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EMIS 8373: Integer Programming. Combinatorial Optimization Problems updated 27 January 2005. Combinatorial Optimization Problems. Input A finite set N = {1, 2, …, n} Weights (costs) c j for all j  N A set F of feasible subsets of N Optimization Problem

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Emis 8373 integer programming

EMIS 8373: Integer Programming

Combinatorial Optimization Problems

updated 27 January 2005


Combinatorial optimization problems
Combinatorial Optimization Problems

  • Input

    • A finite set N = {1, 2, …, n}

    • Weights (costs) cj for all j  N

    • A set F of feasible subsets of N

  • Optimization Problem

    • Find a minimum-weight feasible subset


Cop example minimum spanning tree mst
COP Example: Minimum Spanning Tree (MST)

  • Input

    • A (simple) graph G = (V,E)

    • Edge cost cij for each edge (i,j) E

  • Optimization Problem

    • Find a minimum-cost spanning tree

      • Spanning tree: a set of |V|-1 edges S such that each vertex is incident to at least one edge in S and S contains no cycles.


Mst example input

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MST Example: Input


Mst example some feasible spanning trees
MST Example: Some Feasible Spanning Trees

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cost = 11

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cost = 14

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cost = 12

cost = 9

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Example mst as cop
Example MST as COP

  • N = {(1, 2), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (4, 5)}.

  • c12 = 4, c15 = 2, c23 = 3, c24 = 5, c25 = 7, c34 = 3, c45 =1.

  • F is the set of all spanning trees in G

    • {(1, 2), (2, 3), (3, 4), (4, 5)}  F.

    • {(1, 2), (1, 5), (2, 3), (2, 4)}  F.

    • {(1, 2), (1, 5), (2, 5), (3, 4)}  F.

      • |F| can be very large relative to |E|

  • Optimization Problem

    • Find a minimum-weight feasible subset


The traveling salesman problem tsp
The Traveling Salesman Problem (TSP)

  • Input

    • N is a set of cities {1, 2, …, n}

    • Travel time tij between cities i and j

  • Optimization Problem

    • Find a tour of N that starts at city 1, visits each other city exactly once, and then returns to city 1 in the minimum possible time.


Symmetric tsp example
Symmetric TSP Example

tij Table

Graph Representation

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Stsp example feasible tours

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STSP Example: Feasible Tours

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Tour 1

cost = 13

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Tour 2

cost = 25

Tour 3

cost = 19


Example stsp as cop
Example STSP as COP

  • N = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3,4), (3,5), (4,5)}.

  • c12 = 4, c13 = 7, c12 = 7, etc.

  • F is the set of tours G where a tour is cycle that contains all the vertices of the graph.

    • {(1, 2), (1, 5), (2, 3), (3,4), (4,5)}  F.

  • Optimization Problem

    • Find a minimum-weight feasible subset


Solving tsp by enumeration
Solving TSP by Enumeration

  • Each feasible tour can be represented by as a permutation  of the cites.

    • The salesman visits the cities in the order determined by the permutation.

      • Tour 1 corresponds to  = (1, 2, 3, 4, 5)

      • Tour 2 corresponds to  = (1, 3, 5, 2, 4)

      • Tour 3 corresponds to  = (1, 2, 4, 3, 5)

    • For n cities there are n – 1 choices for the second city, n – 2 for the third, etc. Thus, there are (n-1)! permutations that need be evaluated.

    • Solving a TSP with 11 cities this way requires evaluating 3.6 million permutations!


Formulation tsp as bip
Formulation TSP as BIP

  • Let xij = 1 if the tour goes from city i to city j; and zero, otherwise.

  • Constraints

    • The tour enters city j exactly once

    • The tour leaves city i exactly once

  • Objective function


Bip solution for example stsp

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BIP Solution for Example STSP

  • x12 = x23 = x31 = x45 = x54 =1 and all other variables = 0.

  • Objective function value = 10

  • “Tour”

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Subtour elimination constraints
Subtour Elimination Constraints

  • To eliminate (prevent) the subtour (cycle) of the cities 1, 2, and 3, add the constraint

    x12 + x13 + x21 + x23 + x31 + x32 2.

  • To eliminate (prevent) the subtour (cycle) of the cities 4 and 5, add the constraint

    x45 + x54 1.


Bip with subtour elimination solution for example stsp
BIP with Subtour Elimination: Solution for Example STSP

  • x15 = x54 = x42 = x23 = x31 =1 and all other variables = 0.

  • Objective function value = 12.

  • Need to add 25-6 = 26 subtour elimination constraints.

    • Solving BIP with branch-and-bound may be faster than enumeration, but the number of constraints need grows exponentially with number of cities.


Cop example minimum cost cycle cover mccp
COP Example: Minimum-Cost Cycle Cover (MCCP)

  • Input

    • A (simple) graph G = (V,E)

    • Edge cost cij for each edge (i,j)  E

  • Optimization Problem

    • Find a minimum-cost set of cycles C = {C1, C2, …, Cj} in G such that each edge in E is covered by (contained in) at least one cycle in C.

    • The cost of a cycle is the total cost of the edges it covers.

    • The cost of a cycle cover is the total cost of its cycles.


Mccp example

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MCCP Example

Cycle 1

Cost = 13

Cycle 2

Cost = 13

Cycle 3

Cost = 11

Cost of cycle cover = 13+13+11 =37


Cycles in mccp example

Cycle 1

Cost = 13

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Cycles in MCCP Example

Cycle 3

Cost = 11

Cycle 2

Cost = 13

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Cycle 4

Cost = 13

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Cycle 6

Cost = 14

Cycle 5

Cost = 12


Example mccp as cop
Example MCCP as COP

  • N is the set of all cycles in G

    • N = {Cycle 1, Cycle 2, Cycle 3, Cycle 4, Cycle 5, Cycle 6}

    • Cycle 1 = {(1, 2), (1,5), (2, 5)}, Cycle 2 = {(2,4),(2,5),(4,5)}, etc.

    • In general, |N| >> |E|

  • c[Cycle 1] = 13, c[Cycle 2] = 13, …, c[Cycle 6] = 14.

  • F: a subset S of N is feasible if the union of all the cycles in S = E.

    • S = {Cycle 1, Cycle 2} is not feasible.

    • S = {Cycle 1, Cycle 2, Cycle 3} if feasible.


  • Formulation of mccp as bip
    Formulation of MCCP as BIP

    • Sets

      • Let C ={C1, C2, …C|C|} be the set of all cycles in the graph G=(V,E).

    • Constants

      • Let aij = 1 if cycle j covers edge i; and zero, otherwise.

        • Label the edges 1, 2, 3, …, |E|

      • Let wj = the cost of cycle j

    • Decision variables

      • Let the binary variable xj = 1 iff cycle j is selected as part of the cycle cover.


    Formulation of mccp as bip1
    Formulation of MCCP as BIP

    • Constraints

      • For each edge i, we must select at least one cycle j such where aij = 1. Therefore,

    • Objective Function



    The set covering problem
    The Set Covering Problem

    • Input

      • An m-row, n-column, zero-one matrix A (aij {0,1})

      • A cost cj associated with each column j

    • Optimization Problem


    The set covering problem1
    The Set Covering Problem

    • The matrix A is know as an incidencematrix and the columns are know as incidence vectors.

    • MCCP is a special case of set covering.