New Lower Bounds and Upper Bounds for Listing Avoidable Vertices

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^{?+1}) time, where ? < 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 ? = 2. We then show that if ? > 2, one can obtain an improved algorithm that for the current value of ? 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 ?(n^3.25) time. We obtain further algorithmic improvements for sparse graphs and bounded degree graphs