In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczúr and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).
@InProceedings{cen_et_al:LIPIcs.ICALP.2021.45, author = {Cen, Ruoxu and Cheng, Yu and Panigrahi, Debmalya and Sun, Kevin}, title = {{Sparsification of Directed Graphs via Cut Balance}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {45:1--45:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.45}, URN = {urn:nbn:de:0030-drops-141143}, doi = {10.4230/LIPIcs.ICALP.2021.45}, annote = {Keywords: Graph sparsification, directed graphs, cut sketches, space complexity} }
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