Approximation algorithms for integer covering problems via greedy column generation

Many combinatorial optimization problems can be formulated as covering problems. In some cases, this covering formulation is not polynomial in the input size of the original problem. In order to solve the problem approximately one can apply the greedy algorithm to the covering formulation. In this case, the column generation subproblem is to determine which column the greedy algorithm chooses. This problem might be NP-hard in itself. We propose a modification of the greedy algorithm in which the column generation subproblem is solved approximately, within a factor α. We derive upper bounds for the worst case ratio of this algorithm and related polynomial approximation algorithms.