Lower and Upper Bound for Computing the Size of All Second Neighbourhoods

TL;DR

This paper investigates the computational complexity of determining the size of all second neighborhoods in a graph, establishing lower bounds based on SETH and providing algorithms with specific exponential dependencies on graph parameters.

Contribution

It proves new lower bounds for computing second neighborhood sizes under SETH and presents algorithms with improved exponential dependence on graph parameters.

Findings

Subquadratic algorithms for second neighborhood size are unlikely due to SETH-based lower bounds.

Lower bounds show no algorithms with certain exponential dependencies on vertex cover unless SETH fails.

Algorithms are provided with time complexity depending on vertex cover and treewidth, improving previous results.

Abstract

We consider the problem of computing the size of each $r$-neighbourhood for every vertex of a graph. Specifically, we ask whether the size of the closed second neighbourhood can be computed in subquadratic time. Adapting the SETH reductions by Abboud et al. (2016) that exclude subquadratic algorithms to compute the radius of a graph, we find that a subquadratic algorithm would violate the SETH. On the other hand, a linear fpt-time algorithm by Demaine et al. (2014) parameterized by a certain `sparseness parameter' of the graph is known, where the dependence on the parameter is exponential. We show here that a better dependence is unlikely: for any~$\delta < 2$, no algorithm running in time $O(2^{o({\rm vc}(G))} \, n^{\delta})$, where~${\rm vc}(G)$ is the vertex cover number, is possible unless the SETH fails. We supplement these lower bounds with algorithms that solve the problem in time~$O(2^{{\rm vc}(G)/2} {\rm vc}(G)^2 \cdot n)$ and $O(2^w w \cdot n)$.