Parameterized Complexity of Directed Spanner Problems

Abstract

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most \(m-k\) arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most \(m-k\) arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size \(\mathcal {O}(k^4t^5)\) and can be solved in randomized \((4t)^k\cdot n^{\mathcal {O}(1)}\) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in \(k^{2k}\cdot n^{\mathcal {O}(1)}\) time on directed acyclic graphs, (iii) Directed Additive Spanner is \({{\,\mathrm{\mathsf{W}}\,}}[1]\)-hard when parameterized by k for every fixed \(t\ge 1\) even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is \({{\,\mathrm{\mathsf{FPT}}\,}}\) when parameterized by t and k.