Lower Bounds on 0-Extension with Steiner Nodes

In the 0-Extension problem, we are given an edge-weighted graph G=(V,E,c), a set T⊆ V of its vertices called terminals, and a semi-metric D over T, and the goal is to find an assignment f of each non-terminal vertex to a terminal, minimizing the sum, over all edges (u,v)∈ E, the product of the edge weight c(u,v) and the distance D(f(u),f(v)) between the terminals that u,v are mapped to. Current best approximation algorithms on 0-Extension are based on rounding a linear programming relaxation called the semi-metric LP relaxation. The integrality gap of this LP, with best upper bound O(log |T|/loglog |T|) and best lower bound Ω((log |T|)^2/3), has been shown to be closely related to the best quality of cut and flow vertex sparsifiers. We study a variant of the 0-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 Ω(loglog |T|) even if we allow a super-linear O(|T|log^1-ε|T|) number of Steiner nodes.