A rounding algorithm for approximating minimum Manhattan networks

A rounding algorithm for approximating minimum Manhattan networks

Victor Chepoi, Karim Nouioua, Yann Vaxès

LIF - Université de la Méditerranée Marseille - France

Rectilinear paths

A rectilinear (Manhattan) path between two points p,q of the plane is a path consisting of vertical and horizontal segments.

p

q

l

- Paths

|p_y – q_y|

p

q

An l₁-path is a shortest rectilinear path in the l₁-metric, i.e. a rectilinear path of length ║p-q║₁= |p_x - q_x|+|p_y - q_y|

Manhattan networks

A Manhattan network on a set of n terminals T = { t₁ , …, t_n } is a network containing an l₁-

path

between every pair of terminals of T, i.e. a 1-spanner for T in the l₁-plane.

Minimum Manhattan networks

A minimum Manhattan network (MMN) on T is a Manhattan network of minimum total length.

The MMN problem is to find such a network (introduced by Gudmundsson, Levcopoulos and Narasimhan, APPROX 99).

It is not known whether it is in P or not.P

A MMN on 100

terminals

Applications of MMN

VLSI Circuit Design :

A MMN ensures optimal transmission delay at minimum cost.

Computational biology :

Finding efficient search space in Pair Hidden Markov Models alignment algorithm

[2003] F. Lam, M. Alexanderson and L. Pachter, J. Comput. Biology All areas in which geometric spanners occur (e.g., parallel

computations, distributed, communication networks, wireless networks, etc., etc.)

Previous results

[1999] J. Gudmundsson, C. Levcopoulos and G. Narasimhan Factor 8 approximation algorithm , O(n log n)

Factor 4 approximation algorithm, O(n³) [2002] R. Kato, K. Imai and T. Asano

Factor 2 approximation algorithm, O(n³) [2004] M. Benkert, T. Shirabe and A. Wolf

Factor 3 approximation algorithm, O(n log n) [2004] K. Nouioua

Factor 3 approximation algorithm, O(n log n)

Incomplete proof

• The complete grid on T contains at least one MMN on T.

• The part Г=(V,E) of complete grid inside the Pareto enveloppe contains at least one MMN on T.

• The edges on the boundary of  belong to all MMN inside .

Properties of

MMN

Let le be the length of the edge e.

1. For each edge e in E, we introduce a 0-1 variable xe :

- xe = 1 if e belongs to the solution - x_e = 0 otherwise

2. The objective is to minimize the total length of the network:

Integer Programming Formulation

m i n i m i z e

e E

l x

å

t

t

e

1

e c

x

Î

å ³

Integer Programming Formulation

3. Constraints : for every pair (t_i,t_j) of terminals and for every (t_i,t_j)-cut c of the oriented subgraph Г _ij = Г ∩ R_ij :

Integer programming formulation

minimize

subject to 1

{0,1},

e e e E

e ij F ij

e c e

l x

x c C

x e E

Î Î Î

³ Î

Î Î

å

å 

Let C_ij be the collection of all (t_i,t_j)-cuts in Г_ij.

Integer programming formulation

( ) ( )

( )

minimize

subject to , , { , }

1,

0 , ,

{0,1},

ij ij

ij i

e e e E

ij ij

e e ij i j

e v e v

ij e

e t

ij

e e ij

e

l x

f f ij v V t t

f ij

f x ij e E

x e E

Î

Î Î

Î

   Î 

 

    Î

Î  Î

å

å å å

+ -

-

Г Г

Г

{ 0 , 1 , }

x

Î e Î E

0  x

 1 , e Î E

Linear Relaxation

0,7 0,7

0.3 0.

3 0.

3 0.

Integer programming

formulation

Fractional optimum = 27.5 Integer optimum = 28

Integer programming

formulation

A generating set is a subset F of pairs of terminals such

that a rectilinear network containing l1-paths for all pairs ij in F is a Manhattan network on T.



Generating sets

For example, the set of all pairs ij such that the rectangle R_ij defined by t_i and t_j is empty is a generating set.

t

t_j t_k



Horizontal

strips

Vertical strips

Strips and

Staircases

t

t

t

t

t

Staircase

s

t

t

Strips and

staircases

1. Solve the linear relaxation of the integer program

2. Execute the procedure Round_Strip to connect all pairs in strips.

3. Execute the procedure Round_Staircase to connect all pairs in staircases.

The algorithm

2. Procedure Round_Strip

x

+ x

≥ 1

e e’

t

t

The algorithm

2. Procedure Round_Strip

0.3 0.4 0.7

0.6 0.6 0.6 0.4

0.4

t

t

The algorithm

p’ p

2. Procedure Round_Strip

t

t

+0. 5 0.5

The algorithm

p’ p

3. Procedure Round_Staircase

t

t

’

t

t

The algorithm

3. Procedure Round_Staircase

0. 3

0. 1 0. 1

t

t

’

t

t

The algorithm

3. Procedure Round_Staircase

t

t

’

t

t

t

The algorithm

Analys is

Lemma. The network N(T) returned by the algorithm is a

Manhattan network.

Proof idea.

• N(T) contains an l₁-path between all pairs in strips and staircases.

• The set of all pairs in strips and staircases is a generating set.

Analys is

Lemma. The length of the network N(T) is at most twice the

cost of an optimal fractional solution.

Proof idea. To every rounded up edge e of N(T), we assign

a set E_e of edges parallel to e such that :

' '

(i) 1

e

2

e e E

x

Î

å ³

(ii) E

 E

 

,

Main result

Theorem.

Theorem. The rounding algorithm described in this talk achieves an approximation

guarantee of 2 for the Minimum Manhattan

Network problem.

Open

Questions

• Is the MMN problem NP-hard ?

• What is the worst integrality gap ?

• Does there exist an integrality gap in the case when the terminals define a

staircase ?

F-restricted MMN problem

Let F be a subset of pairs of terminals in T.

The F-restricted MMN problem is to find a rectilinear network of minimum length containing an l₁-path between every pair of terminals in F.

If (T,F) is a complete graph, then we obtain the MMN problem.

The NP-hard Minimum Rectilinear Arborescence Problem

Integrality gap for F-restricted MMN

Fractional optimum = 2

t

t

t

t

1/2 1/2

1/2

1/2

Integer optimum = 3

t

t

t

t

1

1 1

A simple example shows that, in this case, the integrality

gap is at least 3/2 :

Let F={ {t

,t

} ,

{t

,t

}}.