  Asymmetric Travelling Salesman Problem (ATSP) and its special case Directed
Hamiltonicity are among the most fundamental problems in computer science. The
dynamic programming algorithm running in time $O^*(2^n)$ developed almost 60
years ago by Bellman, Held and Karp, is still the state of the art for both of
these problems.
  In this work we focus on sparse digraphs. First, we recall known approaches
for Undirected Hamiltonicity and TSP in sparse graphs and we analyse their
consequences for Directed Hamiltonicity and ATSP in sparse digraphs, either by
adapting the algorithm, or by using reductions. In this way, we get a number of
running time upper bounds for a few classes of sparse digraphs, including
$O^*(2^{n/3})$ for digraphs with both out- and indegree bounded by 2, and
$O^*(3^{n/2})$ for digraphs with outdegree bounded by 3.
  Our main results are focused on digraphs of bounded average outdegree $d$.
The baseline for ATSP here is a simple enumeration of cycle covers which can be
done in time bounded by $O^*(\mu(d)^n)$ for a function
$\mu(d)\le(\lceil{d}\rceil!)^{1/{\lceil{d}\rceil}}$. One can also observe that
Directed Hamiltonicity can be solved in randomized time $O^*((2-2^{-d})^n)$ and
polynomial space, by adapting a recent result of Bj\"{o}rklund [ISAAC 2018]
stated originally for Undirected Hamiltonicity in sparse bipartite graphs.
  We present two new deterministic algorithms for ATSP: the first running in
time $O(2^{0.441(d-1)n})$ and polynomial space, and the second in exponential
space with running time of $O^*(\tau(d)^{n/2})$ for a function $\tau(d)\le d$.
