Integer Programming Formulations for Minimum Spanning Forest
Problem
Mehdi Golari
Systems and Industrial Engineering Department
The University of Arizona
Math 543
November 19, 2015
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
1 / 19
Outline
1
Introduction
2
Minimum Spanning Tree IP Formulations
3
Minimum Spanning Forest IP Formulations
4
Conclusion
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
2 / 19
Introduction
Outline
1
Introduction
2
Minimum Spanning Tree IP Formulations
3
Minimum Spanning Forest IP Formulations
4
Conclusion
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
3 / 19
Introduction
Goals for this talk
Introduce mathematical programming as a general framework to solve decision
making problems
Introduce mathematical programming formulations for minimum spanning tree and
minimum spanning forest problems
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
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Introduction
Operations Research: science of decision making, science of better
Some of the mathematical tools to approach decision making?
Mathematical Programming
Control Theory
Decision Analysis
Game Theory
Queuing Theory
Simulation
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
5 / 19
Introduction
Mathematical Programming
Definition
A general mathematical program has the form
min f (x)
x
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is the
constraint set, {x ∈ X } is the feasible region.
Different assumptions on f (x) and X results in different classes of mathematical programs
Linear Programming (LP): f (x) = cx, X = {Ax ≥ b}, x ∈ Rn , c ∈ Rn , b ∈ Rm .
Nonlinear Programming (NLP): f (x) nonlinear in x, and/or X a nonlinear set.
Integer Linear Programming (ILP): Same assumptions as LP, except x ∈ Zn
Mixed Integer LP (MILP): Same assumptions as LP, except x ∈ Rn1 × Zn2
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
6 / 19
Introduction
Mathematical Programming
Definition
A general mathematical program has the form
min f (x)
x
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is the
constraint set, {x ∈ X } is the feasible region.
Different assumptions on f (x) and X results in different classes of mathematical programs
Linear Programming (LP): f (x) = cx, X = {Ax ≥ b}, x ∈ Rn , c ∈ Rn , b ∈ Rm .
Nonlinear Programming (NLP): f (x) nonlinear in x, and/or X a nonlinear set.
Integer Linear Programming (ILP): Same assumptions as LP, except x ∈ Zn
Mixed Integer LP (MILP): Same assumptions as LP, except x ∈ Rn1 × Zn2
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
6 / 19
Introduction
Mathematical Programming
Definition
A general mathematical program has the form
min f (x)
x
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is the
constraint set, {x ∈ X } is the feasible region.
Different assumptions on f (x) and X results in different classes of mathematical programs
Linear Programming (LP): f (x) = cx, X = {Ax ≥ b}, x ∈ Rn , c ∈ Rn , b ∈ Rm .
Nonlinear Programming (NLP): f (x) nonlinear in x, and/or X a nonlinear set.
Integer Linear Programming (ILP): Same assumptions as LP, except x ∈ Zn
Mixed Integer LP (MILP): Same assumptions as LP, except x ∈ Rn1 × Zn2
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
6 / 19
Introduction
Mathematical Programming
Definition
A general mathematical program has the form
min f (x)
x
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is the
constraint set, {x ∈ X } is the feasible region.
Different assumptions on f (x) and X results in different classes of mathematical programs
Linear Programming (LP): f (x) = cx, X = {Ax ≥ b}, x ∈ Rn , c ∈ Rn , b ∈ Rm .
Nonlinear Programming (NLP): f (x) nonlinear in x, and/or X a nonlinear set.
Integer Linear Programming (ILP): Same assumptions as LP, except x ∈ Zn
Mixed Integer LP (MILP): Same assumptions as LP, except x ∈ Rn1 × Zn2
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
6 / 19
Introduction
Mathematical Programming
Definition
A general mathematical program has the form
min f (x)
x
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is the
constraint set, {x ∈ X } is the feasible region.
Different assumptions on f (x) and X results in different classes of mathematical programs
Linear Programming (LP): f (x) = cx, X = {Ax ≥ b}, x ∈ Rn , c ∈ Rn , b ∈ Rm .
Nonlinear Programming (NLP): f (x) nonlinear in x, and/or X a nonlinear set.
Integer Linear Programming (ILP): Same assumptions as LP, except x ∈ Zn
Mixed Integer LP (MILP): Same assumptions as LP, except x ∈ Rn1 × Zn2
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
6 / 19
Introduction
Mathematical Programming
Definition
A general mathematical program has the form
min f (x)
x
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is the
constraint set, {x ∈ X } is the feasible region.
Different assumptions on f (x) and X results in different classes of mathematical programs
Linear Programming (LP): f (x) = cx, X = {Ax ≥ b}, x ∈ Rn , c ∈ Rn , b ∈ Rm .
Nonlinear Programming (NLP): f (x) nonlinear in x, and/or X a nonlinear set.
Integer Linear Programming (ILP): Same assumptions as LP, except x ∈ Zn
Mixed Integer LP (MILP): Same assumptions as LP, except x ∈ Rn1 × Zn2
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
6 / 19
Minimum Spanning Tree IP Formulations
Outline
1
Introduction
2
Minimum Spanning Tree IP Formulations
3
Minimum Spanning Forest IP Formulations
4
Conclusion
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
7 / 19
Minimum Spanning Tree IP Formulations
Recall: Minimum Spanning Tree
Given a network (G , φ) , we can define the weight of a subgraph H ⊂ G as
X
φ (H) =
φ (e) .
e∈E (H)
Definition
In a connected graph G , a minimal spanning tree T is a tree with minimum value.
MST problem in mathematical programming form:
X
φ (e)
min H(T ) =
T
e∈E (T )
s.t T is a tree in G
How to characterize the set of constraints and objective function explicitly?
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
8 / 19
Minimum Spanning Tree IP Formulations
Recall: Minimum Spanning Tree
Given a network (G , φ) , we can define the weight of a subgraph H ⊂ G as
X
φ (H) =
φ (e) .
e∈E (H)
Definition
In a connected graph G , a minimal spanning tree T is a tree with minimum value.
MST problem in mathematical programming form:
X
φ (e)
min H(T ) =
T
e∈E (T )
s.t T is a tree in G
How to characterize the set of constraints and objective function explicitly?
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
8 / 19
Minimum Spanning Tree IP Formulations
Recall: Minimum Spanning Tree
Given a network (G , φ) , we can define the weight of a subgraph H ⊂ G as
X
φ (H) =
φ (e) .
e∈E (H)
Definition
In a connected graph G , a minimal spanning tree T is a tree with minimum value.
MST problem in mathematical programming form:
X
φ (e)
min H(T ) =
T
e∈E (T )
s.t T is a tree in G
How to characterize the set of constraints and objective function explicitly?
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
8 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Subtour Elimination Formulation
(
Let xij =
1
0
if edge(i, j) is in tree
otherwise
Let x denote the vector formed by xij ’s for all (i, j) ∈ E .
The MST found by optimal x ∗ , denoted T ∗ , will be a subgraph T ∗ = (V , E ∗ ),
where E ∗ = {(i, j) ∈ E : xij∗ = 1} denotes the selected edge into the spanning tree.
Subtour elimination formulation is based on the fact that T has no simple cycles
and has n − 1 edges
X
φij xij
[MST1] min
x
(i,j)∈E
P

P(i,j)∈E xij = n − 1
s.t.
(i,j)∈E (S) xij ≤ |S| − 1, ∀S ⊂ V , S 6= V , S 6= ∅


xij ∈ {0, 1}, ∀(i, j) ∈ E
where E (S) ⊂ E is a subset of edges with both ends in subset S ⊂ V . Constraint
P
(i,j)∈E (S) xij ≤ |S| − 1 ensures that there is no cycles in subset S.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
9 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Subtour Elimination Formulation
(
Let xij =
1
0
if edge(i, j) is in tree
otherwise
Let x denote the vector formed by xij ’s for all (i, j) ∈ E .
The MST found by optimal x ∗ , denoted T ∗ , will be a subgraph T ∗ = (V , E ∗ ),
where E ∗ = {(i, j) ∈ E : xij∗ = 1} denotes the selected edge into the spanning tree.
Subtour elimination formulation is based on the fact that T has no simple cycles
and has n − 1 edges
X
φij xij
[MST1] min
x
(i,j)∈E
P

P(i,j)∈E xij = n − 1
s.t.
(i,j)∈E (S) xij ≤ |S| − 1, ∀S ⊂ V , S 6= V , S 6= ∅


xij ∈ {0, 1}, ∀(i, j) ∈ E
where E (S) ⊂ E is a subset of edges with both ends in subset S ⊂ V . Constraint
P
(i,j)∈E (S) xij ≤ |S| − 1 ensures that there is no cycles in subset S.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
9 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Subtour Elimination Formulation
(
Let xij =
1
0
if edge(i, j) is in tree
otherwise
Let x denote the vector formed by xij ’s for all (i, j) ∈ E .
The MST found by optimal x ∗ , denoted T ∗ , will be a subgraph T ∗ = (V , E ∗ ),
where E ∗ = {(i, j) ∈ E : xij∗ = 1} denotes the selected edge into the spanning tree.
Subtour elimination formulation is based on the fact that T has no simple cycles
and has n − 1 edges
X
φij xij
[MST1] min
x
(i,j)∈E
P

P(i,j)∈E xij = n − 1
s.t.
(i,j)∈E (S) xij ≤ |S| − 1, ∀S ⊂ V , S 6= V , S 6= ∅


xij ∈ {0, 1}, ∀(i, j) ∈ E
where E (S) ⊂ E is a subset of edges with both ends in subset S ⊂ V . Constraint
P
(i,j)∈E (S) xij ≤ |S| − 1 ensures that there is no cycles in subset S.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
9 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Cutset Formulation
Cutset formulation is based on the fact that T is connected and has n − 1 edges
X
[MST2] min
φij xij
x
(i,j)∈E
P

P(i,j)∈E xij = n − 1
s.t.
(i,j)∈δ(S) xij ≥ 1, ∀S ⊂ V , S 6= V , S 6= ∅


xij ∈ {0, 1}, ∀(i, j) ∈ E
where the cutset δ(S) P
⊂ E is a subset of edges with one end in S and the other end
in V \ S. Constraints (i,j)∈δ(S) xij ≥ 1 ensures that subsets S and V \ S are
connected.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
10 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Martin’s formulation
[MST4]
min
X
x,y
φij xij
(i,j)∈E
P

(i,j)∈E xij = n − 1


y k + y k = x , ∀(i, j) ∈ E , k ∈ V
ij
ij
ji
s.t. P
j

y

\{i,j} ik + xij = 1, ∀(i, j) ∈ E

 k∈V
xij , yijk , yjik ∈ {0, 1}, ∀(i, j) ∈ E , k ∈ V
yijk ∈ {0, 1} denotes that edge (i, j) is in the spanning tree and vertex k is on the
side of j
The second constraint for (i, j) ∈ E , k ∈ V guarantees that if (i, j) ∈ E is selected
into the tree (xij = 1), any vertex k ∈ V must be either on the side of j (yijk = 1) or
on the side of i (yjik = 1). If (i, j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (yijk = yjik = 0)
The third constraint for (i, j) ∈ E ensures that
If (i, j) ∈ E is in the tree (xij = 1), edges (i, k) who connects i are on the side of i
If (i, j) ∈ E is not in the tree (xij = 0), there must be an edge (i, k) such that j is on
the side of k (yikj = 1 for some k).
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
11 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Martin’s formulation
[MST4]
min
X
x,y
φij xij
(i,j)∈E
P

(i,j)∈E xij = n − 1


y k + y k = x , ∀(i, j) ∈ E , k ∈ V
ij
ij
ji
s.t. P
j

y

\{i,j} ik + xij = 1, ∀(i, j) ∈ E

 k∈V
xij , yijk , yjik ∈ {0, 1}, ∀(i, j) ∈ E , k ∈ V
yijk ∈ {0, 1} denotes that edge (i, j) is in the spanning tree and vertex k is on the
side of j
The second constraint for (i, j) ∈ E , k ∈ V guarantees that if (i, j) ∈ E is selected
into the tree (xij = 1), any vertex k ∈ V must be either on the side of j (yijk = 1) or
on the side of i (yjik = 1). If (i, j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (yijk = yjik = 0)
The third constraint for (i, j) ∈ E ensures that
If (i, j) ∈ E is in the tree (xij = 1), edges (i, k) who connects i are on the side of i
If (i, j) ∈ E is not in the tree (xij = 0), there must be an edge (i, k) such that j is on
the side of k (yikj = 1 for some k).
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
11 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Martin’s formulation
[MST4]
min
X
x,y
φij xij
(i,j)∈E
P

(i,j)∈E xij = n − 1


y k + y k = x , ∀(i, j) ∈ E , k ∈ V
ij
ij
ji
s.t. P
j

y

\{i,j} ik + xij = 1, ∀(i, j) ∈ E

 k∈V
xij , yijk , yjik ∈ {0, 1}, ∀(i, j) ∈ E , k ∈ V
yijk ∈ {0, 1} denotes that edge (i, j) is in the spanning tree and vertex k is on the
side of j
The second constraint for (i, j) ∈ E , k ∈ V guarantees that if (i, j) ∈ E is selected
into the tree (xij = 1), any vertex k ∈ V must be either on the side of j (yijk = 1) or
on the side of i (yjik = 1). If (i, j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (yijk = yjik = 0)
The third constraint for (i, j) ∈ E ensures that
If (i, j) ∈ E is in the tree (xij = 1), edges (i, k) who connects i are on the side of i
If (i, j) ∈ E is not in the tree (xij = 0), there must be an edge (i, k) such that j is on
the side of k (yikj = 1 for some k).
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
11 / 19
Minimum Spanning Tree IP Formulations
Minimum Spanning Tree: Martin’s formulation
[MST4]
min
X
x,y
φij xij
(i,j)∈E
P

(i,j)∈E xij = n − 1


y k + y k = x , ∀(i, j) ∈ E , k ∈ V
ij
ij
ji
s.t. P
j

y

\{i,j} ik + xij = 1, ∀(i, j) ∈ E

 k∈V
xij , yijk , yjik ∈ {0, 1}, ∀(i, j) ∈ E , k ∈ V
yijk ∈ {0, 1} denotes that edge (i, j) is in the spanning tree and vertex k is on the
side of j
The second constraint for (i, j) ∈ E , k ∈ V guarantees that if (i, j) ∈ E is selected
into the tree (xij = 1), any vertex k ∈ V must be either on the side of j (yijk = 1) or
on the side of i (yjik = 1). If (i, j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (yijk = yjik = 0)
The third constraint for (i, j) ∈ E ensures that
If (i, j) ∈ E is in the tree (xij = 1), edges (i, k) who connects i are on the side of i
If (i, j) ∈ E is not in the tree (xij = 0), there must be an edge (i, k) such that j is on
the side of k (yikj = 1 for some k).
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
11 / 19
Minimum Spanning Forest IP Formulations
Outline
1
Introduction
2
Minimum Spanning Tree IP Formulations
3
Minimum Spanning Forest IP Formulations
4
Conclusion
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
12 / 19
Minimum Spanning Forest IP Formulations
Minimum Spanning Forest
Consider a graph G with m connected components
Assume that the m connected components of G have vertex sets as V1 , V2 · · · , Vm
Also assume Ei is the edge set induced by vertices in Vi from graph G
Thus, each connected component of G can be considered as a subgraph
Gi = (Vi , Ei ) of G .
Proposition
For the graph G with m connected components, denoted by G1 , G2 , · · · , Gm , the forest
F ∗ , consisting of spanning trees T1∗ , T2∗ , · · · , Tm∗ , is a minimum spanning forest of G if
and only if each Ti∗ is a minimum spanning tree for subgraph Gi (i = 1, 2, · · · , m).
Furthermore, the number of edges in a spanning forest of G is n − m.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
13 / 19
Minimum Spanning Forest IP Formulations
Minimum Spanning Forest
Consider a graph G with m connected components
Assume that the m connected components of G have vertex sets as V1 , V2 · · · , Vm
Also assume Ei is the edge set induced by vertices in Vi from graph G
Thus, each connected component of G can be considered as a subgraph
Gi = (Vi , Ei ) of G .
Proposition
For the graph G with m connected components, denoted by G1 , G2 , · · · , Gm , the forest
F ∗ , consisting of spanning trees T1∗ , T2∗ , · · · , Tm∗ , is a minimum spanning forest of G if
and only if each Ti∗ is a minimum spanning tree for subgraph Gi (i = 1, 2, · · · , m).
Furthermore, the number of edges in a spanning forest of G is n − m.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
13 / 19
Minimum Spanning Forest IP Formulations
Adapting Subtour Elimination and Cutset Formulations for MSF
Considering S ⊂ V , S 6= ∅, S 6= V , there are three cases for the subtour elimination
constraints and cutset constraints:
(i) if S ⊂ Vi ,
P
(i,j)∈E (S)
xij ≤ |S| − 1;
P
i∈S,j∈V \S
xij ≥ 1;
(ii) P
if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ VP
ik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,
x
≤
|S|
−
k;
ij
(i,j)∈E (S)
i∈S,j∈V \S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),
Golari (SIE@UA) ()
P
(i,j)∈E (S)
IP Formulations for MSFP
xij ≤ |S| − k;
P
i∈S,j∈V \S
xij ≥ 0
Nov 19, 2015
14 / 19
Minimum Spanning Forest IP Formulations
Adapting Subtour Elimination and Cutset Formulations for MSF
Considering S ⊂ V , S 6= ∅, S 6= V , there are three cases for the subtour elimination
constraints and cutset constraints:
(i) if S ⊂ Vi ,
P
(i,j)∈E (S)
xij ≤ |S| − 1;
P
i∈S,j∈V \S
xij ≥ 1;
(ii) P
if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ VP
ik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,
x
≤
|S|
−
k;
ij
(i,j)∈E (S)
i∈S,j∈V \S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),
Golari (SIE@UA) ()
P
(i,j)∈E (S)
IP Formulations for MSFP
xij ≤ |S| − k;
P
i∈S,j∈V \S
xij ≥ 0
Nov 19, 2015
14 / 19
Minimum Spanning Forest IP Formulations
Adapting Subtour Elimination and Cutset Formulations for MSF
Considering S ⊂ V , S 6= ∅, S 6= V , there are three cases for the subtour elimination
constraints and cutset constraints:
(i) if S ⊂ Vi ,
P
(i,j)∈E (S)
xij ≤ |S| − 1;
P
i∈S,j∈V \S
xij ≥ 1;
(ii) P
if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ VP
ik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,
x
≤
|S|
−
k;
ij
(i,j)∈E (S)
i∈S,j∈V \S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),
Golari (SIE@UA) ()
P
(i,j)∈E (S)
IP Formulations for MSFP
xij ≤ |S| − k;
P
i∈S,j∈V \S
xij ≥ 0
Nov 19, 2015
14 / 19
Minimum Spanning Forest IP Formulations
Adapting Subtour Elimination and Cutset Formulations for MSF
Considering S ⊂ V , S 6= ∅, S 6= V , there are three cases for the subtour elimination
constraints and cutset constraints:
(i) if S ⊂ Vi ,
P
(i,j)∈E (S)
xij ≤ |S| − 1;
P
i∈S,j∈V \S
xij ≥ 1;
(ii) P
if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ VP
ik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,
x
≤
|S|
−
k;
ij
(i,j)∈E (S)
i∈S,j∈V \S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),
Golari (SIE@UA) ()
P
(i,j)∈E (S)
IP Formulations for MSFP
xij ≤ |S| − k;
P
i∈S,j∈V \S
xij ≥ 0
Nov 19, 2015
14 / 19
Minimum Spanning Forest IP Formulations
Minimum Spanning Forest: Subtour Elimination Formulations
[MSF1]
min
X
φij xij
(i,j)∈E
s.t.
X
xij = n − m
(i,j)∈E
X
xij ≤ |S| − 1, ∀S ⊂ V , S 6= V , S 6= ∅
(i,j)∈E (S)
xij ∈ {0, 1}, ∀(i, j) ∈ E
where the first constraint ensures that there are n − m edges in the spanning forest.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
15 / 19
Minimum Spanning Forest IP Formulations
Minimum Spanning Forest: Cutset Formulations
[MSF2]
min
X
φij xij
(i,j)∈E
s.t.
X
xij = n − m
(i,j)∈E
X
i∈S,j∈V \S,(i,j)∈E
xij ≥
max
i∈S,j∈V \S
1{(i,j)∈E } , ∀S ⊂ V , S 6= V , S 6= ∅
xij ∈ {0, 1}, ∀(i, j) ∈ E
where the first constraint ensures that there are n − m edges in the spanning forest.
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
16 / 19
Conclusion
Outline
1
Introduction
2
Minimum Spanning Tree IP Formulations
3
Minimum Spanning Forest IP Formulations
4
Conclusion
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
17 / 19
Conclusion
Conclusion
Covered:
Introduced mathematical programming
IP formulations for MST and MSF
Not covered:
How to solve these problems?
Polyhedral study and comparison of the formulations!
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
18 / 19
Conclusion
Conclusion
Covered:
Introduced mathematical programming
IP formulations for MST and MSF
Not covered:
How to solve these problems?
Polyhedral study and comparison of the formulations!
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
18 / 19
Conclusion
Questions?
Golari (SIE@UA) ()
IP Formulations for MSFP
Nov 19, 2015
19 / 19