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Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs

Authors Syamantak Das , Nikhil Kumar , Daniel Vaz

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Author Details

Syamantak Das
  • IIIT Delhi, India
Nikhil Kumar
  • University of Waterloo, Canada
Daniel Vaz
  • LAMSADE, CNRS, Université Paris-Dauphine, Université PSL, France

Funding

  • Vaz, Daniel: This work was funded by grants ANR-19-CE48-0016 (Algoridam) and ANR-21-CE48-0022 (S-EX-AP-PE-AL) from the French National Research Agency (ANR), and partially done while the author was at École Normale Superieure and IRIF.

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Syamantak Das, Nikhil Kumar, and Daniel Vaz. Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.45

Abstract

Flow sparsification is a classic graph compression technique which, given a capacitated graph G on k terminals, aims to construct another capacitated graph H, called a flow sparsifier, that preserves, either exactly or approximately, every multicommodity flow between terminals (ideally, with size as a small function of k). Cut sparsifiers are a restricted variant of flow sparsifiers which are only required to preserve maximum flows between bipartitions of the terminal set. It is known that exact cut sparsifiers require 2^Ω(k) many vertices [Krauthgamer and Rika, SODA 2013], with the hard instances being quasi-bipartite graphs, where there are no edges between non-terminals. On the other hand, it has been shown recently that exact (or even (1+ε)-approximate) flow sparsifiers on networks with just 6 terminals require unbounded size [Krauthgamer and Mosenzon, SODA 2023, Chen and Tan, SODA 2024]. 
In this paper, we construct exact flow sparsifiers of size 3^k³ and exact cut sparsifiers of size 2^k² for quasi-bipartite graphs. In particular, the flow sparsifiers are contraction-based, that is, they are obtained from the input graph by (vertex) contraction operations. Our main contribution is a new technique to construct sparsifiers that exploits connections to polyhedral geometry, and that can be generalized to graphs with a small separator that separates the graph into small components. We also give an improved reduction theorem for graphs of bounded treewidth [Andoni et al., SODA 2011], implying a flow sparsifier of size O(k⋅w) and quality O((log w)/log log w), where w is the treewidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network flows
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Data compression
Keywords
  • Graph Sparsification
  • Cut Sparsifiers
  • Flow Sparsifiers
  • Quasi-bipartite Graphs
  • Bounded Treewidth

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