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Title: Finding Most-Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time
We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has been unimproved for more than two decades. Our work also removes an important bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95) and finding an even-length directed cycle in a graph, a problem shown to be equivalent to many other fundamental problems (Vazirani and Yannakakis '90, Robertson et al. '99). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain a near-linear time algorithm, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of using connectivity oracles subject to vertex failures to speed up a static graph algorithm.  more » « less
Award ID(s):
2238138
PAR ID:
10536383
Author(s) / Creator(s):
; ; ;
Editor(s):
Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
297
ISSN:
1868-8969
ISBN:
978-3-95977-322-5
Page Range / eLocation ID:
297-297
Subject(s) / Keyword(s):
Graphs Flows Randomized Algorithms Vertex Connectivity Theory of computation → Graph algorithms analysis
Format(s):
Medium: X Size: 19 pages; 948668 bytes Other: application/pdf
Size(s):
19 pages 948668 bytes
Right(s):
Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
Sponsoring Org:
National Science Foundation
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