2 research outputs found
Lower Bounds on -Extension with Steiner Nodes
In the -Extension problem, we are given an edge-weighted graph
, a set of its vertices called terminals, and a
semi-metric over , and the goal is to find an assignment of each
non-terminal vertex to a terminal, minimizing the sum, over all edges , the product of the edge weight and the distance
between the terminals that are mapped to. Current best approximation
algorithms on -Extension are based on rounding a linear programming
relaxation called the \emph{semi-metric LP relaxation}. The integrality gap of
this LP, with best upper bound and best lower bound
, has been shown to be closely related to the best
quality of cut and flow vertex sparsifiers.
We study a variant of the -Extension problem where Steiner vertices are
allowed. Specifically, we focus on the integrality gap of the same semi-metric
LP relaxation to this new problem. Following from previous work, this new
integrality gap turns out to be closely related to the quality achievable by
cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph
compression. Our main result is that the new integrality gap stays
superconstant even if we allow a super-linear
number of Steiner nodes