the traveling salesman problem for cubic graphs
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8/14/2019 The Traveling Salesman Problem for Cubic Graphs
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The Traveling Salesman
Problem for Cubic Graphs
Journal of Graph Algorithms and Applications vol.
11, no. 1, pp. 61–81 (2007)
David Eppstein
Adviser: Yue-Li Wang
As presented by Ying-Jhih Chen
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Outline
• The TSP for cubic graph
• Special Rule
–Triangle
– 4-cycle
• Example
• Analysis• Degree Four
• Hamiltonian Cycles
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Traveling Salesman Problem
• Given a number of cities and the costs of
traveling from any city to any other city.
•What is the least-cost round-trip route thatvisits each city exactly once and then
returns to the starting city?
3
1
1
1
2
4 3
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Cubic
• 3-regular graph.
• E. g.
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This Algorithm
• The solution of TSP in a graph of
degree at most three.
• n vertices, in time O(2n/3) ≈ 1.260n.
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Triangle (1/2)
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Triangle (2/2)
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4-cycle
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Example
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2
2
2
2
1
5
1
1
1
4
33
2
2
2
2
1
5
1
1
1
4
3
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2
2
2
1
5
1
1
1
4
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Case
1:
Case
2:
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Case 2 (1/2)
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2
2
2
2
1
5
1
14
33
3
65
3
3
2
1
2
1
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Case 2 (2/2)
11
3
65
3
3
2
1
2
1
Case 2-1:
3
65
33
2
1
2
1
Case 2-2:
Cost : 21 Cost : 17
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Case 1
12
2
2
2
2
1
5
1
1
1
4
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Case 1-1: Case 1-2:
2
2
2
2
1
5
1
1
1
4
3
3
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Case 1-1
13
2
2
2
2
1
5
1
1
1
4
33
2 4
Cost: 16
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Case 1-2
14
2
2
2
2
1
5
1
1
1
4
3
1
18
16 2
2
2
2
1
5
1
1
1
4
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Cost: 17
Solution: 16 ( case 1-1)
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Algorithm Review
• TSP ( G, F):
• 1. Repeat the following steps until one of the steps returns or none of them
applies:
– (a)~(h)
– (i) If G contains a triangle xyz, …
– (j) If G contains a cycle of four unforced edges, …• 2. If G \ F forms a collection of disjoint 4-cycles, perform the following steps.
• (a)~(e)
• 3. Choose an edge yz according to the following cases:
• (a)~(c)
• 4. TSP(G, F {yz})∪
• 5. TSP(G \ {yz}, F)
• 6. Return the minimum of the numbers returned by the two recursive calls.
(step 4, 5)
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Analysis
• A nonstandard measure of the size of a
graph G: let s(G, F ) = | ν (G)| − |F | − |C |
• C denotes the set of 4-cycles of G that
form connected components of G \ F .
• Clearly, s ≤ n,
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Chain (1/2)
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s(G1,F
1) = s(G, F ) - 2
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Chain (2/2)
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s(G2, F 2) = s(G, F ) - 5
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Cycle Length is Six or More
(1/2)
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s(G1, F
1) = s(G, F ) - 3
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Cycle Length is Six or More
(2/2)
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s( G2 , F
2 ) = s(G, F ) -3
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4-cycle (1/3)
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s ( G1, F
1) = s(G, F ) - 3
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4-cycle (2/3)
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4-cycle (3/3)
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ν (G) - 3
s (G2 , F
2 ) = s(G, F ) - 3
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Case
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s(G1, F
1) = s(G, F ) -3 s(G
2 , F
2 ) = s(G, F ) -3
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Time Complexity
• T(s) ≤ 1 + max{ sO(1),T(s − 1), 2T(s − 3), T(s
− 2) + T(s − 5)}.
• T(s) = O(2s/3).
• s is at most n.
• A bound of O(2n/3)
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Degree Four
• Randomly splitting vertices
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Let f denote the number of degree four vertices in G.
The algorithm has probability (2/3)f of finding the correct TSP solution.
There are (3/2)f repetitions.
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Time for Degree Four
• O((3/2)f 2n/3) = O((3/2)n 2n/3) = O((3/2)3n/3 2n/3) =
O((27/8)n/3 2n/3) = O((27/4)n/3) ≈ 1.890n
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Listing All Hamiltonian Cycles
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To find a Hamiltonian cycle of a 400-vertex 3-regular graph, dual to a
triangulated torus model in approximately two seconds on an 800 MHz
PowerPC G4 computer.
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Graphs with Many Hamiltonian
Cycles (1/2)
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Graphs with Many Hamiltonian
Cycles (2/2)
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Cubic graph with 2n/3 Hamiltonian cycles. (4n/6 =
2n/3 )
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References (1/2)
• [1] E. M. Arkin, M. Held, J. S. B. Mitchell, and S. S. Skiena. Hamiltonian triangulations
for fast rendering. The Visual Computer, 12(9):429–444,1996.
• [2] R. Beigel. Finding maximum independent sets in sparse and general graphs. In Proc.
10th ACM-SIAM Symp. Discrete Algorithms, pages S856–S857, January 1999.
• [3] J. M. Byskov. Chromatic number in time O(2.4023n) using maximal independent
sets. Technical Report RS-02-45, BRICS, December 2002.
• [4] D. Eppstein. Quasiconvex analysis of backtracking algorithms. ACM Trans.
Algorithms. To appear.
• [5] D. Eppstein. Small maximal independent sets and faster exact graph coloring. J.
Graph Algorithms and Applications, 7(2):131–140, 2003.
• [6] D. Eppstein and M. Gopi. Single-strip triangulation of manifolds with arbitrary
topology. Eurographics Forum, 23(3):371–379, 2004.
• [7] M. R. Garey and D. S. Johnson. Computers and Intractability: a Guide to the Theoryof NP-Completeness. W. H. Freeman, 1979.
• [8] D. S. Johnson and L. A. McGeoch. The traveling salesman problem: a case study in
local optimization. In E. H. L. Aarts and J. K. Lenstra, editors, Local Search in
Combinatorial Optimization, pages 215–310. John Wiley and Sons, 1997.
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References (2/2)
• [9] E. L. Lawler. A note on the complexity of the chromatic number problem.
Information Processing Letters, 5(3):66–67, August 1976.
• [10] B. D. McKay and G. F. Royle. Constructing the cubic graphs on up to 20 vertices.
Ars Combinatorica, 21(A):129–140, 1986.
• [11] J. M. Robson. Algorithms for maximum independent sets. J. Algorithms,
7(3):425–440, September 1986.
• [12] N. Schemenauer, T. Peters, and M. L. Hetland. Simple generators. Python
Enhancement Proposal 255, python.org, May 2001.
• [13] R. E. Tarjan and A. E. Trojanowski. Finding a maximum independent set. SIAM
J. Comput., 6(3):537–546, September 1977.
• [14] G. van Rossum et al. Python Language Website. http://www.python.org/.
• [15] B. Vandegriend. Finding Hamiltonian Cycles: Algorithms, Graphs and
Performance. Master’s thesis, Univ. of Alberta, Dept. of Computing Science, 1998.
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