Lower Bounds on $0$-Extension with Steiner Nodes

TL;DR

This work studies a Steiner-augmented variant of the 0-Extension problem, denoted 0EwSN, where a budget of Steiner nodes is allowed and the LP-Metric relaxation remains applicable. The authors prove a near-tight lower bound: for any ε∈(0,1) and f with f(k)=O(k log^{1−ε} k), the integrality gap IG_f(k) is at least Ω(ε log log k), even when a super-linear number of Steiner nodes is permitted. The proof constructs a constant-degree, high-girth expander as a hard instance, shows that canonical (structure-preserving) solutions incur a large cost unless the solution size is large enough, and then extends the lower bound to all solutions via a continuation-based embedding into the graph’s continuization and a tight-span framework. This establishes fundamental limits on improving LP-relaxation quality through Steiner-node augmentation and links the gap to the potential efficiency of Steiner-node flow sparsifiers, highlighting deep connections between compression, sparsification, and metric labeling.

Abstract

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 $Ω((\log |T|)^{2/3})$, 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 $Ω(\log\log |T|)$ even if we allow a super-linear $O(|T|\log^{1-\varepsilon}|T|)$ number of Steiner nodes.

Theorems & Definitions (29)

• Theorem 1.1
• proof
• Lemma 3.4: puder2015expansion
• Corollary 3.5
• proof
• Lemma 3.6: frieze2001hamilton
• Lemma 3.7
• proof
• Claim 3.8
• proof