Subcubic Equivalences Between Graph Centrality Measures and Complementary Problems

TL;DR

This paper explores the computational complexity of graph centrality and related problems, establishing subcubic equivalences between them and fundamental problems like APSP and Diameter, advancing understanding of their inherent difficulty.

Contribution

It introduces vertex versions and complementary problems for centrality measures, proving their equivalence to APSP and Diameter, and advances the subcubic equivalence conjecture.

Findings

APSP and CoDiameter are equivalent under subcubic reductions

Certain centrality problems are equivalent to their complementary versions

Progress towards proving APSP and Diameter equivalence

Abstract

Despite persistent efforts, there is no known technique for obtaining unconditional super-linear lower bounds for the computational complexity of the problems in P. Vassilevska Williams and Williams introduce a fruitful approach to advance a better understanding of the computational complexity of the problems in P. In particular, they consider All Pairs Shortest Paths (APSP) and other fundamental problems such as checking whether a matrix defines a metric, verifying the correctness of a matrix product, and detecting a negative triangle in a graph. Abboud, Grandoni, and Vassilevska Williams study well-known graph centrality problems such as Radius, Median, etc., and make a connection between their computational complexity to that of two fundamental problems, namely APSP and Diameter. They show any algorithm with subcubic running time for these centrality problems, implies a subcubic algorithm for either APSP or Diameter. In this paper, we define vertex versions for these centrality problems and based on that we introduce new complementary problems. The main open problem of Abboud et al. is whether or not APSP and Diameter are equivalent under subcubic reduction. One of the results of this paper is APSP and CoDiameter, which is the complementary version of Diameter, are equivalent. Moreover, for some of the problems in this set, we show that they are equivalent to their complementary versions. Considering the slight difference between a problem and its complementary version, these equivalences give us the impression that every problem has such a property, and thus APSP and Diameter are equivalent. This paper is a step forward in showing a subcubic equivalence between APSP and Diameter, and we hope that the approach introduced in our paper can be helpful to make this breakthrough happen.