s-t path TSP and the randomizedChristofides algorithm

Shatian Wang

URA @ CO, UWaterloo

Oct., 2016

S. Wang (Carleton College) s-t path TSP Oct., 2016 1 / 16

Combinatorial Optimization

Finite set of objects S

Optimal object oopt 2 S

Exhaustive search?

S. Wang (Carleton College) s-t path TSP Oct., 2016 2 / 16

TSP: 49-City Instance

|S| = (49-1)!/2 =

6,206,957,796,268,036,335,431,144,523,686,687,519,260,743,177,338,880,000,000,000

S. Wang (Carleton College) s-t path TSP Oct., 2016 3 / 16

TSP: 49-City Instance

|S| = (49-1)!/2 =

6,206,957,796,268,036,335,431,144,523,686,687,519,260,743,177,338,880,000,000,000

S. Wang (Carleton College) s-t path TSP Oct., 2016 3 / 16

General TSP

NP-hardI no polynomial time algorithm that finds the exactoptimal solution: OPT.

Approximation algorithmI algorithm that finds a near to optimal solution.

Minimization Problem (e.g. TSP)I assign a cost to each objectI optimal object minimize the cost

Approximation ratio ¸I c(output) » ¸ˆ c(OPT).I ¸-approximation algorithm.

S. Wang (Carleton College) s-t path TSP Oct., 2016 4 / 16

General TSP

NP-hardI no polynomial time algorithm that finds the exactoptimal solution: OPT.

Approximation algorithmI algorithm that finds a near to optimal solution.

Minimization Problem (e.g. TSP)I assign a cost to each objectI optimal object minimize the cost

Approximation ratio ¸I c(output) » ¸ˆ c(OPT).I ¸-approximation algorithm.

S. Wang (Carleton College) s-t path TSP Oct., 2016 4 / 16

General TSP

NP-hardI no polynomial time algorithm that finds the exactoptimal solution: OPT.

Approximation algorithmI algorithm that finds a near to optimal solution.

Minimization Problem (e.g. TSP)I assign a cost to each objectI optimal object minimize the cost

Approximation ratio ¸I c(output) » ¸ˆ c(OPT).I ¸-approximation algorithm.

S. Wang (Carleton College) s-t path TSP Oct., 2016 4 / 16

General TSP

NP-hardI no polynomial time algorithm that finds the exactoptimal solution: OPT.

Approximation algorithmI algorithm that finds a near to optimal solution.

Minimization Problem (e.g. TSP)I assign a cost to each objectI optimal object minimize the cost

Approximation ratio ¸I c(output) » ¸ˆ c(OPT).I ¸-approximation algorithm.

S. Wang (Carleton College) s-t path TSP Oct., 2016 4 / 16

metric s-t path TSPGiven a complete graph G = (V ;E)with metric edge costsand two fixed vertices s; t 2 V.

Find a minimum cost Hamiltonian path from s to t.I a path that meets every graph vertex exactly onceI starting from s and ending at t

1

1 p2

p2

1

1

a b

c d

s

t

S. Wang (Carleton College) s-t path TSP Oct., 2016 5 / 16

metric s-t path TSPGiven a complete graph G = (V ;E)with metric edge costsand two fixed vertices s; t 2 V.

Find a minimum cost Hamiltonian path from s to t.I a path that meets every graph vertex exactly onceI starting from s and ending at t

1

1 p2

p2

1

1

a b

c d

s

t

S. Wang (Carleton College) s-t path TSP Oct., 2016 5 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

Definition (T-join)T := set of vertices; |T| = evenF := set of edges, s.t. dgr (v ) = odd w.r.t. F , v 2 T .F is a T-join.

a b c

d e

f g

S. Wang (Carleton College) s-t path TSP Oct., 2016 6 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

Definition (T-join)T := set of vertices; |T| = evenF := set of edges, s.t. dgr (v ) = odd w.r.t. F , v 2 T .F is a T-join.

a b c

d e

f g

c

g

a

f

S. Wang (Carleton College) s-t path TSP Oct., 2016 6 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

Definition (T-join)T := set of vertices; |T| = evenF := set of edges, s.t. dgr (v ) = odd w.r.t. F , v 2 T .F is a T-join.

a b c

d e

f g

c

g

a

f

S. Wang (Carleton College) s-t path TSP Oct., 2016 6 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )

c(J) » c(OPT); c(F ) » 23c(OPT)

53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)

53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

Christofides’ Algorithm(Extended by Hoogeveen 1990)

a b c

d e

f g

s

t

s c

f g

c(HamiltonianPath) » c(EularienWalk) = c(J + F ) =c(J) + c(F )c(J) » c(OPT); c(F ) » 2

3c(OPT)53-approximation algorithm

S. Wang (Carleton College) s-t path TSP Oct., 2016 7 / 16

An, Kleinberg and Shmoys (2012)the Randomized Christofides

First improvement to 53 (1990 Hoogeveen)

LP-based algorithm:I the "Randomized Christofides"

Upper bounded approximation ratio byI

p5+12 = 1:618

S. Wang (Carleton College) s-t path TSP Oct., 2016 8 / 16

LP: Linear Programming

Objective function to minimize/maximize

Linear constraints

Example:

minimize: c1x1 + c2x2

subject to:I a11x1 + a12x2 » b1I a21x1 + a22x2 » b2I a31x1 + a32x2 » b3I x1 – 0

I x2 – 0

minimize: cTxsubject to:I Ax » bI x – 0

where c =

24c1c2

35, x =

24x1x2

35,b =

2664b1b2b3

3775, A =

2664a11a12a13a21a22a23a31a32a33

3775

S. Wang (Carleton College) s-t path TSP Oct., 2016 9 / 16

LP: Linear Programming

Objective function to minimize/maximize

Linear constraints

Example:

minimize: c1x1 + c2x2

subject to:I a11x1 + a12x2 » b1I a21x1 + a22x2 » b2I a31x1 + a32x2 » b3I x1 – 0

I x2 – 0

minimize: cTxsubject to:I Ax » bI x – 0

where c =

24c1c2

35, x =

24x1x2

35,b =

2664b1b2b3

3775, A =

2664a11a12a13a21a22a23a31a32a33

3775

S. Wang (Carleton College) s-t path TSP Oct., 2016 9 / 16

LP: Linear Programming

Objective function to minimize/maximize

Linear constraints

Example:

minimize: c1x1 + c2x2

subject to:I a11x1 + a12x2 » b1I a21x1 + a22x2 » b2I a31x1 + a32x2 » b3I x1 – 0

I x2 – 0

minimize: cTxsubject to:I Ax » bI x – 0

where c =

24c1c2

35, x =

24x1x2

35,b =

2664b1b2b3

3775, A =

2664a11a12a13a21a22a23a31a32a33

3775

S. Wang (Carleton College) s-t path TSP Oct., 2016 9 / 16

s-t path TSP: IP Formulation

minimize: Xe2Ecexe = cTx

subject to:I x(‹(s)) = x(‹(t)) = 1

I x(‹(v )) = 2 ————— 8 v 6= s; tI x(‹(S)) – 1 ————— 8 jS \ fs; tgj = 1

I x(‹(S)) – 2 ————— 8 ; ( S ( V ; jS \ fs; tgj evenI xe = 0 or xe = 1 ———— 8 e 2 E

Also NP-hard!

S. Wang (Carleton College) s-t path TSP Oct., 2016 10 / 16

s-t path TSP: LP-relaxations

L.P.1 (Held-Karp Relaxation for s-t path TSP)I minimize: X

e2Ecexe = cTx

I subject to:F x(‹(s)) = x(‹(t)) = 1

F x(‹(v )) = 2 ——————– 8 v 6= s; tF x(‹(S)) – 1 ——————– 8 jS \ fs; tgj = 1

F x(‹(S)) – 2 ——————– 8 ; ( S ( V ; jS \ fs; tgj evenF 1 – xe – 0 ——————— 8 e 2 E

Can compute optimal solution x˜ in polytime.

x˜ might contain fractional values.

S. Wang (Carleton College) s-t path TSP Oct., 2016 11 / 16

The Randomized Christofides Algorithm

1 Find the optimal solution to LP1: x˜

2 Write x˜ as a convex combination of spanning trees:

x˜ =XpiX Ji

I X Ji is the incidence vector of spanning tree Ji

I pi – 0;Ppi = 1

3 Sample a spanning tree Jaccording to the probability defined by pi ’s.

4 Perform Christofides algorithm on J.

S. Wang (Carleton College) s-t path TSP Oct., 2016 12 / 16

The Randomized Christofides:Made-up Example

X J1 :[1; 0; 1; 0; 1; 0]T

a b

c d

s

t

X J2 :[0; 1; 1; 1; 0; 0]T

a b

c d

s

t

X J3 :[0; 1; 1; 0; 1; 0]T

a b

c d

s

t

x˜ = 0:3X J1 + 0:2X J2 + 0:5X J3

S. Wang (Carleton College) s-t path TSP Oct., 2016 13 / 16

The Randomized Christofides:Made-up Example

X J1 :[1; 0; 1; 0; 1; 0]T

a b

c d

s

t

X J2 :[0; 1; 1; 1; 0; 0]T

a b

c d

s

t

X J3 :[0; 1; 1; 0; 1; 0]T

a b

c d

s

t

b

c

x˜ = 0:3X J1 + 0:2X J2 + 0:5X J3

S. Wang (Carleton College) s-t path TSP Oct., 2016 13 / 16

The Randomized Christofides:Made-up Example

X J1 :[1; 0; 1; 0; 1; 0]T

a b

c d

s

t

X J2 :[0; 1; 1; 1; 0; 0]T

a b

c d

s

t

X J3 :[0; 1; 1; 0; 1; 0]T

a b

c d

s

t

b

c

x˜ = 0:3X J1 + 0:2X J2 + 0:5X J3

S. Wang (Carleton College) s-t path TSP Oct., 2016 13 / 16

The Randomized Christofides:Made-up Example

X J1 :[1; 0; 1; 0; 1; 0]T

a b

c d

s

t

X J2 :[0; 1; 1; 1; 0; 0]T

a b

c d

s

t

X J3 :[0; 1; 1; 0; 1; 0]T

a b

c d

s

t

b

c

x˜ = 0:3X J1 + 0:2X J2 + 0:5X J3

S. Wang (Carleton College) s-t path TSP Oct., 2016 13 / 16

The Randomized Christofides:Made-up Example

X J1 :[1; 0; 1; 0; 1; 0]T

a b

c d

s

t

X J2 :[0; 1; 1; 1; 0; 0]T

a b

c d

s

t

X J3 :[0; 1; 1; 0; 1; 0]T

a b

c d

s

t

b

c

x˜ = 0:3X J1 + 0:2X J2 + 0:5X J3

S. Wang (Carleton College) s-t path TSP Oct., 2016 13 / 16

The Randomized Christofides: Analysis

E(c(HamPath)) » E(c(J)) + E(c(F))

x˜ =XpiX Ji

E(c(J)) = c(x˜)

E(c(HamPath)) » c(x˜) + E(c(F))

c(x˜) » c(OPT)

S. Wang (Carleton College) s-t path TSP Oct., 2016 14 / 16

The Randomized Christofides: Analysis

E(c(HamPath)) » E(c(J)) + E(c(F))

x˜ =XpiX Ji

E(c(J)) = c(x˜)

E(c(HamPath)) » c(x˜) + E(c(F))

c(x˜) » c(OPT)

S. Wang (Carleton College) s-t path TSP Oct., 2016 14 / 16

The Randomized Christofides: Analysis

E(c(HamPath)) » E(c(J)) + E(c(F))

x˜ =XpiX Ji

E(c(J)) = c(x˜)

E(c(HamPath)) » c(x˜) + E(c(F))

c(x˜) » c(OPT)

S. Wang (Carleton College) s-t path TSP Oct., 2016 14 / 16

The Randomized Christofides: Analysis

E(c(HamPath)) » E(c(J)) + E(c(F))

x˜ =XpiX Ji

E(c(J)) = c(x˜)

E(c(HamPath)) » c(x˜) + E(c(F))

c(x˜) » c(OPT)

S. Wang (Carleton College) s-t path TSP Oct., 2016 14 / 16

The Randomized Christofides: Analysis

E(c(HamPath)) » E(c(J)) + E(c(F))

x˜ =XpiX Ji

E(c(J)) = c(x˜)

E(c(HamPath)) » c(x˜) + E(c(F))

c(x˜) » c(OPT)

S. Wang (Carleton College) s-t path TSP Oct., 2016 14 / 16

Towards bounding E(c(F))

An, Kleinberg, Shmoys (2012):I E(c(F)) » 0:618c(x˜) » 0:618c(OPT)

I E(c(HamPath)) » c(x˜) + 0:618c(x˜) » 1:618c(OPT)

I 1.618-approximation algorithm

Sebo (2013)I E(c(F)) » 0:6c(x˜).

I 1.6-approximation algorithm

Gottschalk, Vygen (2015)I Better convex combination.

I 1.566-approximation algorithm

Sebo (2016)I 1.53-approximation algorithm

LP-based 1.5-approximation algorithm?

S. Wang (Carleton College) s-t path TSP Oct., 2016 15 / 16

Towards bounding E(c(F))

An, Kleinberg, Shmoys (2012):I E(c(F)) » 0:618c(x˜) » 0:618c(OPT)

I E(c(HamPath)) » c(x˜) + 0:618c(x˜) » 1:618c(OPT)

I 1.618-approximation algorithm

Sebo (2013)I E(c(F)) » 0:6c(x˜).

I 1.6-approximation algorithm

Gottschalk, Vygen (2015)I Better convex combination.

I 1.566-approximation algorithm

Sebo (2016)I 1.53-approximation algorithm

LP-based 1.5-approximation algorithm?

S. Wang (Carleton College) s-t path TSP Oct., 2016 15 / 16

Towards bounding E(c(F))

An, Kleinberg, Shmoys (2012):I E(c(F)) » 0:618c(x˜) » 0:618c(OPT)

I E(c(HamPath)) » c(x˜) + 0:618c(x˜) » 1:618c(OPT)

I 1.618-approximation algorithm

Sebo (2013)I E(c(F)) » 0:6c(x˜).

I 1.6-approximation algorithm

Gottschalk, Vygen (2015)I Better convex combination.

I 1.566-approximation algorithm

Sebo (2016)I 1.53-approximation algorithm

LP-based 1.5-approximation algorithm?

S. Wang (Carleton College) s-t path TSP Oct., 2016 15 / 16

Towards bounding E(c(F))

An, Kleinberg, Shmoys (2012):I E(c(F)) » 0:618c(x˜) » 0:618c(OPT)

I E(c(HamPath)) » c(x˜) + 0:618c(x˜) » 1:618c(OPT)

I 1.618-approximation algorithm

Sebo (2013)I E(c(F)) » 0:6c(x˜).

I 1.6-approximation algorithm

Gottschalk, Vygen (2015)I Better convex combination.

I 1.566-approximation algorithm

Sebo (2016)I 1.53-approximation algorithm

LP-based 1.5-approximation algorithm?

S. Wang (Carleton College) s-t path TSP Oct., 2016 15 / 16

Towards bounding E(c(F))

An, Kleinberg, Shmoys (2012):I E(c(F)) » 0:618c(x˜) » 0:618c(OPT)

I E(c(HamPath)) » c(x˜) + 0:618c(x˜) » 1:618c(OPT)

I 1.618-approximation algorithm

Sebo (2013)I E(c(F)) » 0:6c(x˜).

I 1.6-approximation algorithm

Gottschalk, Vygen (2015)I Better convex combination.

I 1.566-approximation algorithm

Sebo (2016)I 1.53-approximation algorithm

LP-based 1.5-approximation algorithm?

S. Wang (Carleton College) s-t path TSP Oct., 2016 15 / 16

Thank you! Questions?

S. Wang (Carleton College) s-t path TSP Oct., 2016 16 / 16