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New Lower Bounds and Upper Bounds for Listing Avoidable Vertices

Authors Mingyang Deng, Virginia Vassilevska Williams, Ziqian Zhong

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Mingyang Deng
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Virginia Vassilevska Williams
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Ziqian Zhong
  • Massachusetts Institute of Technology, Cambridge, MA, USA

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Mingyang Deng, Virginia Vassilevska Williams, and Ziqian Zhong. New Lower Bounds and Upper Bounds for Listing Avoidable Vertices. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.41

Abstract

We consider the problem of listing all avoidable vertices in a given n vertex graph. A vertex is avoidable if every pair of its neighbors is connected by a path whose internal vertices are not neighbors of the vertex or the vertex itself. Recently, Papadopolous and Zisis showed that one can list all avoidable vertices in O(n^{ω+1}) time, where ω < 2.373 is the square matrix multiplication exponent, and conjectured that a faster algorithm is not possible. 
In this paper we show that under the 3-OV Hypothesis, and thus the Strong Exponential Time Hypothesis, n^{3-o(1)} time is needed to list all avoidable vertices, and thus the current best algorithm is conditionally optimal if ω = 2. We then show that if ω > 2, one can obtain an improved algorithm that for the current value of ω runs in O(n^3.32) time. We also show that our conditional lower bound is actually higher and supercubic, under a natural High Dimensional 3-OV hypothesis, implying that for our current knowledge of rectangular matrix multiplication, the avoidable vertex listing problem likely requires Ω(n^3.25) time. We obtain further algorithmic improvements for sparse graphs and bounded degree graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Avoidable Vertex
  • Fine-Grained Complexity

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