Approximation and parameterized algorithms for covering disjointness-compliable set families

A set-family ${\cal F}$ is disjointness-compliable if $A' \subseteq A \in {\cal F}$ implies $A' \in {\cal F}$ or $A \setminus A' \in {\cal F}$; if ${\cal F}$ is also symmetric then ${\cal F}$ is proper. A classic result of Goemans and Williamson [SODA 92:307-316] states that the problem of covering a proper set-family by a min-cost edge set admits approximation ratio $2$, by a classic primal-dual algorithm. However, there are several famous algorithmic problems whose set-family ${\cal F}$ is disjointness-compliable but not symmetric -- among them $k$-Minimum Spanning Tree ($k$-MST), Generalized Point-to-Point Connection (G-P2P), Group Steiner, Covering Steiner, multiroot versions of these problems, and others. We will show that any such problem admits approximation ratio $O(α\log τ)$, where $τ$ is the number of inclusion-minimal sets in the family ${\cal F}$ that models the problem and $α$ is the best known approximation ratio for the case when $τ=1$. This immediately implies several results, among them the following two. (i) The first deterministic polynomial time $O(\log n)$-approximation algorithm for the G-P2P problem. Here the $τ=1$ case is the $k$-MST problem. (ii) Approximation ratio $O(\log^4 n)$ for the multiroot version of the Covering Steiner problem, where each root has its own set of groups. Here the $τ=1$ case is the Covering Steiner problem. We also discuss the parameterized complexity of covering a disjointness-compliable family ${\cal F}$, when parametrized by $τ$. We will show that if ${\cal F}$ is proper then the problem is fixed parameter tractable and can be solved in time $O^*(3^τ)$. For the non-symmetric case we will show that the problem admits approximation ratio between $α$ and $α+1$ in time $O^*(3^τ)$, which is essentially the best possible.