MATP6620/ISYE6760 Combinatorial Optimization & Integer Programming
Homework 3.
Solutions

Due: Thursday, March 4, 2021, by end of day.
Penalty for late homeworks: 10% for each day or part of a day.

The following graph G = (V,E) is used in question 1.

PICT

  1. Consider a node packing problem on the above graph, with each vertex having weight equal to 2 more than its degree. The LP relaxation includes the clique constraints vCxv 1 for each maximal clique C in the graph. The point xA = 0.4, xB = 0.6, xC = 0.4, xD = 0.5, xE = 0.4, xF = 0.5, xG = 0.1, xH = 0.4, and xJ = 0.1 is feasible in the LP relaxation.
    1. Show that the given point is not in the convex hull of feasible solutions, by giving a valid constraint that is violated by this point.
    2. Find an optimal solution to the node packing problem for this graph. Prove your solution is optimal.

    Solution:

    1. Vertices A,B,C,H,F constitute an odd hole, giving the valid constraint
      xA +xB + xC + xH + xF ≤ 2.

      The left hand side of this constraint evaluates to 2.3 for the given point.

    2. If xG = 1 then the largest possible value for the left hand side of the constraint in part (a) is 1, so the constraint can be lifted to
      xA + xB + xC + xH +xF + xG ≤ 2.
      (1)

      The vertices D,E,J form a clique, so we also have the valid constraint

      xD +xE + xJ ≤ 1.

      Thus, any packing can use at most 3 vertices.

      (We could also argue this using the odd hole constraint formed by {C,D,E,F,H}, which can be lifted with both G and J, so at most 2 of the 7 vertices {C,D,E,F,G,H,J} can be used. Also, at most one of {A,B} can be used.)

      The vertices and their degrees are:

      degreevertices


      5 C
      4 F,G,H,J
      3 A,D,E
      2 B

      We claim the node packing A,C,E with value 17 is optimal.

      First note, any packing with just 2 vertices has value at most 13. So the optimal packing must contain 3 vertices.

      Vertices C and F constitute a maximal packing, so they cannot both be part of an optimal packing. Vertex C is adjacent to all of G, H, and J, so any packing containing C with 3 nodes cannot have value larger than 17.

      F, G, H form a clique, so no packing can contain more than 2 of the vertices of degree 4.

      So no packing has value larger than 17.

      Alternatively:

      Solving the LP relaxation of the node packing problem with the lifted odd hole constraint (1) and the clique inequalities gives the integral solution. Here are the AMPL model and data files, and the AMPL output.

  2. Consider the constraints
       t1  ≥  |x1 - x2|
   t2  ≥  |x1 + x2 - 1|
 t1,t2     integer
x1,x2     binary

    1. By considering the different possibilities for x, show that t1 + t2 1.
    2. The constraints can be modeled equivalently as
         t1  ≥    x1 -   x2
   t1  ≥  - x1 +   x2
   t2  ≥    x1 +   x2  -  1
   t2  ≥  - x1 -   x2  +  1
 t1,t2     integer
x1,x2     binary

      Show that the valid constraint t1 + t2 1 has Chvatal rank equal to 2.

    Solution:

    1. Four choices for x, with the minimum possible corresponding choices for t:
             |      |
x1--x2-|t1--t2-|t1 +-t2
 0   0 |0   1 |  1
 1   0 |1   0 |  1
 0   1 |1   0 |  1
 1   1  0   1    1

    2. Rank is no larger than 2:

      Write constraints as:

       x1 - x2 - t1 ≤  0                                   (2)

- x1 + x2 - t1 ≤ 0                                   (3)
 x1 + x2 - t2 ≤  1                                   (4)
- x1 - x2 - t2 ≤ - 1.                                (5)

      0.5(2) + 0.5(4) implies

      x1 - 0.5t1 - t2 ≤ 0.5  =⇒   x1 - t1 - t2 ≤ 0.
      (6)

      0.5(3) + 0.5(5) implies

      - x1 - 0.5t1 - t2 ≤ - 0.5 =⇒   - x1 - t1 - t2 ≤ - 1.
      (7)

      In the second round, 0.5(6) + 0.5(7) implies

      - t1 - t2 ≤ - 0.5  =⇒   - t1 - t2 ≤ - 1

      as required.

      Rank is at least 2:

      The point x = (0.5,0.5) and t = (0,0) satisfies (2)–(5). This point is a convex combination of the integer points x = (0,0), t = (0,0) and x = (1,1), t = (0,0). So any valid linear combination of (2)–(5) must be satisfied by at least one of these points, and so the rounded version must also be satisfied by at least one of the points.

      Alternatively, since the point x = (0.5,0.5) and t = (0,0) is feasible in the LP relaxation, the maximum value of -t1 - t2 in the LP relaxation is 0. Thus, by the proposition in the notes, the best rank one inequality is only -t1 - t2 0, so the desired inequality has CG rank at least two.

  3. The optimal tableau to the linear programming relaxation of the integer program
    min   - x1 -  10x2                     min  - 36    +  32x3  +   1x4     17
s.t.    x1  +   2x2  ≤  14              s.t.  x1      +  4x3  -   4x4  =   2
      - x1 +   6x2  ≤  8          is            x2  +  18x3  +   18x4  =  114
             x1,x2  ≥  0,integer                          x,x ,x ,x   ≥  0
                                                          1 2  3  4

    where x3 and x4 are the slack variables in the two constraints. Find the Gomory and strong Gomory cutting planes implied by the two constraints. Express these constraints in terms of the original variables x1 and x2 and draw them on a graph of the feasible region.

    Solution:

    First constraint: Fractional parts: f3 = 3
4, f4 = 3
4, f0 = 1
2.

    Gomory cut:

    3x3 + 3 x4 ≥ 1 .
4    4     2

    In terms of the original variables, x3 = 14 - x1 - 2x2 and x4 = 8 + x1 - 6x2, so we have

    161- 6x2 ≥ 1,
  2        2

    or

          2
x2 ≤ 2 3
    (8)

    Mixed integer Gomory cut:

    1( 1- 34)      1( 1- 34 )     1
2  1--1- x3 + 2  1--1- x4 ≥ 2
      2             2

    or equivalently

    1x3 + 1x4 ≥ 1   or equivalently x3 + x4 ≥ 2.
4    4     2

    In terms of the original variables, we have

    22 - 8x2 ≥ 2,

    or

         1
x2 ≤ 2-.
     2
    (9)

    Second constraint: Fractional parts: f3 = 1
8, f4 = 1
8, f0 = 3
4.

    Gomory cut:

    1x3 + 1x4 ≥ 3
8     8     4

    In terms of the original variables, we have

    11- x2 ≥ 3,
4        4

    or

    x  ≤ 2.
 2
    (10)

    Mixed integer Gomory cut: Since f3 f0 and f4 f0, the mixed integer Gomory cut is identical to the original Gomory cut.

    Graph:

  4. Let x Bn satisfy the constraints
    xi + xj ≥ 1 for 1 ≤ i < j ≤ n.
    (11)

    Show that the constraint

    ∑n
   xi ≥ n- 1
 i=1
    (12)

    is valid. Give a fractional point with 0 x e that satisfies the original n(n - 1)2 constraints but violates the new constraint. Show that the new constraint has Chvatal rank no larger than O(log n).

    Solution:

  5. The AMPL model of the LP relaxation of a random weighted node packing problem with 15 nodes is contained in the file
    http://www.rpi.edu/~mitchj/matp6620/hw3/nodepack.mod

    The initial model contains only the adjacency constraint that just one endpoint of an edge can appear in the node packing. Pick a seed and then solve the problem using a cutting plane algorithm:

    1. Solve the LP relaxation.
    2. If the solution is integral, STOP.
    3. If necessary, add one or more valid inequalities to the LP. These inequalities can be clique inequalities or odd hole inequalities.
    4. Return to Step (a).

    (It is highly likely that you will need to use both clique inequalities and odd hole inequalities, and that these inequalities will be sufficient to solve the problem.)

    (Hint: The graph consists of the cycle 1 - 2 - 3 - 4 -- 14 - 15 - 1, plus some extra edges. You might be able to see the structure by displaying adjacency.)

    Solution:

    Here is the model file with the seed 3242. Solving this problem required the addition of two clique constraints and 4 odd hole constraints, with each clique containing 3 nodes and each odd hole containing 5 nodes. Here is the output file with the seed 3242.

  6. The Project:
    Along with your solutions to this homework, hand in a brief description of what you would like to do for the project part of this course.

References

[1]   R. Müller and A. S. Schulz. Transitive packing: a unifying concept in combinatorial optimization. SIAM Journal on Optimization, 13(2):335–367, 2002.

    John Mitchell
    Amos Eaton 325
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    mitchj at rpi dot edu
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