License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2020.11
URN: urn:nbn:de:0030-drops-116969
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11696/
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Lincoln, Andrea ; Vyas, Nikhil

Algorithms and Lower Bounds for Cycles and Walks: Small Space and Sparse Graphs

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LIPIcs-ITCS-2020-11.pdf (0.6 MB)

Abstract

We consider space-efficient algorithms and conditional time lower bounds for finding cycles and walks in graphs. We give a reduction that connects the running time of undirected 2k-cycle to finding directed odd cycles, s-t connectivity in directed graphs, and Max-3-SAT. For example, we show that if 2k-cycle on O(n)-edge graphs can be solved in O(n^(1.5-ε)) time for some ε>0 then, a 2^(n(1-ε')) time algorithm exists for Max-3-SAT for some ε'>0. Additionally, we give a tight combinatorial lower bound for 2k-cycle detection, specifically when k is odd, of m^{2k/(k+1) +o(1)} given the Combinatorial k-Clique Hypothesis.
On the algorithms side, we present a randomized algorithm for directed s-t connectivity using O(lg(n)^2) space and O(n^{lg(n)/2 + o(lg(n))}) expected time, giving a time improvement over Savitch’s famous algorithm, which takes at least n^{lg(n) - o(lg(n))} time. Under the conjecture that every O(lg(n)^2)-space algorithm for directed s-t connectivity requires n^Ω(lg(n)) time, we show that undirected 2k-cycle in O(lg(n)) space requires n^Ω(lg(k)) time.

BibTeX - Entry

@InProceedings{lincoln_et_al:LIPIcs:2020:11696,
  author =	{Andrea Lincoln and Nikhil Vyas},
  title =	{{Algorithms and Lower Bounds for Cycles and Walks: Small Space and Sparse Graphs}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11696},
  URN =		{urn:nbn:de:0030-drops-116969},
  doi =		{10.4230/LIPIcs.ITCS.2020.11},
  annote =	{Keywords: k-cycle, Space, Savitch, Sparse Graphs, Max-3-SAT}
}

Keywords: k-cycle, Space, Savitch, Sparse Graphs, Max-3-SAT
Collection: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 06.01.2020

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