1. Not(Pij & Pjk) or Pik (see implication in PL) NotPij or NotPjk or Pik
If Pij & Pjk then Pik
Not(Pij & Pjk) or Pik (see implication in PL)
NotPij or NotPjk or Pik
1-xij + 1-xjk + yik >= 1
xij + xjk + (1-yik) <=2
This works with objective function
Min sum of yik’s
So object function forces yik to be 0
Unless both xij,xjk are 1 then constraint forces yik to be 1
So a quadratic term is represented by yik and problem remains linear, i.e. ILP

2. Graph coloring Xv,c and Yc
Xv,c + Xu,c <= 1 for all (u,v) edges, c in other words vertices u,v must be assigned different colors if they share an edge.
We need Yc to represent the objective function which is to minimize total number of colors. If Xu,c=1 then Yc =1
Using PL, Puc=>Pc, 1-Xuc + Yc >=1
Yc >= Xuc for all u nodes, c

3. GUB constraint … Yc <= sum u Xuc for all colors
This provides an upper bound on Yc
Although objective function of sum over c Yc will force Yc’s to zero anyways if the color is not being used.

4. Set packing problem Let Xi represent a vertex i of the graph
Select max number of vertices
No two vertices can share an edge
Ax <= 1
Represents Xi + Xj <=1 for all (i,j) edges