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

Abstract

In this paper we investigate the computational complexity 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 denote 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 e 2 ≈ 7.39 , and for smaller values of a / k it decreases towards 1 as a linear function of a / k with lim a / k → 0 E [ r ( a , k ) ] = 1 . 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. We also comment about the impossibility of a significantly faster convergence of E [ r ( a , k ) ] towards 1 for any polynomial-time approximation algorithm.