Edge Covers of Setpairs and the Iterative Rounding Method

Abstract

Given a digraph G = (V;E), we study a linear programming relaxation of the problem of finding a minimum-cost edge cover of pairs of sets of nodes (called setpairs), where each setpair has a nonnegative integer-valued demand. Our results are as follows: (1) An extreme point of the LP is characterized by a noncrossing family of tight setpairs, $\begin{gathered} \mathcal{L} \hfill \\ (where |\mathcal{L}| \leqslant |E|). \hfill \\ \end{gathered}$ . (2) In any extreme point x, there exists an edge e with $x_e \geqslant \Theta (1)/\sqrt {|\mathcal{L}}$ , and there is an example showing that this lower bound is best possible. (3) The iterative rounding method applies to the LP and gives an integer solution of cost $O(\sqrt {|\mathcal{L}} ) = O(\sqrt {|E|} )$ times the LP’s optimal value. The proofs rely on the fact that $\mathcal{L}$ can be represented by a special type of partially ordered set that we call diamond-free.