Subcubic Equivalences Between APSP, Co-Diameter, and Other Complementary Problems∗†

Despite persistent effort, there is no known technique for obtaining super-linear lower bounds for the computational complexity of the problems in P. Vassilevska Williams and Williams [38] 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. They show if there is a truly subcubic algorithm (an O(n3− ) time algorithm for a constant > 0) for any of these problems, then there exist truly subcubic algorithms for other problems as well. Abboud, Grandoni, and Vassilevska Williams [1] 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 truly subcubic running time for these centrality problems, implies a truly 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 [1] 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, 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. 1998 ACM Subject Classification Complexity Measures and Classes