Lower Bounds on $0$-Extension with Steiner Nodes

• Main Authors: Chen, Yu, Tan, Zihan
• Format: Journal Article
• Language: English
• Published: 17-01-2024
• Subjects: Computer Science - Data Structures and Algorithms
• DOI: 10.48550/arxiv.2401.09585

In the $0$-Extension problem, we are given an edge-weighted graph
$G=(V,E,c)$, a set $T\subseteq V$ 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 $(u,v)\in
E$, 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 \emph{semi-metric LP relaxation}. The integrality gap of
this LP, with best upper bound $O(\log |T|/\log\log |T|)$ and best lower bound
$\Omega((\log |T|)^{2/3})$, 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 $\Omega(\log\log |T|)$ even if we allow a super-linear
$O(|T|\log^{1-\varepsilon}|T|)$ number of Steiner nodes.