eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-07-02
47:1
47:18
10.4230/LIPIcs.ICALP.2024.47
article
Lower Bounds on 0-Extension with Steiner Nodes
Chen, Yu
1
https://orcid.org/0009-0006-3595-1297
Tan, Zihan
2
https://orcid.org/0000-0003-4844-8480
EPFL, Lausanne, Switzerland
Rutgers University, Piscataway, NJ, USA
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, is upper bounded by O(log|T|/log log|T|) and lower bounded by Ω((log|T|)^{2/3}), has been shown to be closely related to the 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. We show that the new integrality gap stays superconstant Ω(log log |T|) even if we allow a super-linear O(|T|log^{1-ε}|T|) number of Steiner nodes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol297-icalp2024/LIPIcs.ICALP.2024.47/LIPIcs.ICALP.2024.47.pdf
Graph Algorithms
Zero Extension
Integrality Gap