NearlyTight Bounds for Flow Sparsifiers in QuasiBipartite Graphs
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 $2^{\Omega(k)}$ many vertices [Krauthgamer and Rika, SODA 2013], with the hard instances being quasibipartite graphs, where there are no edges between nonterminals. On the other hand, it has been shown recently that exact (or even $(1+\varepsilon)$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 $3^{k^{3}}$ and exact cut sparsifiers of size $2^{k^2}$ for quasibipartite graphs. In particular, the flow sparsifiers are contractionbased, 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 $O(k\cdot w)$ and quality $O\bigl(\frac{\log w}{\log \log w}\bigr)$, where $w$ is the treewidth.
 Publication:

arXiv eprints
 Pub Date:
 July 2024
 DOI:
 10.48550/arXiv.2407.09433
 arXiv:
 arXiv:2407.09433
 Bibcode:
 2024arXiv240709433D
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 doi:10.4230/LIPIcs.MFCS.2024.45