integer programming formulations for minimum spanning forest
TRANSCRIPT
Integer Programming Formulations for Minimum Spanning ForestProblem
Mehdi Golari
Systems and Industrial Engineering DepartmentThe University of Arizona
Math 543November 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 decisionmaking problems
Introduce mathematical programming formulations for minimum spanning tree andminimum spanning forest problems
Golari (SIE@UA) () IP Formulations for MSFP Nov 19, 2015 4 / 19
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
minx
f (x)
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is theconstraint 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
minx
f (x)
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is theconstraint 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
minx
f (x)
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is theconstraint 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
minx
f (x)
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is theconstraint 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
minx
f (x)
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is theconstraint 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
minx
f (x)
s.t x ∈ X
where x is the vector of decision variables, f (x) is the objective function, X is theconstraint 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
φ (H) =∑
e∈E(H)
φ (e) .
Definition
In a connected graph G , a minimal spanning tree T is a tree with minimum value.
MST problem in mathematical programming form:
minT
H(T ) =∑
e∈E(T )
φ (e)
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
φ (H) =∑
e∈E(H)
φ (e) .
Definition
In a connected graph G , a minimal spanning tree T is a tree with minimum value.
MST problem in mathematical programming form:
minT
H(T ) =∑
e∈E(T )
φ (e)
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
φ (H) =∑
e∈E(H)
φ (e) .
Definition
In a connected graph G , a minimal spanning tree T is a tree with minimum value.
MST problem in mathematical programming form:
minT
H(T ) =∑
e∈E(T )
φ (e)
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 if edge(i , j) is in tree
0 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 : x∗ij = 1} denotes the selected edge into the spanning tree.
Subtour elimination formulation is based on the fact that T has no simple cyclesand has n − 1 edges
[MST1] minx
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1∑(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∑(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 if edge(i , j) is in tree
0 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 : x∗ij = 1} denotes the selected edge into the spanning tree.
Subtour elimination formulation is based on the fact that T has no simple cyclesand has n − 1 edges
[MST1] minx
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1∑(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∑(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 if edge(i , j) is in tree
0 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 : x∗ij = 1} denotes the selected edge into the spanning tree.
Subtour elimination formulation is based on the fact that T has no simple cyclesand has n − 1 edges
[MST1] minx
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1∑(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∑(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
[MST2] minx
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1∑(i,j)∈δ(S) xij ≥ 1, ∀S ⊂ V , S 6= V , S 6= ∅
xij ∈ {0, 1}, ∀(i , j) ∈ E
where the cutset δ(S) ⊂ E is a subset of edges with one end in S and the other endin 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] minx,y
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1
y kij + y k
ji = xij , ∀(i , j) ∈ E , k ∈ V∑k∈V\{i,j} y
jik + xij = 1, ∀(i , j) ∈ E
xij , ykij , y
kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V
y kij ∈ {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 selectedinto the tree (xij = 1), any vertex k ∈ V must be either on the side of j (y k
ij = 1) or
on the side of i (y kji = 1). If (i , j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (y kij = y k
ji = 0)
The third constraint for (i , j) ∈ E ensures thatIf (i , j) ∈ E is in the tree (xij = 1), edges (i , k) who connects i are on the side of iIf (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 (y jik = 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] minx,y
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1
y kij + y k
ji = xij , ∀(i , j) ∈ E , k ∈ V∑k∈V\{i,j} y
jik + xij = 1, ∀(i , j) ∈ E
xij , ykij , y
kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V
y kij ∈ {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 selectedinto the tree (xij = 1), any vertex k ∈ V must be either on the side of j (y k
ij = 1) or
on the side of i (y kji = 1). If (i , j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (y kij = y k
ji = 0)
The third constraint for (i , j) ∈ E ensures thatIf (i , j) ∈ E is in the tree (xij = 1), edges (i , k) who connects i are on the side of iIf (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 (y jik = 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] minx,y
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1
y kij + y k
ji = xij , ∀(i , j) ∈ E , k ∈ V∑k∈V\{i,j} y
jik + xij = 1, ∀(i , j) ∈ E
xij , ykij , y
kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V
y kij ∈ {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 selectedinto the tree (xij = 1), any vertex k ∈ V must be either on the side of j (y k
ij = 1) or
on the side of i (y kji = 1). If (i , j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (y kij = y k
ji = 0)
The third constraint for (i , j) ∈ E ensures thatIf (i , j) ∈ E is in the tree (xij = 1), edges (i , k) who connects i are on the side of iIf (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 (y jik = 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] minx,y
∑(i,j)∈E
φijxij
s.t.
∑
(i,j)∈E xij = n − 1
y kij + y k
ji = xij , ∀(i , j) ∈ E , k ∈ V∑k∈V\{i,j} y
jik + xij = 1, ∀(i , j) ∈ E
xij , ykij , y
kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V
y kij ∈ {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 selectedinto the tree (xij = 1), any vertex k ∈ V must be either on the side of j (y k
ij = 1) or
on the side of i (y kji = 1). If (i , j) ∈ E is not in the tree (xij = 0), any vertex k
cannot be on the side of j nor i (y kij = y k
ji = 0)
The third constraint for (i , j) ∈ E ensures thatIf (i , j) ∈ E is in the tree (xij = 1), edges (i , k) who connects i are on the side of iIf (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 (y jik = 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 subgraphGi = (Vi ,Ei ) of G .
Proposition
For the graph G with m connected components, denoted by G1,G2, · · · ,Gm, the forestF ∗, consisting of spanning trees T ∗1 ,T
∗2 , · · · ,T ∗m, is a minimum spanning forest of G if
and only if each T ∗i 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 subgraphGi = (Vi ,Ei ) of G .
Proposition
For the graph G with m connected components, denoted by G1,G2, · · · ,Gm, the forestF ∗, consisting of spanning trees T ∗1 ,T
∗2 , · · · ,T ∗m, is a minimum spanning forest of G if
and only if each T ∗i 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 eliminationconstraints and cutset constraints:
(i) if S ⊂ Vi ,∑
(i,j)∈E(S) xij ≤ |S | − 1;∑
i∈S,j∈V\S xij ≥ 1;
(ii) if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ Vik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,∑(i,j)∈E(S) xij ≤ |S | − k;
∑i∈S,j∈V\S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),∑
(i,j)∈E(S) xij ≤ |S | − k;∑
i∈S,j∈V\S xij ≥ 0
Golari (SIE@UA) () IP Formulations for MSFP 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 eliminationconstraints and cutset constraints:
(i) if S ⊂ Vi ,∑
(i,j)∈E(S) xij ≤ |S | − 1;∑
i∈S,j∈V\S xij ≥ 1;
(ii) if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ Vik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,∑(i,j)∈E(S) xij ≤ |S | − k;
∑i∈S,j∈V\S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),∑
(i,j)∈E(S) xij ≤ |S | − k;∑
i∈S,j∈V\S xij ≥ 0
Golari (SIE@UA) () IP Formulations for MSFP 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 eliminationconstraints and cutset constraints:
(i) if S ⊂ Vi ,∑
(i,j)∈E(S) xij ≤ |S | − 1;∑
i∈S,j∈V\S xij ≥ 1;
(ii) if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ Vik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,∑(i,j)∈E(S) xij ≤ |S | − k;
∑i∈S,j∈V\S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),∑
(i,j)∈E(S) xij ≤ |S | − k;∑
i∈S,j∈V\S xij ≥ 0
Golari (SIE@UA) () IP Formulations for MSFP 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 eliminationconstraints and cutset constraints:
(i) if S ⊂ Vi ,∑
(i,j)∈E(S) xij ≤ |S | − 1;∑
i∈S,j∈V\S xij ≥ 1;
(ii) if S ⊂ Vi1 ∪ Vi2 ∪ · · · ∪ Vik (2 ≤ k ≤ m) and S ∩ Vi1 6= ∅, · · · , S ∩ Vik 6= ∅,∑(i,j)∈E(S) xij ≤ |S | − k;
∑i∈S,j∈V\S xij ≥ k;
(iii) if S = Vi1 ∪ Vi2 ∪ · · · ∪ Vik (1 ≤ k < m),∑
(i,j)∈E(S) xij ≤ |S | − k;∑
i∈S,j∈V\S xij ≥ 0
Golari (SIE@UA) () IP Formulations for MSFP Nov 19, 2015 14 / 19
Minimum Spanning Forest IP Formulations
Minimum Spanning Forest: Subtour Elimination Formulations
[MSF1] min∑
(i,j)∈E
φijxij
s.t.∑
(i,j)∈E
xij = n −m
∑(i,j)∈E(S)
xij ≤ |S | − 1, ∀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 15 / 19
Minimum Spanning Forest IP Formulations
Minimum Spanning Forest: Cutset Formulations
[MSF2] min∑
(i,j)∈E
φijxij
s.t.∑
(i,j)∈E
xij = n −m
∑i∈S,j∈V\S,(i,j)∈E
xij ≥ maxi∈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