Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks

In this paper we investigate the computational complexities of a combinatorial problem that arises in the reverse engineering of protein and gene networks. Our contributions are as follows: - We abstract a combinatorial version of the problem and observe that this is "equivalent" to the set multicover problem when the "coverage" factor k is a function of the number of elements n of the universe. An important special case for our application is the case in which k = n-1. - We observe that the standard greedy algorithm produces an approximation ratio of Ω(log n) even if k is "large" i.e. k = n-c for some constant c ≥ 0. - Let 1 ≤ a ≤ n denotes the maximum number of elements in any given set in our set multicover problem. Then, we show that a non-trivial analysis of a simple randomized polynomial-time approximation algorithm for this problem yields an expected approximation ratio E[r(a, k)] that is an increasing function of a/k. The behavior of E[r(a,k)] is "roughly" as follows: it is about ln(a/k) when a/k is at least about e² ≈7.39, and for smaller values of a/k it decreases towards 2 exponentially with increasing k with lim _a/k→0 E[r(a, k)] ≤ 2. Our randomized algorithm is a cascade of a deterministic and a randomized rounding step parameterized by a quantity β followed by a greedy solution for the remaining problem.