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
_{1} Paths
p_{y} – q_{y}
p
q
An l_{1}path is a shortest rectilinear path in the l_{1}metric, i.e. a rectilinear path of length ║pq║_{1}= p_{x } q_{x}+p_{y } q_{y}
Manhattan networks
A Manhattan network on a set of n terminals T = { t_{1} , …, t_{n} } is a network containing an l_{1}
path
between every pair of terminals of T, i.e. a 1spanner for T in the l_{1}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(n^{3}) [2002] R. Kato, K. Imai and T. Asano
Factor 2 approximation algorithm, O(n^{3}) [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 01 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}e E
l x
å
Ît
_{i}t
_{j}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
eÎ e Î E
0 x
_{e} 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 l1paths 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
_{i}t
_{i'}t
_{j'}t
_{j}t
_{k}Staircase
s
t
_{i}t
_{j}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
_{e}+ x
_{e’}≥ 1
e e’
t
_{i}t
The algorithm
2. Procedure Round_Strip
0.3 0.4 0.7
0.6 0.6 0.6 0.4
0.4
t
_{i}t
_{j}The algorithm
p’ p
2. Procedure Round_Strip
t
_{i}t
_{j}+0. 5 0.5
The algorithm
p’ p
3. Procedure Round_Staircase
t
_{k}t
_{j}’
t
_{i}t
_{j}The algorithm
3. Procedure Round_Staircase
0. 3
0. 1 0. 1
t
_{k}t
_{j}’
t
_{i}t
_{j}The algorithm
3. Procedure Round_Staircase
t
_{k}t
_{j}’
t
_{l}t
_{i}t
_{j}The algorithm
Analys is
Lemma. The network N(T) returned by the algorithm is a
Manhattan network.
Proof idea.
• N(T) contains an l_{1}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} E
_{f}
for any two rounded up edges e, f of N(T),
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 NPhard ?
• What is the worst integrality gap ?
• Does there exist an integrality gap in the case when the terminals define a
staircase ?
Frestricted MMN problem
Let F be a subset of pairs of terminals in T.
The Frestricted MMN problem is to find a rectilinear network of minimum length containing an l_{1}path between every pair of terminals in F.
If (T,F) is a complete graph, then we obtain the MMN problem.
The NPhard Minimum Rectilinear Arborescence Problem
Integrality gap for Frestricted MMN
Fractional optimum = 2
t
_{1}t
_{3}t
_{2}t
_{4}1/2 1/2
1/2
1/2
Integer optimum = 3
t
_{1}t
_{3}t
_{2}t
_{4}1
1 1
A simple example shows that, in this case, the integrality
gap is at least 3/2 :