Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for [Formula: see text] Boolean matrices of the form [Formula: see text] has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the classical computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time [Formula: see text], where [Formula: see text] satisfies the equation [Formula: see text] and [Formula: see text] is the exponent of the multiplication of an [Formula: see text] matrix by an [Formula: see text] matrix. Next, we consider a relaxed version of the MW problem (in the classical model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most [Formula: see text] in time [Formula: see text] Then, by reducing the relaxed problem to the so called [Formula: see text]-witness problem, we provide an algorithm that reports for each non-zero entry [Formula: see text] of the product matrix [Formula: see text] a witness of rank [Formula: see text], where [Formula: see text] is the number of witnesses for [Formula: see text], with high probability. The algorithm runs in [Formula: see text] time, where [Formula: see text].