Some practice problems. Answers will be provided.
1 Give a 1/2-approximation algorithm for the following problem:
Acyclic subgraph:
Instance:
Directed graph G = (V, A)
Solution:
A0 ⊆ A such that G0 = (V, A0 ) does not have a directed cycle.
Cost:
Number of arcs |A0 |
Goal :
Maximize |A0 |.
Hint: Arbitrarily number the vertices and pick the bigger of the two sets: the
forward-going arcs (from low to high index) and the backward-going arcs.
2 Give a 2-approximation algorithm for the problem of finding a minimum
maximal matching in an undirected graph.
Figure 1: (a) The maximum matching, (b) A maximal matching, and (c) the
minimum maximal matching.
3 A tournament G = (V, A) is a complete, directed graph. That means, for
every pair of vertices i, j, either arc (i, j) ∈ A or arc (j, i) ∈ A and not both. A
feedback vertex set S for G is a subset of the vertices such that removing S and
its adjacent arcs leaves an acyclic graph (a graph without directed cycles).
Feedback vertex set:
Instance:
Tournament G = (V, A)
Solution:
S ⊂ V such that the graph induced by V − S is acyclic
Cost:
|S|
Goal :
Minimize |S|
1
Give a 3-approximation for finding a minimum feedback vertex set.
Hint: First show that if the remaining graph has no directed cycles of length 3,
then it has no directed cycles at all. Then, formulate the problem as a set cover
problem and use the fact that there is an f -approximation for set cover.
4 We showed that the LPT algorithm has an approximation ratio of 4/3. That
means, for any instance, the LPT algorithm finds a schedule for which the length
is at most 4/3 times the optimal length. In fact, by a refined analysis one can
show that the ratio is even
1
4
−
.
3 3m
Give an example which shows that the analysis is tight. That means, give an
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example for which the length of the LPT-schedule is exactly 34 − 3m
times the
optimal length.
(If you can’t find an example yourself, you can certainly find one on the internet.)
5 The book (Section 2.1) mentions that the scheduling problem of section 2.1
(1|rj |Lmax ) is an NP-hard problem but it gives no proof. Show that the problem is NP-hard by reducing the 3-Partition problem to this scheduling problem.
The 3-partion problem is known to be NP-hard (NP-complete).
3-Partition
P3n
Instance: Integers, a1 , a2 , . . . , a3n and an integer A such that i=1 ai = nA.
Question:
Can we partition 1, 2, . . . , 3n into n sets S1 , S2 , . . . , Sn such that
P
i∈Sj ai = A for all j = 1, .., n?
Example:
P If the instance is a1 = 2, a2 = 3, . . . , a9 = 10 then the answer is yes.
Note, i ai = 54 = 3 · 18 and 2 + 7 + 9 = 3 + 5 + 10 = 4 + 6 + 8 = 18.
The right way to look at such a reduction is as follows. Assume you do have
some algorithm that can solve the scheduling problem in polynomial time. How
can you use this (as a black box) to solve some instance of the 3-Partition
problem?
Hint: Given an instance of the 3-Partition problem construct an instance of the
scheduling problem and show that the optimal value L∗max = 0 if and only if a
3-Partition exists.
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