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.
@InProceedings{das_et_al:LIPIcs.MFCS.2024.45, author = {Das, Syamantak and Kumar, Nikhil and Vaz, Daniel}, title = {{Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {45:1--45:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.45}, URN = {urn:nbn:de:0030-drops-206018}, doi = {10.4230/LIPIcs.MFCS.2024.45}, annote = {Keywords: Graph Sparsification, Cut Sparsifiers, Flow Sparsifiers, Quasi-bipartite Graphs, Bounded Treewidth} }
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