Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Asymmetric Travelling Salesman Problem in Sparse Digraphs   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 $\tau(d)\le d$.

23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
2
The categories of this paper are: math
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci