Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A linear-time parameterized algorithm for computing the width of a DAG   The width $k$ of a directed acyclic graph (DAG) $G = (V, E)$ equals the
largest number of pairwise non-reachable vertices. Computing the width dates
back to Dilworth's and Fulkerson's results in the 1950s, and is doable in
quadratic time in the worst case. Since $k$ can be small in practical
applications, research has also studied algorithms whose complexity is
parameterized on $k$. Despite these efforts, it is still open whether there
exists a linear-time $O(f(k)(|V| + |E|))$ parameterized algorithm computing the
width. We answer this question affirmatively by presenting an $O(k^24^k|V| +
k2^k|E|)$ time algorithm, based on a new notion of frontier antichains. As we
process the vertices in a topological order, all frontier antichains can be
maintained with the help of several combinatorial properties, paying only
$f(k)$ along the way. The fact that the width can be computed by a single
$f(k)$-sweep of the DAG is a new surprising insight into this classical
problem. Our algorithm also allows deciding whether the DAG has width at most
$w$ in time $O(f(\min(w,k))(|V|+|E|))$.

30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin