Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP
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1 Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP Matthias Englert Heiko Röglin Berthold Vöcking Department of Computer Science RWTH Aachen Oberwolfach 2007 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
2 Traveling Salesperson Problem Traveling Salesperson Problem (TSP) Input: weighted (complete) graph G = (V, E, d) with d : E R. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
3 Traveling Salesperson Problem Traveling Salesperson Problem (TSP) Input: weighted (complete) graph G = (V, E, d) with d : E R. Goal: Find Hamiltonian cycle of minimum length. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
4 Theoretical Results General TSP Strongly NP-hard. Not approximable within any polynomial factor. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
5 Theoretical Results a b c a + b c General TSP Strongly NP-hard. Not approximable within any polynomial factor. Metric TSP Strongly NP-hard. 3/2-approximation [Christofides, (1976)] APX-hard: lower bound of 220/219 [Papadimitriou, Vempala (2000)] Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
6 Theoretical Results a b c a + b c (x 1, y 1 ) (x 2, y 2 ) d(p 1, P 2 ) = (x1 -x 2 ) 2 +(y 1 -y 2 ) 2 General TSP Strongly NP-hard. Not approximable within any polynomial factor. Metric TSP Strongly NP-hard. 3/2-approximation [Christofides, (1976)] APX-hard: lower bound of 220/219 [Papadimitriou, Vempala (2000)] Euclidean TSP Cities R d Strongly NP-hard ( no FPTAS) [Papadimitriou (1977)] PTAS exists [Arora (1996), Mitchell (1996)]. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
7 Experimental Results Numerous experimental studies. TSPLIB contains real-world and random (Euclidean) instances. DIMACS Implementation Challenge [Johnson and McGeoch (2002)]. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
8 Experimental Results Numerous experimental studies. TSPLIB contains real-world and random (Euclidean) instances. DIMACS Implementation Challenge [Johnson and McGeoch (2002)]. Some conclusions: Worst-case results are often too pessimistic. The PTAS is too slow on large scale instances. The most successful algorithms (w. r. t. quality and running time) in practice rely on local search. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
9 2-Opt Heuristic 1 Start with an arbitrary tour. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
10 2-Opt Heuristic 1 Start with an arbitrary tour. 2 Remove two edges from the tour. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
11 2-Opt Heuristic 1 Start with an arbitrary tour. 2 Remove two edges from the tour. 3 Complete the tour by two other edges. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
12 2-Opt Heuristic 1 Start with an arbitrary tour. 2 Remove two edges from the tour. 3 Complete the tour by two other edges. 4 Repeat steps 2 and 3 until no local improvement is possible anymore. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
13 Why 2-Opt? Experiments on Random Euclidean Instances [Johnson and McGeoch (2002)] Approximation Ratio Christofides (for n 10 5 ): Opt (for n 10 6 ): 1.05 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
14 Why 2-Opt? Experiments on Random Euclidean Instances [Johnson and McGeoch (2002)] Approximation Ratio Christofides (for n 10 5 ): Opt (for n 10 6 ): 1.05 Number of Local Improvements of 2-Opt Greedy Starts: Probably O(n) Random Starts: Probably O(n log n) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
15 Running time of 2-Opt: Known and New Results General TSP Euclidean metric Manhattan metric average smoothed worst-case 2 Ω(n) Average-case results: [Chandra, Karloff, Tovey (1999)]. Worst-case results: [Lueker (1975)]. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
16 Running time of 2-Opt: Known and New Results General TSP Euclidean metric Manhattan metric average smoothed worst-case 2 Ω(n) 2 Ω(n) 2 Ω(n) Average-case results: [Chandra, Karloff, Tovey (1999)]. Worst-case results: [Lueker (1975)]. Our results. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
17 Running time of 2-Opt: Known and New Results General TSP Euclidean metric Manhattan metric average Õ(n 10 ) Õ(n 6 ) smoothed worst-case 2 Ω(n) 2 Ω(n) 2 Ω(n) Average-case results: [Chandra, Karloff, Tovey (1999)]. Worst-case results: [Lueker (1975)]. Our results. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
18 Running time of 2-Opt: Known and New Results General TSP Euclidean metric Manhattan metric average Õ(n 10 ) Õ(n 6 ) n 3+o(1) Õ(n 4.33 ) [Õ(n3.83 )] Õ(n 4 ) [Õ(n3.5 )] smoothed worst-case 2 Ω(n) 2 Ω(n) 2 Ω(n) Average-case results: [Chandra, Karloff, Tovey (1999)]. Worst-case results: [Lueker (1975)]. Our results. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
19 Lower Bound 1 Introduction 2 Lower Bound 3 Upper Bound 4 Extensions and Open Problems Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
20 Lower Bound Theorem For every n N, there is a graph in the Euclidean plane with 8n vertices on which 2-Opt can make 2 n+3 14 steps. Gadget G 1 Gadget G n Possible States of a Gadget: (Long,Long), (Long,Short), (Short,Long), (Short,Short) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
21 Lower Bound (Long,Long) = 0 (Long,Short) = 1 (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
22 Lower Bound (Long,Long) = (Long,Short) = 1 (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
23 Lower Bound (Long,Long) = (Long,Short) = 1 (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
24 Lower Bound (Long,Long) = Trigger 1 0 (Long,Short) = 1 (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
25 Lower Bound (Long,Long) = Trigger (Long,Short) = 1 2 (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
26 Lower Bound (Long,Long) = Trigger (Long,Short) = (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
27 Lower Bound (Long,Long) = Trigger (Long,Short) = Trigger 0 (Short,Short) = 2 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
28 Lower Bound (Long,Long) = Trigger (Long,Short) = Trigger 0 (Short,Short) = Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
29 Lower Bound (Long,Long) = Trigger (Long,Short) = Trigger 0 (Short,Short) = Gadget G i is reset 2 i 1 times to (Long,Long) = 0. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
30 Euclidean Embedding of the Gadgets Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
31 Upper Bound 1 Introduction 2 Lower Bound 3 Upper Bound 4 Extensions and Open Problems Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
32 Upper Bound Theorem Assume that n points are placed independently, uniformly at random in the unit square [0, 1] 2. The expected number of 2-Opt steps is bounded by O(n 4+1/3 log n) (for every initial tour and every pivot rule). Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
33 Simple Polynomial Bound Theorem The expected number of 2-Opt steps is bounded by O(n 7 log 2 n). Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
34 Simple Polynomial Bound Theorem The expected number of 2-Opt steps is bounded by O(n 7 log 2 n). Proof. Consider a 2-Opt step (e 1, e 2 ) (e 3, e 4 ). (e 1, e 2, e 3, e 4 ) = l(e 1 ) + l(e 2 ) l(e 3 ) l(e 4 ). Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
35 Simple Polynomial Bound Theorem The expected number of 2-Opt steps is bounded by O(n 7 log 2 n). Proof. Consider a 2-Opt step (e 1, e 2 ) (e 3, e 4 ). (e 1, e 2, e 3, e 4 ) = l(e 1 ) + l(e 2 ) l(e 3 ) l(e 4 ). = min (e 1, e 2, e 3, e 4 ). e 1,e 2,e 3,e 4 Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
36 Simple Polynomial Bound Theorem The expected number of 2-Opt steps is bounded by O(n 7 log 2 n). Proof. Consider a 2-Opt step (e 1, e 2 ) (e 3, e 4 ). (e 1, e 2, e 3, e 4 ) = l(e 1 ) + l(e 2 ) l(e 3 ) l(e 4 ). = min (e 1, e 2, e 3, e 4 ). e 1,e 2,e 3,e 4 # 2-Opt Steps 2n. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
37 Simple Polynomial Bound Theorem The expected number of 2-Opt steps is bounded by O(n 7 log 2 n). Proof. Consider a 2-Opt step (e 1, e 2 ) (e 3, e 4 ). (e 1, e 2, e 3, e 4 ) = l(e 1 ) + l(e 2 ) l(e 3 ) l(e 4 ). = min (e 1, e 2, e 3, e 4 ). e 1,e 2,e 3,e 4 # 2-Opt Steps 2n. Bound by a union bound: There are O(n 4 ) different 2-Opt steps, analyze (e 1, e 2, e 3, e 4 ) for one of them. 1/(n 4 log n). Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
38 Idea for Improvement The bound is too pessimistic: Not every step yields the smallest possible improvement 1/(n 4 log n). Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
39 Idea for Improvement The bound is too pessimistic: Not every step yields the smallest possible improvement 1/(n 4 log n). Consider two consecutive steps: They yield + 2 > 2. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
40 Idea for Improvement The bound is too pessimistic: Not every step yields the smallest possible improvement 1/(n 4 log n). Consider two consecutive steps: They yield + 2 > 2. Consider linked pair: (e 1, e 2 ) (e 3, e 4 ) and (e 3, e 5 ) (e 6, e 7 ). Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
41 Idea for Improvement The bound is too pessimistic: Not every step yields the smallest possible improvement 1/(n 4 log n). Consider two consecutive steps: They yield + 2 > 2. Consider linked pair: (e 1, e 2 ) (e 3, e 4 ) and (e 3, e 5 ) (e 6, e 7 ). Sequence of t consecutive steps, contains Ω(t) linked pairs: S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 (S 1, S 4 ) (S 2, S 5 ) (S 6, S 9 ) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
42 Idea for Improvement The bound is too pessimistic: Not every step yields the smallest possible improvement 1/(n 4 log n). Consider two consecutive steps: They yield + 2 > 2. Consider linked pair: (e 1, e 2 ) (e 3, e 4 ) and (e 3, e 5 ) (e 6, e 7 ). Sequence of t consecutive steps, contains Ω(t) linked pairs: S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 Linked 1/(n 3+1/3 log 2/3 n). (S 1, S 4 ) (S 2, S 5 ) (S 6, S 9 ) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
43 Extensions and Open Problems 1 Introduction 2 Lower Bound 3 Upper Bound 4 Extensions and Open Problems Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
44 Smoothed Analysis Smoothed Analysis Each point i {1,..., n} is chosen independently according to a probability density f i : [0, 1] 2 [0, φ]. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
45 Smoothed Analysis Smoothed Analysis Each point i {1,..., n} is chosen independently according to a probability density f i : [0, 1] 2 [0, φ]. 1/ φ 1/ φ Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
46 Smoothed Analysis General TSP Euclidean metric Manhattan metric average n 3+o(1) Õ(n 4.33 ) Õ(n 4 ) smoothed m n 1+o(1) φ Õ(n 4.33 φ 2.67 ) Õ(n 4 φ) worst-case 2 Ω(n) 2 Ω(n) 2 Ω(n) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
47 Approximation Ratio Worst Case: O(log n) Worst Case: Ω(log n/ log log n) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
48 Approximation Ratio Worst Case: O(log n) Worst Case: Ω(log n/ log log n) Average Case: O(1) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
49 Approximation Ratio Worst Case: O(log n) Worst Case: Ω(log n/ log log n) Average Case: O(1) Smoothed: O( φ) Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
50 Open Problems Worst-Case Analysis Analyze the diameter of the 2-Opt state graph. Analyze particular pivot rules like largest improvement. Probabilistic Analysis Show exact bounds on the running time of 2-Opt and k-opt. Show small constant approximation ratio for 2-Opt on random Euclidean instances. Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
51 The End Thanks! Questions? Heiko Röglin (RWTH Aachen) Worst Case and Prob. Analysis of 2-Opt Oberwolfach / 21
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