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, 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.
Graph Algorithms
Zero Extension
Integrality Gap
Theory of computation~Sparsification and spanners
47:1-47:18
Track A: Algorithms, Complexity and Games
https://arxiv.org/pdf/2401.09585.pdf
We would like to thank Julia Chuzhoy for many helpful discussions. We want to thank Arnold Filtser for pointing to us some previous works on similar problems.
Yu
Chen
Yu Chen
EPFL, Lausanne, Switzerland
https://orcid.org/0009-0006-3595-1297
Zihan
Tan
Zihan Tan
Rutgers University, Piscataway, NJ, USA
https://orcid.org/0000-0003-4844-8480
Supported by a grant to DIMACS from the Simons Foundation (820931).
10.4230/LIPIcs.ICALP.2024.47
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Yu Chen and Zihan Tan
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