CS 880: Approximation Algorithms Homework 1
Out: 2/01/07 Due: 2/22/07
1. Approximating the TSP on directed graphs is mu ch harder than on undirected graphs. The best known algorithm
for the former achieves an O(log n)-approximation. In this question we will consider th e TSP in special directed
graphs that we will call {1,2}-graphs. These are complete graphs with each directed edge of length 1or 2.
(Convince yourself that these lengths satisfy the triangle inequality.)
(a) First we will look at a problem closely related to the TSP. In the cycle-cover problem our goal is to find
a minimum weight collection of simple (directed) cycles in the graph such that each vertex in the graph
is contained in exactly one cycle. Prove that the minimum weight cycle cover in a graph can be found in
polynomial time. (Cycles of length 2are allowed.)
(b) Use the algorithm from part (a) to give a 3/2-approximation for the TSP on {1,2}-graphs.
(c) Can you improve your algorithm from part (b) to obtain a 4/3-approximation? What property do you
require from the cycle cover algorithm in order to obtain this improvement?
2. In the k-cut problem, we are given a weighted graph, and asked to return a cut of minimum weight that sep arates
the graph into at least kconnected components. Note that this problem is very similar to min multiway cut,
except that we are not given specific terminals to separate.
Consider the following algorithm for k-cut: at every step, find the min-cut in every component in the graph,
remove the smallest weight min-cut, and repeat until kcomponents are formed.
Prove that this algorithm achieves a 2(1 −1/k)-approximation to k-cut.
3. Recall that in the set cover problem our goal is to cover a givenset of elements using a collection of subsets of
the least total cost. In the maximum coverage problem our goal is complementary—we want to cover the most
number of elements using a collection of subsets with total cost no larger than a given bound B.
(a) Give a constant factor approximation for the maximum coverage problem. Try to obtain as good an ap-
proximation as possible.
(b) Prove that maximum coverage cannot be approximated within a factor of 1−1/e +for any constant
> 0unless NP⊂DTIME(nlog log n).(Hint: Use Feige’s hardness result for set cover.)
4. Recall the min makespan scheduling algorithm from class, and consider the following variant of the algorithm—
schedule jobs in decreasing order of size and send each job to the shortest queue so far.
(a) Prove that this algorithm obtains a 3/2-approximation.
(b) Give the worst gap example you can think of for this algorithm.
(c) (This part is not for credit.) Can you prove that the algorithm achieves a 4/3-approximation?
5. A legal k-coloring of a graph is an assignment of colors 1,2,···, k to the vertices of the graph such that no two
adjacent vertices receive the same color. A graph is k-colorable if there exists a legal k-coloring of its vertices.
The problem of finding a legal k-coloring of a k-colorable graph is NP-complete for k≥3.
(a) Prove that graphs with maximum degree ∆are (∆ + 1 )-colorable. Also give a polynomial time algorithm
for finding a (∆ + 1)-coloring.
(b) Give a polynomial time algorithm for 2-coloring a bipartite graph.
(c) Using parts (a) and (b) above, give a polynomial time algorithm for finding an O(√n)-coloring of a 3-
colorable graph.
(Hint: Verify and use the fact that the neighborhood of any vertex in a 3-colorable graph is 2-colorable.)
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