Integer Programming Formulations for Minimum Spanning Forest ...

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


Introduction

Outline

1 Introduction



4 Conclusion


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


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


Introduction


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


Introduction


Definition


minx

f (x)

s.t x ∈ X








Introduction


Definition


minx

f (x)

s.t x ∈ X








Introduction


Definition


minx

f (x)

s.t x ∈ X








Introduction


Definition


minx

f (x)

s.t x ∈ X








Introduction


Definition


minx

f (x)

s.t x ∈ X








Minimum Spanning Tree IP Formulations

Outline

1 Introduction



4 Conclusion



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?





φ (H) =∑

e∈E(H)

φ (e) .

Definition



minT

H(T ) =∑

e∈E(T )

φ (e)







φ (H) =∑

e∈E(H)

φ (e) .

Definition



minT

H(T ) =∑

e∈E(T )

φ (e)





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 .




Let xij =


0 otherwise




[MST1] minx

∑(i,j)∈E

φijxij

s.t.

∑


xij ∈ {0, 1}, ∀(i , j) ∈ E





Let xij =


0 otherwise




[MST1] minx

∑(i,j)∈E

φijxij

s.t.

∑


xij ∈ {0, 1}, ∀(i , j) ∈ E




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.



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).




[MST4] minx,y

∑(i,j)∈E

φijxij

s.t.

∑

(i,j)∈E xij = n − 1

y kij + y k


jik + xij = 1, ∀(i , j) ∈ E

xij , ykij , y

kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V


side of j


ij = 1) or



ji = 0)






[MST4] minx,y

∑(i,j)∈E

φijxij

s.t.

∑

(i,j)∈E xij = n − 1

y kij + y k


jik + xij = 1, ∀(i , j) ∈ E

xij , ykij , y

kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V


side of j


ij = 1) or



ji = 0)






[MST4] minx,y

∑(i,j)∈E

φijxij

s.t.

∑

(i,j)∈E xij = n − 1

y kij + y k


jik + xij = 1, ∀(i , j) ∈ E

xij , ykij , y

kji ∈ {0, 1}, ∀(i , j) ∈ E , k ∈ V


side of j


ij = 1) or



ji = 0)




Minimum Spanning Forest IP Formulations

Outline

1 Introduction



4 Conclusion



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.



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.



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





(i) if S ⊂ Vi ,∑

(i,j)∈E(S) xij ≤ |S | − 1;∑





(i,j)∈E(S) xij ≤ |S | − k;∑






(i) if S ⊂ Vi ,∑

(i,j)∈E(S) xij ≤ |S | − 1;∑





(i,j)∈E(S) xij ≤ |S | − k;∑






(i) if S ⊂ Vi ,∑

(i,j)∈E(S) xij ≤ |S | − 1;∑





(i,j)∈E(S) xij ≤ |S | − k;∑




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.



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.


Conclusion

Outline

1 Introduction



4 Conclusion


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!


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!


Conclusion

Questions?


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