Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Disjoint Shortest Paths with Congestion on DAGs   In the k-Disjoint Shortest Paths problem, a set of terminal pairs of vertices
$\{(s_i,t_i)\mid 1\le i\le k\}$ is given and we are asked to find paths
$P_1,\ldots,P_k$ such that each path $P_i$ is a shortest path from $s_i$ to
$t_i$ and every vertex of the graph routes at most one of them. We introduce a
generalization of the problem, namely, $k$-Disjoint Shortest Paths with
Congestion-$c$ where every vertex is allowed to route up to $c$ paths.
  We provide a simple algorithm to solve the problem in time $f(k) n^{O(k-c)}$
on DAGs. Using the techniques for DAGs, we show the problem is solvable in time
$f(k) n^{O(k)}$ on general undirected graphs. Our algorithm for DAGs is based
on the earlier algorithm for $k$-Disjoint Paths with Congestion-$c$[IPL2019],
but we significantly simplify their argument.
  Then we prove that it is not possible to improve the algorithm significantly
by showing that for every constant $c$ the problem is W[1]-hard w.r.t.\
parameter $k-c$. We also consider the problem on acyclic planar graphs, but
this time we restrict ourselves to the edge-disjoint shortest paths problem. We
show that even on acyclic planar graphs there is no $f(k)n^{o(k)}$ algorithm
for the problem unless ETH fails.

14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
2
The categories of this paper are: math
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex