Modeling problems
with Integer Linear Programming
(ILP)
Name: Chuan Huang
Use of ILP

Solution to network or graph problems.
 Traveling salesman problem
 Shortest path
 Minimum spanning tree
 Knapsack
Definition of ILP
• Linear programming (LP) : mathematical method for determining a
way to achieve the best outcome in a given mathematical model for some
list of requirements represented as linear equations.
• Integer linear programming(ILP) : LP with some or all of the
variables are restricted to be integers.
• Canonical form:
minimize
cTx
subject to Ax  b
Modeling with ILP

Problem description

Formulation
 Input
 Output: Identify the decision variable which is binary;
 Objective function: Minimize/Maximize a function under
the constraints.
 Constraints: Rules that the feasible solutions will satisfy.

Solution
Solution to ILP

Package
 GNU Linear Programming Kit (GLPK)
 Cplex: mathematical optimizer

AMPL: A modeling language for Mathematical
programming developed by BELL Laboratories.

Gusek - LP IDE in Windows
 http://gusek.sourceforge.net/gusek.html
Example 1 - Vertex Cover
• Vertex cover: Is there a vertex cover of size K or less for G(V, E) such that for
each edge {u, v} E at least one of u and v belongs to V.
B
C
D

A
E
•
F
Minimum vertex cover: Find a smallest vertex cover in a given graph.
B
C
E
F
A
D
Example 1 - Input and Output
Minimum vertex cover
B
C
E
F
D
A
 Input:
(1) Set of vertexes V;
(2) Set of edges E;
(3) Costs of each vertex, c(i) = 1, where i  V;
 Output:
The set of the vertexes; x(i) = 1 if ith vertex
belongs to the set; otherwise x(i) = 0;
 Objective: Minimum costs,  c(i) x(i) .
iV
Example 1 – Input and Output(Cont)
Input
/* Number of blocks */
param m, integer, > 0;
/* Set of basic blocks */
set V, default {1..m};
/* Set of edges */
set E, within V cross V;
/*Cost of vertex*/
param c{i in V}, default 0;
Output
param m:=6;
param : V : c :=
1 1, 2 1,
3 1, 4 1,
5 1, 6 1;
param : E : a :=
1 2 1, 1 5 1 , 2 3 1,
2 5 1, 3 4 1 , 6 3 1,
6 5 1;
var x{i in V}, binary, >=0;
/*Decision*/
Vertex_cover.mod
Vertex_cover.dat
Example 1 – Objective and Constraints
Mathematic description:
AMPL:
Objective:
Minimize
 c(v)xv
minimize obj: sum{i in V} x[i]*c[i];
vV
(Minimize the
total cost)
Constraint:
xv  {0, 1}
for all v  E
(Vertex is either in the vertex cover or not)
xu+xv ≥ 1
s.t. phi{(i,j) in E}: x[i]+x[j] >= 1;
Solve;
for all {u, v} E
(Cover every edge of the graph)
Vertex_cover.mod
Example 2
- Vertex Cover with non-linear Costs
B
C
E
F
D
A
Assume input, output and objective are same except
Cost of each vertex: c(i) 
 x( j)
( i , j )E
Objective : Minimize
 c(i) x(i)
iV
Decided by
output x(i)
X2!
Not Linear!
Example 2
- Vertex Cover with non-linear Costs
 Linearise the equations by using new variable
Define variable cost(i), i  B which represents the final
cost of ith node.
 Cost of each
vertex:
c(i) 
 x( j)
( i , j )E
 Cost of each vertex cost(i )  c(i )  (1  x(i )) * 
when x(i) is decided: cost(i )  0
 Equivalent objective : Minimize
 cost(i)
iV
Linear!
Example 2
- Vertex Cover with non-linear Costs
B
C
E
F
D
A
 Output:
x = [1, 1, 0, 1, 0, 1];
cost = [1, 1, 0, 0, 0, 0];
Total cost = 2