Summary: The k-path tree matroid and its applications to
survivable network design
Esther M. Arkin
September 19, 2007
We define the k-path tree matroid, and use it to solve network design problems in which
the required connectivity is arbitrary for a given pair of nodes, and 1 for the other pairs. We
solve the problems for undirected and directed graphs. We then use these exact algorithms to
give improved approximation algorithms for problems in which the weights satisfy the triangle
inequality and the connectivity requirement is either 2 among at most five nodes and 1 for the
other nodes, or it is 3 among a set of three nodes and 1 for all other nodes.
Mathematics subject classification: Discrete mathematics 68R99, connectivity 05C40.
Consider a complete graph G = (V, E) with nonnegative edge weights wij for each (i, j) E.
Denote the weight of a subgraph G
of G by w(G
) = (i,j)G wij. The survivable network
design problem (SND), seeks a minimum weight subgraph of G such that each pair of nodes
i, j has a pre-specified requirement rij of edge-disjoint i - j paths. When rij = 1 for all i, j, this