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

IEOR269 notes, Prof. Hochbaum, 2010 21
(l − 1) + (k − 1) − ∑ e∈P
X e ≤ k + l − (p + 1) − 1
p ≤ ∑ e∈P
X e
⇒ X e = 1 ∀e ∈ P
Done!
Step 2:
There is no fractional tight cycles.
Proof of Step 2: Suppose there were fractional tight cycles in (V, E ∗ ). Consider the one containing
the least number of fractional edges. Each such cycle contains at least two fractional edges e 1 &e 2
(else, the corresponding constraint is not tight).
Let c e1 ≤ c e2 and θ = min {1 − X e1 , X e2 }.
The solution {X ′ e} with:
(X ′ e 1
= X e1 + θ), (X ′ e 2
= X e2 − θ) and (X ′ e = X e , ∀e ≠ e 1 , e 2 )
is feasible and at least as good as the optimal solution {X e }. The feasibility comes from the Lemma
12.3. We are sure that we do not violate any other nontight fractional cycles which share the edge
e 1 . Otherwise, we should have selected θ as the minimum slack on that nontight fractional cycle
and update X e accordingly. But, this would incur a solution where two fractional tight cycle share
the fractional edge e ′ 1 which cannot be the case due to Lemma 12.3.
If there were nontight fractional cycles, then we could repeat the same modification for the fractional
edges without violating feasibility. Therefore, there are no fractional cycles in (V, E ∗ ).
The last step to prove that (LP) is the correct form will be to show that the resulting graph is
connected. (Assignment 2)
Lec4
13 Branch-and-Bound
13.1 The Branch-and-Bound technique
The Branch-and-Bound Technique is a method of implicitly (rather than explicitly) enumerating
all possible feasible solutions to a problem.
Summary of LP-based Branch-and-Bound Technique (for maximization of pure integer linear programs)
Step 1 Initialization: Begin with the entire set of solutions under consideration as the only remaining
set. Set Z L = −∞. Throughout the algorithm, the best know feasible solution is referred
to as the incumbent solution, and its objective value Z L is a lower bound on the maximal