CS614. Advanced Algorithms. L06 Quiz.
CS614. Advanced Algorithms.
L06 Solutions
The questions in this problem set are adapted from the Coursera course on Approximation Algorithms taught by Mark de Berg.
Consider the algorithm LPAPX-WVC
from the class.
Suppose that instead of putting a vertex
- We still get a valid solution, and the algorithm remains a 2-approximation.
- We still get a valid solution, and the algorithm becomes a (3/2)-approximation.
- We still get a valid solution, and the algorithm becomes a 3-approximation.
- We may no longer get a valid solution.
Suppose that instead of putting a vertex
- We still get a valid solution, and the algorithm remains a 2-approximation.
- We still get a valid solution, and the algorithm becomes a (3/2)-approximation.
- We still get a valid solution, and the algorithm becomes a 3-approximation.
- We may no longer get a valid solution.
If we increase the threshold
If we decrease the threshold
Consider a different rounding strategy for the LP relaxation of the vertex cover problem. Instead of rounding up every vertex whose value is at least
We look at every edge, and then we round up the variable of the endpoint with the highest value, where in case of ties we take the endpoint with the highest index.
In other words, if the vertex set is
Which statement is true?
- This does not work, because we might report an invalid solution.
- This gives a valid solution, but the approximation ratio becomes worse.
- This gives a valid solution, and in fact the solution is always exactly the same as in the original rounding scheme.
- This gives a valid solution. We sometimes report a better solution than in the original rounding scheme, but the approximation ratio of the algorithm is still more than
for any . - This gives a valid solution, and the approximation ratio of the algorithm becomes
.
The solution is valid: indeed, let
We look at every edge,
and then we round up the variable of the endpoint with the highest value,
where in case of ties we take the endpoint with the highest index.
Since one of the endpoints was rounded up, the edge is covered; and this is evidently true of every edge.
The solution with respect to this rounding may be better than the threshold-based rounding: for example, consider again a complete graph where the LPOPT sets all variables to
However, to see that the approximation ratio of the algorithm is still more than
Suppose you have created an algorithm for a certain problem using LP relaxation and you want to say something about its approximation ratio. Which lower bound on the optimal solution can you use?
- The solution to the 0/1-LP.
- The solution to the relaxed LP.
- Depends on the problem.
As was clarified in class, this question is in the context of minimization problems.
The solution to the relaxed LP is a useful lower bound for the OPT. The value of the OPT for the 0/1-LP is exactly equal to the OPT (in a presumed exact formulation of the problem) and does not, by itself, provide information about the behavior of the relaxed LP.
What is the integrality gap of the vertex-cover LP for the complete graph on
ILPOPT =