On 2-Coverings and 2-Packings of Laminar Families

Abstract

Let H be a laminar family of subsets of a groundset V. A k-cover of H is a multiset C of edges on V such that for every subset S in H, C has at least k edges that have exactly one end in S. A k-packing of H is a multiset P of edges on V such that for every subset S in H, P has at most k · u(S) edges that have exactly one end in S. Here, u assigns an integer capacity to each subset in H. Our main results are: (a) Given a k-cover C of H, there is an efficient algorithm to find a 1-cover contained in C of size ≤ k|C|=(2k - 1). For 2-covers, the factor of 2=3 is best possible. (b) Given a 2-packing P of H, there is an efficient algorithm to find a 1-packing contained in P of size ≥ |P|/3. The factor of 1/3 for 2-packings is best possible.

These results are based on efficient algorithms for finding appropriate colorings of the edges in a k-cover or a 2-packing, respectively, and they extend to the case where the edges have nonnegative weights. Our results imply approximation algorithms for some NP-hard problems in connectivity augmentation and related topics. In particular, we have a 4/3-approximation algorithm for the following problem: Given a tree T and a set of nontree edges E that forms a cycle on the leaves of T, find a minimum-size subset E′ of E such that T + E′ is 2-edge connected.

Supported in part by NSERC research grant OGP0138432.

Supported in part by the Hungarian Scientific Research Fund no. OTKA T29772 and T30059.

Supported in part by NSF career grant CCR-9625297.