Set Cover

# Problem definition.

In weighted set cover problem we are given Universe $$U$$ and family $$S$$ of subsets of U, a cover is a subfamily $$C\subseteq S$$ of sets whose union is $$U$$. The goal is to find minimum cost cover.

# Solution

Here we implement the following algorithm:

At each stage we add to $$C$$ a set which maximizes(number of uncovered elements)/(cost of set).

We repeat the step as long as uncovered elements exist. When the algorithm finishes we return selected sets in $$C$$.

example:

#include <iostream>
#include <vector>
#include <iterator>
#include <boost/range/irange.hpp>
int main() {
std::vector<std::vector<int>> set_to_elements = {
{ 1, 2 },
{ 3, 4, 5, 6 },
{ 7, 8, 9, 10, 11, 12, 13, 0 },
{ 1, 3, 5, 7, 9, 11, 13 },
{ 2, 4, 6, 8, 10, 12, 0 }
};
std::vector<int> costs = { 1, 1, 1, 1, 1 };
auto sets = boost::irange(0, 5);
std::vector<int> result;
auto element_index = [](int el){return el;};
auto cost = paal::greedy::set_cover(sets,
[&](int set){return costs[set];},
[&](int set){return set_to_elements[set];},
back_inserter(result),
element_index);
std::cout << "Cost: " << cost << std::endl;
}

complete example is set_cover_example.cpp

## Parameters

IN: SetIterator sets_begin

IN: SetIterator sets_eEnd,

IN: GetCostOfSet set_to_cost

IN: GetElementsOfSet set_to_elements

OUT: OutputIterator result

The Iterators of selected Sets will be output to the output iterator result

The iterator type must be a model of Output Iterator

IN: GetElementIndex get_el_index we need in algorithm map elements to small unique integers

## Approximation Ratio

equals to $$H(s')$$ where $$s'$$ is the maximum cardinality set of $$S$$ and $$H(n)$$ is n-th harmonic number.

## The complexity

Complexity of the algorithm is $$|I|*log(|I|)$$. where $$I$$ is number of elements in all sets

Memory complexity of the algorithm is $$|I|$$. where $$I$$ is number of elements in all sets

## References

The algorithm analysis is described in [22]