IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

IEOR269 notes, Prof. Hochbaum, 2010 13
Then the formulation is
min
∑
w ij x ij
s.t.
[i,j]∈E
∑
x ij ≤ |S| − 1
∀ S ⊂ V
i∈S,j∈S,[i,j]∈E
∑
x ij = |V | − 1
[i,j]∈E
x ij ≥ 0 ∀ [i, j] ∈ E.
We mention two things here:
• Originally, the last constraint x ij ≥ 0 should be x ij ∈ {0, 1}. It can be shown that this binary
constraint can be relaxed without affecting the optimal solution.
• For each subset of V , we have a constraint. Therefore, the number of constraints is O(2 n ),
which means we have an exponential number of constraints.
The discussion on MST will be continued in the next lecture.
Lec3
9 The Minimum Spanning Tree (MST) Problem
Definition: Let there be a graph G = (V, E) with weight c ij assigned to each edge e = (i, j) ∈ E,
and let n = |V |.
The MST problem is finding a tree that connects all the vertices of V and is of minimum total edge
cost.
The MST has the following linear programming (LP) formulation:
Decision Variables:
{ 1 if edge e=(i,j) is selected
X ij =
0 o/w
min
s.t
∑
(i,j)∈E
∑
(i,j)∈E
∑
i,j∈S
(i,j)∈E
X ij ≥ 0
c ij .X ij
X ij = n − 1
X ij ≤ |S| − 1 (∀S ⊂ V )