Modeling problems with Integer Linear Programming (ILP)

1. Modeling problems with Integer Linear Programming (ILP) Name: Chuan Huang

2. Use of ILP • Solution to network or graph problems. • Traveling salesman problem • Shortest path • Minimum spanning tree • Knapsack

3. Definition of ILP minimize cTx subject to Ax b • 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:

4. 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

5. Solution to ILP • Package • GNU Linear Programming Kit (GLPK) • Cplex: mathematical optimizer • AMPL:A modeling language for Mathematical programmingdeveloped by BELL Laboratories. • Gusek - LP IDE in Windows • http://gusek.sourceforge.net/gusek.html

6. 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 D A E F

7. Example 1 - Input and Output Minimum vertex cover B C D A E F • Input: Set of vertexes V; Set of edges E; 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, .

8. Example 1 – Input and Output(Cont) 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; 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 var x{i in V}, binary, >=0; /*Decision*/ Vertex_cover.mod Vertex_cover.dat

9. Example 1 – Objective and Constraints Mathematic description: AMPL: Objective: minimize obj: sum{i in V} x[i]*c[i]; s.t. phi{(i,j) in E}: x[i]+x[j] >= 1; Solve; Minimize c(v)xv (Minimize the total cost) Constraint: xv {0, 1} for all v E (Vertex is either in the vertex cover or not) xu+xv ≥ 1 for all {u, v} E (Cover every edge of the graph) Vertex_cover.mod

10. Example 2- Vertex Cover with non-linear Costs B C D A E F Assume input, output and objective are same except Cost of each vertex: Decided by output x(i) Objective : Minimize X2! Not Linear!

11. 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: • Cost of each vertex • when x(i) is decided: • Equivalent objective : Minimize Linear!

12. Example 2- Vertex Cover with non-linear Costs B C D A E F • Output: x = [1, 1, 0, 1, 0, 1]; cost = [1, 1, 0, 0, 0, 0]; Total cost = 2