Approximate Counting of k-Paths: Simpler, Deterministic, and in Polynomial Space

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4^kmϵ^-2-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ϵ based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: •We present a deterministic 4^{k+ O}(^{√k(log k+log2ϵ-1}))m-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. •Additionally, we present a randomized 4^{k+mathcal O(logk(logk+logϵ-1}))m-time polynomial-space algorithm. Our algorithm is simple - we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q-dimensional p-matchings).