Randomized Approximation of Bounded Multicovering Problems

The problem of finding approximate solutions for a subclass of multicovering problems denoted by ILP(k, b) is considered. The problem involves finding x ∈ {0, 1}ⁿ that minimizes ∑_jx_j subject to the constraint Ax ≥ b, where A is a 0-1 m × n matrix with at most k ones per row, b is an integer vector, and b is the smallest entry in b. This subclass includes, for example, the Bounded Set Cover problem when b = 1, and the Vertex Cover problem when k = 2 and b=1. An approximation ratio of k - b + 1 is achievable by known deterministic algorithms. A new randomized approximation algorithm is presented, with an approximation ratio of (k - b + 1)(1 - (c/m)^{1/(k - b + 1)}) for a small constant c > 0. The analysis of this algorithm relies on the use of a new bound on the sum of independent Bernoulli random variables, that is of interest in its own right.