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VC14_02_08

VIEWS: 6 PAGES: 34

									A rounding algorithm for approximating
    minimum Manhattan networks

   Victor Chepoi, Karim Nouioua, Yann Vaxès


     LIF -   Université de la Méditerranée
             Marseille - France




                   Seminaire du LIRMM, Montpellier, 14 Février 2008
                     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




                           Seminaire du LIRMM, Montpellier, 14 Février 2008
                           l1- Paths
An l1-path is a shortest rectilinear path in the l1-metric,
i.e. a rectilinear path of length ║p-q║1= |px - qx|+|py - qy|


p

                                                            |py – qy|

                                                    q

                     |px - qx|


                             Seminaire du LIRMM, Montpellier, 14 Février 2008
                Manhattan networks
A Manhattan network on a set of n terminals
T = { t1 , …, tn } 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.




                        Seminaire du LIRMM, Montpellier, 14 Février 2008
           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.




                            Seminaire du LIRMM, Montpellier, 14 Février 2008
A MMN on 100 terminals




         Seminaire du LIRMM, Montpellier, 14 Février 2008
                  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.)




                                  Seminaire du LIRMM, Montpellier, 14 Février 2008
                         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       Incomplete proof
   Factor 2 approximation algorithm, O(n3)

[2004] M. Benkert, T. Shirabe and A. Wolff
   Factor 3 approximation algorithm, O(n log n)

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




                                Seminaire du LIRMM, Montpellier, 14 Février 2008
                   Properties of MMN
• 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 .




                             Seminaire du LIRMM, Montpellier, 14 Février 2008
       Integer Programming Formulation

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
        - xe = 0 otherwise

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

                 minimize          ål x
                                   eÎE
                                         e e




                            Seminaire du LIRMM, Montpellier, 14 Février 2008
          Integer Programming Formulation
3. Constraints : for every pair (ti,tj) of terminals and for
   every (ti,tj)-cut c of the oriented subgraph Г ij = Г ∩ Rij :

                             åx
                             eÎc
                                   e   ³1


                                                                ti


            tj

                              Seminaire du LIRMM, Montpellier, 14 Février 2008
       Integer programming formulation


Let Cij be the collection of all (ti,tj)-cuts in Гij.


        mm
         n z 
        i i ielx
                        
                        eE
                          ee


        sbct  1
        u e o x
         j t                 e             jF i
                                         c iC j
                        
                        ec

                          0, 
                        x,} e E
                        e {1




                           Seminaire du LIRMM, Montpellier, 14 Février 2008
   Integer programming formulation

mi i e
i m
 n z   lx
         
         eE
              e e


uj t o 
   e
s b ct    f        i
                   e
                    j
                        ej, , Vt tj}
                          fi   j 
                               i v ij {,
                                       i
         j v
           +
         eГ ( )
           i            j v
                         -
                        eГ( )
                         i


          e 1
           fij ,                       
                                        ij
         j t
            -
         e Г (i)
            i


          e x
         0 fij  e,                      j 
                                        , E
                                         i e ij
         x , }
         e {1
           0 ,                           
                                         E
                                        e



                        Seminaire du LIRMM, Montpellier, 14 Février 2008
 Integer programming formulation

                        xe Î {0,1} ,         eÎ E
  Linear
Relaxation              0 £ xe £ 1,         eÎ E
               0,7
                                                  0,7

  0.3
         0.3
                          0.3
                                           0.3


                     Seminaire du LIRMM, Montpellier, 14 Février 2008
    Integer programming formulation




Integer optimum = 28             Fractional optimum = 27.5




                       Seminaire du LIRMM, Montpellier, 14 Février 2008
                    Generating sets

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.

For example, the set of all pairs ij such that the rectangle Rij
defined by ti and tj is empty is a generating set.

                                                      tk
                             tj     

                


       ti

                           Seminaire du LIRMM, Montpellier, 14 Février 2008
Horizontal strips




      Seminaire du LIRMM, Montpellier, 14 Février 2008
Vertical strips




     Seminaire du LIRMM, Montpellier, 14 Février 2008
Strips and Staircases




       Seminaire du LIRMM, Montpellier, 14 Février 2008
                 Staircases
            ti

                         tk


                                                                 tj
tj'

      ti'



                  Seminaire du LIRMM, Montpellier, 14 Février 2008
     Strips and staircases

                                            ti




tj



            Seminaire du LIRMM, Montpellier, 14 Février 2008
                     The algorithm

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.




                        Seminaire du LIRMM, Montpellier, 14 Février 2008
                  The algorithm

2. Procedure Round_Strip
                   ti


      e           e’           xe + xe’ ≥ 1



       tj

                        Seminaire du LIRMM, Montpellier, 14 Février 2008
              The algorithm

2. Procedure Round_Strip
                ti
    0.3        0.7
    0.4        0.6
      p’       p
    0.6        0.4
    0.6        0.4
      tj

                     Seminaire du LIRMM, Montpellier, 14 Février 2008
              The algorithm

2. Procedure Round_Strip

                    0.5

                            ti

                    +0.5 p
               p’


               tj



                     Seminaire du LIRMM, Montpellier, 14 Février 2008
              The algorithm

3. Procedure Round_Staircase
              ti

                        tk



                                                         tj
t j’
       ti’

                  Seminaire du LIRMM, Montpellier, 14 Février 2008
                   The algorithm

3. Procedure Round_Staircase
              ti

             0.3             tk
             0.1

             0.1

                                                              tj
t j’
       ti’

                       Seminaire du LIRMM, Montpellier, 14 Février 2008
                  The algorithm

3. Procedure Round_Staircase
             ti

                            tk
                                     tl


                                                             tj
t j’
       ti’

                      Seminaire du LIRMM, Montpellier, 14 Février 2008
                   Analysis

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.




                         Seminaire du LIRMM, Montpellier, 14 Février 2008
                   Analysis

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 Ee of edges parallel to e such that :

           1
()  ' 
 i      xe
    
   e E
    ' e    2

i e 
( )E E , for any two rounded up edges e, f of N(T)
 i    f




                         Seminaire du LIRMM, Montpellier, 14 Février 2008
                Main result


Theorem. The rounding algorithm described in this
  talk achieves an approximation guarantee of 2 for
  the Minimum Manhattan Network problem.




                     Seminaire du LIRMM, Montpellier, 14 Février 2008
              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 ?




                      Seminaire du LIRMM, Montpellier, 14 Février 2008
         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
can also be viewed as a F-restricted MMN problem.
In this case, (T,F) is a star.

                            Seminaire du LIRMM, Montpellier, 14 Février 2008
    Integrality gap for F-restricted MMN

A simple example shows that, in this case, the integrality
gap is at least 3/2 :

Let F={ {t1,t3} , {t2,t4}}.

     t1     1/2     t2                    t1        1       t2

     1/2           1/2                      1

     t4     1/2      t3                   t4         1       t3
   Fractional optimum = 2                Integer optimum = 3



                              Seminaire du LIRMM, Montpellier, 14 Février 2008

								
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