PTAS for Euclidean Traveling Salesman and Other Geometric Problems - PowerPoint PPT Presentation

CS468, Wed Feb 15th 2006 PTAS for Euclidean Traveling Salesman and Other Geometric Problems Sanjeev Arora Journal of the ACM, 45(5):753–782, 1998

PTAS → same as LTAS, with ”Linear” replaced by ”Polynomial” Given a problem P and a cost function | . | , a PTAS of P is a one- Def parameter family of PT algorithms, { A ε } ε> 0 , such that, for all ε > 0 and all instance I of P , | A ε ( I ) | ≤ (1 + ε ) | OPT( I ) | . 1 S. Arora — Euclidean TSP and other related problems

PTAS → same as LTAS, with ”Linear” replaced by ”Polynomial” Given a problem P and a cost function | . | , a PTAS of P is a one- Def parameter family of PT algorithms, { A ε } ε> 0 , such that, for all ε > 0 and all instance I of P , | A ε ( I ) | ≤ (1 + O ( ε )) | OPT( I ) | . • PT means time complexity n O (1) , where the constant may depend on 1 / ε and on the dimension d (when pb in R d ) • As far as we get n O (1) , we do not care about the constant • the constant in (1 + O ( ε )) must not depend on I nor on ε 1 S. Arora — Euclidean TSP and other related problems

TSP 2 Given a complete graph G = ( V, E ) with non- 5 negative weights, find the Hamiltonian tour of 3 7 minimum total cost. 1 17 0 . 2 8 K 7 2 S. Arora — Euclidean TSP and other related problems

TSP | OPT | = 36 . 2 2 Given a complete graph G = ( V, E ) with non- 5 negative weights, find the Hamiltonian tour of 3 7 minimum total cost. 1 17 0 . 2 8 OPT K 7 2 S. Arora — Euclidean TSP and other related problems

TSP | OPT | = 36 . 2 2 Given a complete graph G = ( V, E ) with non- 5 negative weights, find the Hamiltonian tour of 3 7 minimum total cost. 1 TSP is NP-hard ⇒ no PT algorithm, unless P = NP . 17 0 . 2 8 OPT K 7 2 S. Arora — Euclidean TSP and other related problems

TSP | OPT | = 36 . 2 2 Given a complete graph G = ( V, E ) with non- 5 negative weights, find the Hamiltonian tour of 3 7 minimum total cost. 1 TSP is NP-hard ⇒ no PT algorithm, unless P = NP . 17 0 . 2 8 OPT K 7 For all PT computable function α ( n ) , TSP cannot be approxi- Thm mated in PT within a factor of (1 + α ( n )) , unless P = NP . 2 S. Arora — Euclidean TSP and other related problems

TSP Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost. TSP is NP-hard ⇒ no PT algorithm, unless P = NP . K 7 For all PT computable function α ( n ) , TSP cannot be approxi- Thm mated in PT within a factor of (1 + α ( n )) , unless P = NP . Reduction of Hamiltonian Cycle: Proof Let G = ( V, E ) unweighted, incomplete → G ′ = ( V ′ , E ′ ) where: • V ′ = V • ∀ e ∈ E , add ( e, 1) to E ′ ∈ E , add ( e, (1 + α ( n )) n ) to E ′ • ∀ e / 2 S. Arora — Euclidean TSP and other related problems

TSP Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of 1 minimum total cost. TSP is NP-hard ⇒ no PT algorithm, unless P = NP . n (1 + α ( n )) K 7 For all PT computable function α ( n ) , TSP cannot be approxi- Thm mated in PT within a factor of (1 + α ( n )) , unless P = NP . Reduction of Hamiltonian Cycle: Proof Let G = ( V, E ) unweighted, incomplete → G ′ = ( V ′ , E ′ ) where: • V ′ = V • ∀ e ∈ E , add ( e, 1) to E ′ ∈ E , add ( e, (1 + α ( n )) n ) to E ′ • ∀ e / 2 S. Arora — Euclidean TSP and other related problems

Metric TSP The weights of G ( V, E ) now satisfy the triangle 2 inequality 5 3 7 5 1 17 0 . 2 8 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: 2 (1) build MST M of G (Kruskal) 5 3 5 1 17 0 . 2 8 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: (1) build MST M of G (Kruskal) (2) double edges → M + Eulerian 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: 1 T + 2 (1) build MST M of G (Kruskal) 12 3 (2) double edges → M + Eulerian 11 4 5 10 (3) build greedily a Eulerian tour T + on M + 6 9 8 7 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: 1 T + 2 (1) build MST M of G (Kruskal) 12 T 3 (2) double edges → M + Eulerian 11 4 5 10 (3) build greedily a Eulerian tour T + on M + 6 9 8 (4) Trim edges of T + → T 7 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: 1 T + 2 (1) build MST M of G (Kruskal) 12 T 3 (2) double edges → M + Eulerian 11 4 5 10 (3) build greedily a Eulerian tour T + on M + 6 9 8 (4) Trim edges of T + → T 7 | T | ≤ 2 | OPT | Thm | T | ≤ | T + | proof tri. ineq. 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: 1 T + 2 (1) build MST M of G (Kruskal) 12 3 (2) double edges → M + Eulerian 11 4 5 10 (3) build greedily a Eulerian tour T + on M + 6 9 8 (4) Trim edges of T + → T 7 | T | ≤ 2 | OPT | Thm | T | ≤ | T + | = | M + | proof tri. ineq. 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: (1) build MST M of G (Kruskal) (2) double edges → M + Eulerian (3) build greedily a Eulerian tour T + on M + (4) Trim edges of T + → T | T | ≤ 2 | OPT | Thm | T | ≤ | T + | = | M + | = 2 | M | proof tri. ineq. 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: 2 (1) build MST M of G (Kruskal) 5 3 (2) double edges → M + Eulerian 5 1 (3) build greedily a Eulerian tour T + on M + 17 0 . 2 8 (4) Trim edges of T + → T | T | ≤ 2 | OPT | Thm | T | ≤ | T + | = | M + | = 2 | M | ≤ 2 | OPT | proof tri. ineq. OPT =”tree+edge” 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: (1) build MST M of G (Kruskal) (2) double edges → M + Eulerian (3) build greedily a Eulerian tour T + on M + (4) Trim edges of T + → T Replace (2) by adding to M a min cost perfect matching of its odd-valenced vertices → 3 2 -approximation [Christofides76] Can we do better? Q 3 S. Arora — Euclidean TSP and other related problems

Metric TSP 2 -approximation algorithm: (1) build MST M of G (Kruskal) (2) double edges → M + Eulerian (3) build greedily a Eulerian tour T + on M + (4) Trim edges of T + → T Replace (2) by adding to M a min cost perfect matching of its odd-valenced vertices → 3 2 -approximation [Christofides76] Can we do better? Q Thm [ALMSS92] There is no PTAS for Metric TSP, unless P = NP best approximation factor: 4 / 3 Conjecture 3 S. Arora — Euclidean TSP and other related problems

Euclidean TSP V ⊂ R d , E is the set of all pairs weighted by Euclidean distances 4 S. Arora — Euclidean TSP and other related problems

Euclidean TSP Thm [Arora96] Euclidean TSP admits a PTAS Let n = | V | Overview 4 S. Arora — Euclidean TSP and other related problems

Euclidean TSP 1 Thm [Arora96] Euclidean TSP admits a PTAS 1 Let n = | V | Overview (1) rescale/snap V n 2 √ 2 4 S. Arora — Euclidean TSP and other related problems

Euclidean TSP 1 Thm [Arora96] Euclidean TSP admits a PTAS 1 Let n = | V | Overview (1) rescale/snap V (2) subdivide the grid with a quadtree n 2 √ 2 level 1 level 2 level 3 4 S. Arora — Euclidean TSP and other related problems

Euclidean TSP 1 Thm [Arora96] Euclidean TSP admits a PTAS 1 Let n = | V | Overview (1) rescale/snap V (2) subdivide the grid with a quadtree n 2 √ 2 (3) place portals on grid lines level 1 level 2 level 3 4 S. Arora — Euclidean TSP and other related problems

Euclidean TSP 1 Thm [Arora96] Euclidean TSP admits a PTAS 1 Let n = | V | Overview (1) rescale/snap V (2) subdivide the grid with a quadtree n 2 √ 2 (3) place portals on grid lines level 1 (4) compute the smallest portal-respecting tour OPT p level 2 level 3 4 S. Arora — Euclidean TSP and other related problems

Euclidean TSP 1 Thm [Arora96] Euclidean TSP admits a PTAS 1 Let n = | V | Overview (1) rescale/snap V (2) subdivide the grid with a quadtree n 2 √ 2 (3) place portals on grid lines level 1 (4) compute the smallest portal-respecting tour OPT p level 2 level 3 (5) Trim the edges of OPT p and output the result T 4 S. Arora — Euclidean TSP and other related problems

(1) rescale V Let V s be V scaled by a factor of s . ∀ T , | T | s = s | T | ⇒ OPT for V s is the same as OPT for V ⇒ solving the pb for V s is the same as solving the pb for V 5 S. Arora — Euclidean TSP and other related problems

(1) rescale V Let V s be V scaled by a factor of s . ∀ T , | T | s = s | T | ⇒ OPT for V s is the same as OPT for V n 2 √ 2 ⇒ solving the pb for V s is the same as solving the pb for V → wlog, we assume that the smallest square containing V has sidelength n 2 √ 2 5 S. Arora — Euclidean TSP and other related problems

(1) snap V 1 g : v ∈ V �→ v g ∈ grid closest to v 1 6 S. Arora — Euclidean TSP and other related problems