An almost always polynomial time algorithm for the (α, β)-cover problem in bipartite graphs

The (α, β)-cover problem is the problem of finding a vertex cover S_X∪S_Y with S_X⊆X and S_Y⊆Y, in a bipartite graph G=(X , Y, E) that satisfies the constraints |S_x|⩽α|X| and |S_Y |⩽β|Y| where α, β∈(0, 1). This problem has applications in the repair of large memory chips and has been shown to be NP-complete. The authors a new algorithm for finding (α, β)-covers, improvements to the probabilistic analysis given by W.P. Shi and W.K. Fuchs (1989), and a new probabilistic algorithm which runs almost always in O(|E|√n) on any edge probability. The authors note that the probabilistic algorithm works for any edge probability p(n) while the algorithm of Shi and Fuchs works only when p(n)⩽0.5/n. In particular, the result shows that the (α, β)-cover problem is almost always solvable in polynomial time