ILP Formulations
Spanning Trees
Cuttingplane separation
kMST
C/G Cutting Planes
ILP Formulation Properties and Strenghtening
Techniques
Version 2009.2
Andreas M. Chwatal
Algorithms and Data Structures Group
Institute of Computer Graphics and Algorithms
Vienna University of Technology
VU Fortgeschrittene Algorithmen und Datenstrukturen
Mai 2009
1
ILP Formulations
Spanning Trees
Cuttingplane separation
kMST
C/G Cutting Planes
What is a strong/good (I)LP formulation?
Linear program:
low number of variables
low number of constraints
complexity grows polynomially in these entities
Integer Linear Program:
Let
F
=
{
x
1
, . . . ,
x
k
}
be an feasible integer solution to some ILP
Convex hull
conv
(
F
) =
k
i
=1
λ
i
x
i
k
i
=1
λ
i
= 1
, λ
i
≥
0
Let
P
denote the polyhedron associated to the linear relaxation of
the ILP:
conv
(
F
)
⊆
P
Optimal case:
conv
(
F
) =
P
, but this is often not achievable
In a strong formulation
P
closely approximates
conv
(
F
)
2
ILP Formulations
Spanning Trees
Cuttingplane separation
kMST
C/G Cutting Planes
The pigeonhole
Suppose, we want to place
n
+ 1 items into
n
holes, such that each hole
contains exactly one item. (This is clearly infeasible)
n
j
=1
x
ij
= 1
,
i
= 1
, . . . ,
n
+ 1
(1a)
x
ij
∈ {
0
,
1
}
i
= 1
, . . . ,
n
+ 1
,
j
= 1
, . . . ,
n
(1b)
Two alternatives for the additional constraints:
x
ij
+
x
kj
≤
1
,
j
= 1
, . . . ,
n
,
i
=
k
,
i
,
k
= 1
, . . . ,
n
+ 1
(2)
n
+1
i
=1
x
ij
≤
1
,
j
= 1
, . . . ,
n
(3)
Linear relaxation of formulation with (2) is feasible (with
x
ij
= 1
/
n
), but
infeasible with (3).
3
ILP Formulations
Spanning Trees
Cuttingplane separation
kMST
C/G Cutting Planes
Example: Facility Location
Given:
n
potential facility locations with opening costs
c
j
,
m
clients with
service costs
d
ij
(for client
i
from facility
j
)
Formulation 1:
min
n
j
=1
c
j
y
j
+
m
i
=1
n
j
=1
d
ij
x
ij
(4a)
s
.
t
.
n
j
=1
x
ij
= 1
for all
i
(4b)
x
ij
≤
y
j
for all
i
,
j
(4c)
0
≤
x
ij
≤
1
,
y
i
∈ {
0
,
1
}
for all
i
,
j
(4d)
4
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ILP Formulations
Spanning Trees
Cuttingplane separation
kMST
C/G Cutting Planes
Facility Location (cont.)
Formulation 2:
min
n
j
=1
c
j
y
j
+
m
i
=1
n
j
=1
d
ij
x
ij
(5a)
s
.
t
.
n
j
=1
x
ij
= 1
for all
i
(5b)
m
i
=1
x
ij
≤
m
·
y
j
for all
j
(5c)
0
≤
x
ij
≤
1
,
y
i
∈ {
0
,
1
}
for all
i
,
j
(5d)
Which formulation is better?
F1 has
n
+
nm
constraints, whereas F2 has
n
+
m
constraints
But,
P
F
1
⊂
P
F
2
!! (where
P
X
denotes the polyhedron corresponding
to the LP relaxation of formulation
X
)
5
ILP Formulations
Spanning Trees
Cuttingplane separation
kMST
C/G Cutting Planes
Spanning Trees
In this section we study various ST formulations.
Why?
Spanning tree (ST) problems arise in various contexts, often as
subproblems to more complex problems.
While the MST is solvable in polynomial time, further constraints usually
make the problem NPhard.
Variants:
Minimum (Weight) Spanning Tree
Steiner Tree, Price Collecting ST
{
Delay, Resource, Hop, Diameter, ...
}
 Constrained S.T.
Minimum Label Spanning Tree
. . .
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 Spring '97
 RAVINDRAN
 Linear Programming, Linear Programming Relaxation, Spanning tree, C/G Cutting Planes, ILP Formulation Properties, ILP Formulations

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