Consider the following optimization problem.
- Instance: positive integers .
- Find two disjoint nonempty subsets with , such that the ratio is minimized.
Give a pseudo-polynomial time algorithm for the problem, and then give an FPTAS for the problem based on the pseudo-polynomial time algorithm.
In the maximum directed cut (MAX-DICUT) problem, we are given as input a directed graph . The goal is to partition into disjoint and so that the number of edges in is maximized. The following is the integer program for MAX-DICUT:
Let denote the optimal solution to the LP-relaxation of the above integer program.
- Apply the randomized rounding such that for every , independently with probability . Analyze the approximation ratio (between the expected size of the random cut and OPT).
- Apply another randomized rounding such that for every , independently with probability . Analyze the approximation ratio for this algorithm.
Recall the MAX-SAT problem and its integer program:
Recall that are the respective sets of variables appearing positively and negatively in clause .
Let denote the optimal solution to the LP-relaxation of the above integer program. In our class we learnt that if is round to 1 independently with probability , we have approximation ratio .
We consider a generalized rounding scheme such that every is round to 1 independently with probability for some function to be specified.
- Suppose is an arbitrary function satisfying that for any . Show that with this rounding scheme, the approximation ratio (between the expected number of satisfied clauses and OPT) is at least .
- Derandomize this algorithm through conditional expectation and give a deterministic polynomial time algorithm with approximation ratio .
- Is it possible that for some more clever we can do better than this? Try to justify your argument.
The following is the weighted version of set cover problem:
Given subsets , where is a universe of size , and each subset is assigned a positive weight , the goal is to find a such that and the total weight is minimized.
- Give an integer program for the problem and its LP relaxation.
- Consider the following idea of randomized rounding: independently round each fractional value to with the probability of the fractional value itself; and repeatedly apply this process to the variables rounded to 0 in previous iterations until is fully covered. Show that this can return a set cover with approximation ratio with probability at least .
Recall that the instance of set cover problem is a collection of subsets , where is a universe of size . The goal is to find the smallest such that . The frequency is defined to be .
- Give the primal integer program for set cover, its LP-relaxation and the dual LP.
- Describe the complementary slackness conditions for the problem.
- Give a primal-dual algorithm for the problem. Present the algorithm in the language of primal-dual scheme (alternatively raising variables for the LPs). Analyze the approximation ratio in terms of the frequency .