Beyond the BEST Theorem: Fast Assessment of Eulerian Trails

Given a directed multigraph G = (V, E), with vertical bar V vertical bar = n nodes and vertical bar E vertical bar = m edges, and an integer z, we are asked to assess whether the number #ET(G) of node-distinct Eulerian trails of G is at least z; two trails are called node-distinct if their node sequences are different. This problem has been formalized by Bernardini et al. [ALENEX 2020] as it is the core computational problem in several string processing applications. It can be solved in O(n(omega)) arithmetic operations by applying the well-known BEST theorem, where omega < 2.373 denotes the matrix multiplication exponent. The algorithmic challenge is: Can we solve this problem faster for certain values of m and z? Namely, we want to design a combinatorial algorithm for assessing whether #ET(G) >= z, which does not resort to the BEST theorem and has a predictably bounded cost as a function of m and z. We address this challenge here by providing a combinatorial algorithm requiring O(m . min{z, #ET(G)}) time.