Greedy TSP from Test 1 - Problem 1. tour length = 40 (assumes 0's = no connection). A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1

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Algorithm Design 6

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  1. Algorithm Design 6

  2. Greedy TSP from Test 1 - Problem 1 tour length = 40 (assumes 0's = no connection) A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0

  3. Improved TSP from Test 1 - Problem 1 tour length = 39 (assumes 0's = no connection) A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0

  4. Greedy TSP from Test 1 - Problem 1 tour length = 33 (assumes 0's = zero cost transition) A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0

  5. Improved TSP from Test 1 - Problem 1 tour length = 31 (assumes 0's = zero cost transition) A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0

  6. Alternative Improved TSP from Test 1 - Problem 1 tour length = 31 (assumes 0's = zero cost transition) A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0

  7. Problem Analysis A combined effort of 1000's of trials were conducted. No one found a tour length shorter than 31. What can be deduce from this? Is there a better tour? There are 20! possible tours of these 21 cities. A brute-force evaluation of all of the tours would require apx. 220 operations. Can we reduce the complexity (size) of this problem, assuming that the 0's are valid zero-cost transitions? Since these transitions do not add to the total tour length and the remaining paths are of similar values (i.e. choosing these transitions does not force excessively large distances for the remaining transitions), we can force the zero-cost transitions before handing to problem over to our algorithms.

  8. Forcing Transitions A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0 MC ND OE PF QG RH SI TJ UK

  9. Forcing Transitions A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0 C D E F G H I J K M N O P Q R S T U

  10. MC ND OE PF QG RH SI TJ UK 9 3 5 6 4 2 7 5 1 5 3 7 6 4 2 1 4 2 6 8 7 9 8 6 4 1 3 4 3 5 4 3 2 4 6 7 6 4 3 6 5 7 6 5 8 2 4 3 2 3 4 3 5 4 9 8 7 6 5 8 4 6 3 3 4 2 6 6 6 6 8 7 8 9 8 7 1 1 1 1 9 5 4 3 7 5 6 4 5 7 4 2 2 3 1 1 7 8 6 7 5 4 6 5 7 8 6 9 Forcing Transitions A B C D E F G H I J K L A 0 4 3 5 7 5 3 1 6 4 5 7 B 5 0 3 2 1 5 4 3 2 4 5 6 C 7 6 0 8 7 6 9 8 7 6 5 4 D 5 4 7 0 8 3 9 3 9 4 3 2 E 4 5 3 5 0 6 7 5 6 5 4 2 F 4 3 2 3 4 0 4 5 4 3 2 1 G 2 3 2 1 2 3 0 3 2 3 2 3 H 9 8 7 8 9 8 7 0 7 7 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 J 1 3 2 6 1 3 2 9 4 0 7 8 K 7 5 4 7 8 7 6 8 7 9 0 9 L 1 2 1 1 2 1 2 1 1 2 1 0 A B MC ND OE PF QG RH SI TJ UK L A 0 4 9 3 5 6 4 2 7 5 1 7 B 5 0 5 3 7 6 4 2 1 4 2 6 MC 7 6 0 8 7 9 8 6 4 1 3 4 ND 5 4 4 0 5 4 3 2 4 6 7 2 OE 4 5 6 4 0 6 5 7 6 5 8 2 PF 4 3 2 4 3 0 3 4 3 5 4 1 QG 2 3 9 8 7 6 0 8 4 6 3 3 RH 9 8 3 4 2 6 6 0 6 8 7 7 SI 3 6 8 9 8 7 1 1 0 1 9 2 TJ 1 3 5 4 3 7 5 6 4 0 7 8 UK 7 5 4 2 2 3 1 1 7 8 0 9 L 1 2 7 5 4 6 5 7 8 6 9 0 M N O P Q R S T U

  11. TSP Reduced from 21 Cities to 12 Cities A B MC ND OE PF QG RH SI TJ UK L A0 4 9 3 5 6 4 2 7 5 1 7 B 5 0 5 3 7 6 4 2 1 4 2 6 MC 7 6 0 8 7 9 8 6 4 1 3 4 ND 5 4 4 0 5 4 3 2 4 6 7 2 OE 4 5 6 4 0 6 5 7 6 5 8 2 PF 4 3 2 4 3 0 3 4 3 5 4 1 QG 2 3 9 8 7 6 0 8 4 6 3 3 RH 9 8 3 4 2 6 6 0 6 8 7 7 SI 3 6 8 9 8 7 1 1 0 1 9 2 TJ 1 3 5 4 3 7 5 6 4 0 7 8 UK 7 5 4 2 2 3 1 1 7 8 0 9 L 1 2 7 5 4 6 5 7 8 6 9 0

  12. Starting from City A Enter file name...tsp_12_reduced.txt Enter name of starting city... A A is city # 0 A B MC ND OE PF QG RH SI TJ UK L A 0 4 9 3 5 6 4 2 7 5 1 7 B 5 0 5 3 7 6 4 2 1 4 2 6 MC 7 6 0 8 7 9 8 6 4 1 3 4 ND 5 4 4 0 5 4 3 2 4 6 7 2 OE 4 5 6 4 0 6 5 7 6 5 8 2 PF 4 3 2 4 3 0 3 4 3 5 4 1 QG 2 3 9 8 7 6 0 8 4 6 3 3 RH 9 8 3 4 2 6 6 0 6 8 7 7 SI 3 6 8 9 8 7 1 1 0 1 9 2 TJ 1 3 5 4 3 7 5 6 4 0 7 8 UK 7 5 4 2 2 3 1 1 7 8 0 9 L 1 2 7 5 4 6 5 7 8 6 9 0 used = [ 1 1 1 1 1 1 1 1 1 1 1 1 ] itour = [ 0 10 6 1 8 7 4 11 3 2 9 5 ] tour = [ A UK QG B SI RH OE L ND MC TJ PF ] Tour Length = 32

  13. Starting from City MC Enter file name...tsp_12_reduced.txt Enter name of starting city... MC MC is city # 2 A B MC ND OE PF QG RH SI TJ UK L 0 4 9 3 5 6 4 2 7 5 1 7 5 0 5 3 7 6 4 2 1 4 2 6 7 6 0 8 7 9 8 6 4 1 3 4 5 4 4 0 5 4 3 2 4 6 7 2 4 5 6 4 0 6 5 7 6 5 8 2 4 3 2 4 3 0 3 4 3 5 4 1 2 3 9 8 7 6 0 8 4 6 3 3 9 8 3 4 2 6 6 0 6 8 7 7 3 6 8 9 8 7 1 1 0 1 9 2 1 3 5 4 3 7 5 6 4 0 7 8 7 5 4 2 2 3 1 1 7 8 0 9 1 2 7 5 4 6 5 7 8 6 9 0 used = [ 1 1 1 1 1 1 1 1 1 1 1 1 ] itour = [ 2 9 0 10 6 1 8 7 4 11 3 5 ] tour = [ MC TJ A UK QG B SI RH OE L ND PF ] Tour Length = 24 A B MC ND 0E PF QG RH SI TJ UK L

  14. Apply Tour to Original 21 City Problem A B C D E F G H I J K L M N O P Q R S T U A 0 4 3 5 7 5 3 1 6 4 5 7 9 3 5 6 4 2 7 5 1 B 5 0 3 2 1 5 4 3 2 4 5 6 5 3 7 6 4 2 1 4 2 C 7 6 0 8 7 6 9 8 7 6 5 4 6 8 7 9 8 6 4 1 3 D 5 4 7 0 8 3 9 3 9 4 3 2 4 3 5 4 3 2 4 6 7 E 4 5 3 5 0 6 7 5 6 5 4 2 6 4 3 6 5 7 6 5 8 F 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 2 3 4 3 5 4 G 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 5 8 4 6 3 H 9 8 7 8 9 8 7 0 7 7 8 7 3 4 2 6 6 6 6 8 7 I 3 6 5 4 7 3 1 8 0 1 4 2 8 9 8 7 1 1 1 1 9 J 1 3 2 6 1 3 2 9 4 0 7 8 5 4 3 7 5 6 4 5 7 K 7 5 4 7 8 7 6 8 7 9 0 9 4 2 2 3 1 1 7 8 6 L 1 2 1 1 2 1 2 1 1 2 1 0 7 5 4 6 5 7 8 6 9 M 7 6 0 8 7 6 9 8 7 6 5 4 0 8 7 9 8 1 4 1 3 N 5 4 7 0 8 3 7 3 9 4 3 2 4 0 5 4 3 2 4 6 7 O 4 5 3 5 0 6 7 5 6 5 4 2 6 4 0 6 5 7 6 5 8 P 4 3 2 3 4 0 4 5 4 3 2 1 2 4 3 0 3 4 3 5 4 Q 2 3 2 1 2 3 0 3 2 3 2 3 9 8 7 6 0 8 4 6 3 R 9 8 7 8 9 1 7 0 7 7 8 7 3 4 2 6 6 0 6 8 7 S 3 6 5 4 7 3 1 8 0 1 4 2 8 8 8 7 1 9 0 1 9 T 1 3 2 6 1 3 5 9 4 0 7 8 5 4 3 7 5 6 4 0 7 U 7 5 4 7 8 7 6 8 7 9 0 9 4 4 2 3 1 1 4 8 0 tour = [ A UK QG B SI RH OE L ND PF MC TJ ]

  15. Discussion Could this method be applied to preselect smaller non-zero values? Would this work with an undirected TSP (i.e. A->B = B->A)? Would this work with Euclidean TSP? Does this method guarantee a reduced tour length (if one exists)? Does this guarantee optimality? Is there a shorter tour than 24? Under what conditions could this method be applied effectively?

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