Lecture Notes Discrete Optimization - Applied Mathematics

Note that these Lecture Notes have been last modified on January 10, 2012. I guess most
of you will have printed the Lecture Notes of December 6, 2011 (see “document last
modified” date on your printout). I therefore keep track of the changes that I made since
then below. I also mark these changes as “[major]” or “[minor]”, depending on their
importance.
Changes made with respect to Lecture Notes of December 6, 2011:
• [major] Introducing the notion of a pseudoflow on page 42: corrected “it need
to satisfy the flow balance constraints” to “it need not satisfy the flow balance
constraints”.
Changes made with respect to Lecture Notes of December 20, 2011: (Thanks to Rutger
Kerkkamp for pointing these out.)
• [minor] Symbol for symmetric difference on page 4 is now the same as the one
used in Chapter 7 (“△”).
• [minor] Strictly speaking we would have to add a constraint x j ≤ 1 for every j ∈
{1,...,n} to the LP relaxation (2) of the integer linear program (1) on page 5.
However, these constraints are often (but not always) redundant because of the
minimization objective. Note that the discussion that follows refers to the LP
relaxation as stated in (2) (see also remark after statement of LP (2)).
• [minor] Corrected “multiplies” to “multipliers” in last paragraph on page 5.
• [minor] Lower part of Figure 2 on page 2: changed edge e from solid to dotted
line.
• [minor] At various places in Chapter 3, Condition (M1) of the independent set system
has not been mentioned explicitly (Example 3.1, Examples 3.3–3.5, Theorem
3.1). Mentioned now.
• [minor] Corrected “many application” to “many applications” in first paragraph
on page 27.
• [minor] Corrected “δ(v)” and “δ(u)” to “δ(s,v)” and “δ(s,u)” in last paragraph
of the proof of Lemma 5.5. on page 34 (3 occurrences).
• [minor] Algorithms 7 and 8 should mention that the capacity function is nonnegative.
• [major] There was a mistake in the objective function of the dual linear program
(6) on page 46: The sign in front of the second summation must be negative.
• [minor] Algorithms 9, 10 and 11 should mention that the capacity function is nonnegative.
• [minor] Proof of Theorem 7.3 on page 56: corrected “O(mα(n,m))” to
“O(mα(n, m n ))”.
• [minor] Caption of the illustration on page 71 has been put on the same page.
• [minor] Typo in the second last sentence of the proof of Theorem 6.7 on page 47:
“c π (u,v)