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.
- 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 ∑
v∈Cxv ≤ 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.
- 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.
- Find an optimal solution to the node packing problem for this graph. Prove your solution is
- Consider the constraints
- By considering the different possibilities for x, show that t1 + t2 ≥ 1.
- The constraints can be modeled equivalently as
Show that the valid constraint t1 + t2 ≥ 1 has Chvatal rank equal to 2.
- The optimal tableau to the linear programming relaxation of the integer program
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.
- Let x ∈ Bn satisfy the constraints
Show that the constraint ∑
i=1nxi ≥ n - 1 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).
- The AMPL model of the LP relaxation of a random weighted node packing problem with 15 nodes is
contained in the file
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:
- Solve the LP relaxation.
- If the solution is integral, STOP.
- If necessary, add one or more valid inequalities to the LP. These inequalities can be
clique inequalities or odd hole inequalities.
- 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.)
- 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.