PTAS for Euclidean Traveling Salesman and Other Geometric Problems

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CS468, Wed Feb 15th 2006

PTAS for Euclidean Traveling Salesman and Other Geometric Problems

Sanjeev Arora

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PTAS

→ same as LTAS, with ”Linear” replaced by ”Polynomial”

Def Given a problem P and a cost function |.|, a PTAS of P is a one- 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)|.

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PTAS

→ same as LTAS, with ”Linear” replaced by ”Polynomial”

Def Given a problem P and a cost function |.|, a PTAS of P is a one- 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

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TSP

K7

2

3 7

Given a complete graph G = (V, E) with non- negative weights, find the Hamiltonian tour of minimum total cost.

5

1

0.2

8

17

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TSP

K7

2

3 7

OPT 5

1

0.2

8

17

|OPT| = 36.2

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TSP

K7

2

3 7

OPT 5

1

0.2

8

17

|OPT| = 36.2

TSP is NP-hard ⇒ no PT algorithm, unless P = N P .

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TSP

K7

2

3 7

OPT 5

1

0.2

8

17

|OPT| = 36.2

Thm For all PT computable function α(n), TSP cannot be approxi- mated in PT within a factor of (1 + α(n)), unless P = N P .

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TSP

K7

Proof Reduction of Hamiltonian Cycle:

Let G = (V, E) unweighted, incomplete → G⁰ = (V ⁰, E⁰) where:

• V ⁰ = V

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TSP

K7

Proof Reduction of Hamiltonian Cycle:

Let G = (V, E) unweighted, incomplete → G⁰ = (V ⁰, E⁰) where:

• V ⁰ = V

1

n(1 + α(n))

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Metric TSP

2

5 3 1

0.2

8

17 7 5

The weights of G(V, E) now satisfy the triangle inequality

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Metric TSP

2

5 3 1

0.2

8

17 5

2-approximation algorithm:

(1) build MST M of G (Kruskal)

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Metric TSP

(2) double edges → M⁺ Eulerian

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Metric TSP

1

2 3

4 5

6 7

9 8 10

11 12

(3) build greedily a Eulerian tour T ⁺ on M⁺

T ⁺

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Metric TSP

1

2 3

4 5

6 7

9 8 10

11 12

(3) build greedily a Eulerian tour T ⁺ on M⁺ (4) Trim edges of T ⁺ → T

T ⁺ T

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Metric TSP

1

2 3

4 5

6 7

9 8 10

11 12

T ⁺ T

Thm |T | ≤ 2|OPT|

proof |T | ≤ |T ⁺|

tri. ineq.

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Metric TSP

1

2 3

4 5

6 7

9 8 10

11 12

T ⁺

Thm |T | ≤ 2|OPT|

proof |T | ≤ |T ⁺| = |M⁺|

tri. ineq.

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Metric TSP

Thm |T | ≤ 2|OPT|

proof |T | ≤ |T ⁺| = |M⁺| = 2|M |

tri. ineq.

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Metric TSP

2

5 3 1

0.2

8

17 5

Thm |T | ≤ 2|OPT|

proof |T | ≤ |T ⁺| = |M⁺| = 2|M | ≤ 2|OPT|

OPT=”tree+edge”

tri. ineq.

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Metric TSP

Replace (2) by adding to M a min cost perfect matching of its odd-valenced vertices → ³₂-approximation [Christofides76]

Q Can we do better?

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Metric TSP

Replace (2) by adding to M a min cost perfect matching of its odd-valenced vertices → ³₂-approximation [Christofides76]

Q Can we do better?

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Euclidean TSP

V ⊂ R^d, E is the set of all pairs weighted by Euclidean distances

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Euclidean TSP

Thm [Arora96] Euclidean TSP admits a PTAS Overview Let n = |V |

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

(2) subdivide the grid with a quadtree

level 1 level 2 level 3

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

(3) place portals on grid lines

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

(4) compute the smallest portal-respecting tour OPT_p

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

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(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

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(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

√

n²√ 2

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(1) snap V

1

g : v ∈ V 7→ v_g ∈ grid closest to v

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(1) snap V

1

∀T = (v₁, v₂, · · · , v_n), g(T) := (g(v₁), g(v₂), · · · , g(v_n))

⇒ ∀T , ||g(T )| − |T || ≤ n√ 2

Through g, a vertex is moved by at most ^√²/²

⇒ an edge is elongated/shortened by at most √ 2

⇒ |OPT_g| ≤ |g(OPT)| ≤ |OPT| + n√ 2

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(1) snap V

Q How to construct a path for V from OPT_g? g⁻¹(OPT_g) is not defined uniquely

(several nodes of V may be mapped to a same grid point)

OPT_g

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(1) snap V

OPT_g

2 1

3

4

→ Define g⁻¹(OPT_g) as follows: for each vertex v_g of OPT_g,

- order the vertices of V mapped to v_g and connect them to v_g twice

≤ ²

√2 2

(+n√ 2)

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(1) snap V

2 1

3

4

→ Define g⁻¹(OPT_g) as follows: for each vertex v_g of OPT_g,

- order the vertices of V mapped to v_g and connect them to v_g twice

g⁻¹(OPT_g)

(+n√ 2)

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(1) snap V

|g⁻¹(OPT_g)| ≤ |OPT_g| + n√

2 ≤ |g(OPT)| + n√

2 ≤ |OPT| + 2n√ 2

≤ |OPT| 1 + _n¹

|OPT| ≥ 2n²√ 2

n²√ 2

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(1) snap V

|g⁻¹(OPT_g)| ≤ |OPT_g| + n√

2 ≤ |g(OPT)| + n√

2 ≤ |OPT| + 2n√ 2

≤ |OPT| 1 + _n¹

|OPT| ≥ 2n²√ 2

n²√ 2

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(2) Grid subdivision

2^k ≤ 2n²√ 2

Let k s.t. 2^k−1 ≤ n²√

2 ≤ 2^k ≤ 2n²√ 2

1

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(2) Grid subdivision

2^k ≤ 2n²√ 2

Let k s.t. 2^k−1 ≤ n²√

2 ≤ 2^k ≤ 2n²√ 2

1

level 1

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(2) Grid subdivision

2^k ≤ 2n²√ 2

Let k s.t. 2^k−1 ≤ n²√

2 ≤ 2^k ≤ 2n²√ 2

1

level 1 level 2

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(2) Grid subdivision

2^k ≤ 2n²√ 2

Let k s.t. 2^k−1 ≤ n²√

2 ≤ 2^k ≤ 2n²√ 2

1

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(3) Portals

Let m =

j log n ε

k

On each level i line, place 2ⁱm equally-spaced portals, plus one at each grid point

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(3) Portals

Let m =

j log n ε

k

On each level i line, place 2ⁱm equally-spaced portals, plus one at each grid point

Each level i line is incident to 2ⁱ pairs of level i squares ⇒ m portals per pair (w/o corners)

Each level i square has a boundary made of level j ≤ i lines

⇒ at most 4m + 4 portals per square

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(4) Portal-respecting tours

Def A tour is portal-respecting if it crosses the grid only at portals

Pb: an exhaustive search has considers infinitely many instances, since the number of passes through a portal is unbounded

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(4) Portal-respecting tours

Def a tour is k-light if each portal is visited at most k times

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(4) Portal-respecting tours

a

b

Prop OPT_p is 2-light

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(4) Portal-respecting tours

a

b

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(4) Portal-respecting tours

a

b c

a⁰

b⁰ c⁰

(a, a⁰, b⁰, b, c, c⁰) Prop OPT_p is 2-light

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(4) Portal-respecting tours

a

c⁰

a⁰

c

(a, a⁰, c, c⁰)

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(4) Portal-respecting tours

a

c⁰

a⁰

c

(a, c, a⁰, c⁰)

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(4) Portal-respecting tours

a

c⁰

a⁰

c

(a, c, a⁰, c⁰)

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(4) Portal-respecting tours

a

c⁰

a⁰

c

Prop OPT_p does not self-intersect, except at portals

(a, a⁰, c, c⁰)

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(4) Portal-respecting tours

a

c⁰

a⁰

c

Prop OPT_p does not self-intersect, except at portals

(a, c, a⁰, c⁰)

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(4) Portal-respecting tours

Goal: find shortest tour that is:

- portal-respecting - 2-light

- non self-intersecting (except at portals)

→ divide-and-conquer approach, using the quadtree

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(4) Portal-respecting tours

For any square s, interface is defined by:

- a number of passes through each portal of s

- a paring between selected portals ³^O(m) ⁼ ⁿ^O(¹^/^ε⁾

Ω(m!) = Ω(n^log ⁿ)

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(4) Portal-respecting tours

O(C_m) = O `

2^2m´

= n^O(¹^/^ε⁾

( ( ) ) ( ( ) ) ( )

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(4) Portal-respecting tours

Pb: a simple recursion is not sufficient (optimum for square s is not concatenation of optima of sons of s)

→ dynamic programming

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(4) Portal-respecting tours

Lookup table:

R²

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(4) Portal-respecting tours

Lookup table:

R²

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(4) Portal-respecting tours

Lookup table:

R²

Fill the table ”in depth”

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(4) Portal-respecting tours

Lookup table:

R²

∀ (leaf,interface), ∀ (node, interface),

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(4) Portal-respecting tours

Lookup table:

R²

total running time: O

“

n⁴ n^O(¹^/^ε⁾

”

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

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Euclidean TSP

(1) rescale/snap V

n²√ 2

1

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Structure theorem

Pb: |OPT_p| can be made arbitrarily large compared to |OPT|

n

1 1

|V | = 2n

n

2 n + n 2

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Structure theorem

n

1 1

|V | = 2n

|OPT| ≤ 2 ⁿ₂ n + 2ⁿ₂ 2√

2 + 2n²

√2

2 = n²(1 + √

2) + 2n√ 2

n

2 n + n 2

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Structure theorem

n

1 1

|V | = 2n

|OPT| ≤ 2 ⁿ₂ n + 2ⁿ₂ 2√

2 + 2n²

√2

2 = n²(1 + √

2) + 2n√ 2

One crossing every n ⇒ overhead per consecutive portals ≥ 2^δ₄ = ^δ₂

⇒ total overhead ≥ 4m^δ₂ =

2 2

n

2 n + n 2

At level 2, 4m portals ⇒ interportal distance δ = ⁿ²_8m⁺²ⁿ >> n

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Structure theorem

Patch: randomize the algorithm:

Choose random integers 0 ≤ x, y ≤ 2^k, then apply (2)-(5) to square of sidelength 2^k+1 shifted by (−x, −y).

2^k

x

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Structure theorem

2^k

For any vertical line l in domain, P_x(l is at level i) = ²ⁱ⁻²

1+2^k

x

2ⁱ⁻¹ level i lines, half of which reach l 1 + 2^k possible values for x

Thm The expectation (over x, y) of |OPT_g| − |OPT| is at most

k+1

m |OPT|

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Structure theorem

2^k

x

k+1

m |OPT|

→ transform OPT into a portal- respecting tour:

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Structure theorem

2^k

x

k+1

m |OPT|

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Structure theorem

k+1

m |OPT|

(level i)

P_x(level i) = ²ⁱ⁻²

1+2^k (same for y) Expected overhead: P_k+1

i=1

2ⁱ⁻² 1+2^k

2^k+1 m 2ⁱ

≤ P_k+1

i=1

2ⁱ⁻² 2^k

2^k+1

m 2ⁱ = ^k+1_2m

For every crossing, overhead ≤ 2 times half the interportal distance = ²^k+1

m 2ⁱ

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Structure theorem

2^k

x

k+1

m |OPT|

P_x(level i) = ²ⁱ⁻²

1+2^k (same for y) Expected overhead: P_k+1

i=1

2ⁱ⁻² 1+2^k

2^k+1 m 2ⁱ

≤ P_k+1

i=1

2ⁱ⁻² 2^k

2^k+1

m 2ⁱ = ^k+1_2m

For every crossing, overhead ≤ 2 times half the interportal distance = ²^k+1

m 2ⁱ

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Structure theorem

k+1

m |OPT| ≤ ^{2 log}_logⁿ⁺_n_/³^/²⁺¹

2ε |OPT| ≤ (4 + ⁵/^log ⁿ) ε |OPT| ≤ 9ε |OPT|.

2^k ≤ 2n²√ 2 m =

jlog n ε

k ≥ ^log_2εⁿ

(n ≥ 2)

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Structure theorem

k+1

m |OPT|

Corollary P_x,y (|OPT_g| − |OPT| ≤ 18ε |OPT|) ≥ ¹/²

≤ ^{2 log}_logⁿ⁺_n_/³^/²⁺¹

2ε |OPT| ≤ (4 + ⁵/^log ⁿ) ε |OPT| ≤ 9ε |OPT|.

→ Monte-Carlo procedure given a constant 0 < c < 1, repeat dlog(1/c)e times the process ”randomization + (2)-(5)” and keep the best computed tour T . Then, P (|OPT_g| − |OPT| ≤ 18ε |OPT|) ≥ 1 − c

→ Derandomization try all possible choices of (x, y) (there are O(n⁴)

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Higher dimensions

The analysis extends to higher dimensions, except for the valid pair- ing argument.

O(C_m) = O `

2^2m´

= n^O(¹^/^ε⁾

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Higher dimensions

Patch: instead of considering all 2-light tours, consider only those that intersect each side of the boundary of a given square at most l times.

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Higher dimensions

→ for l = Θ `₁

ε

´ and m = ¨_log _n

ε

˝:

• E_x,y [|OPT_p(l)| − |OPT|] ≤ O(ε) |OPT|

• ∀ square, #{interfaces} ≤ m^O(l) l! ≤ (log n)^O(¹^/^ε⁾

R

²

Thm E_x,y [|OPT_p(l)| − |OPT|] ≤

log (n)+1

m + _l−5¹²

|OPT|

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Higher dimensions

→ for l = Θ “

`^√_d

/^ε´^d−1”

and m = Θ “

`_{log (n)}^√_d

/^ε´^d−1” :

• E_x,y [|OPT_p(l)| − |OPT|] ≤ O(ε) |OPT|

• ∀ square, #{interfaces} ≤ m^O(2dl) l! ≤ O

„

(log n)^O

“(^√^d^/^ε)^d−1^”«

R

^d

Thm E_x,y [|OPT_p(l)| − |OPT|] ≤ O

log (n)√ d m

d−11 + ^(l+1)

1− 1 d−1

l+1−2^d+1

|OPT|

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Higher dimensions

Proof → key ingredient: patching lemma.

log (n)+1

m + _l−5¹²

|OPT|

- reduce the # of crossings by dealing w/

several portals at once

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Higher dimensions

Proof → key ingredient: patching lemma.

log (n)+1

m + _l−5¹²

|OPT|

→ use patching lemma repeat- edly, to reduce the total # of crossings of OPT when made

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Other norms

• Cannot reduce pb to Euclidean TSP:

C₁ |.|_E ≤ |.| ≤ C₂ |.|_E

→ get T s.t. |T |_E ≤ (1 + ε)|OPT|_E

|T | ≤ C₂|T |_E ≤ C₂(1 + ε)|OPT|_E ≤ ^C_C²

1 (1 + ε)|OPT|

Euclidean

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Other norms

• Cannot reduce pb to Euclidean TSP:

C₁ |.|_E ≤ |.| ≤ C₂ |.|_E

→ get T s.t. |T |_E ≤ (1 + ε)|OPT|_E

|T | ≤ C₂|T |_E ≤ C₂(1 + ε)|OPT|_E ≤ ^C_C²

1 (1 + ε)|OPT|

Euclidean

• Algorithm and its analysis hold for any other geometric norm (modulo some constants factors in the optimal values of m and l).

norm (6= metric) is important for scaling phase

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Recap

• Euclidean TSP admits a PTAS. Idem for TSP in (R^d, |.|).

• In R^d, the PTAS given has space and time complexities of O

n^2d(log n)^O

“(^√^d^/^ε)^d−1^”

• Complexity is reduced to O

n(log n)^O

“(^√^d^/^ε)^d−1^”

if a reduced quadtree is used

• By using a (1 + ε)-spanner of the input nodes to give better

”hints” of what portals to use, one reduces the complexity to