Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $r$-Simple $k$-Path and Related Problems Parameterized by $k/r$   Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems.
In the $r$-Simple $k$-Path problem, given a digraph $G$ on $n$ vertices and
integers $r,k$, decide whether $G$ has an $r$-simple $k$-path, which is a walk
where every vertex occurs at most $r$ times and the total number of vertex
occurrences is $k$. In the $(r,k)$-Monomial Detection problem, given an
arithmetic circuit that encodes some polynomial $P$ on $n$ variables and
integers $k,r$, decide whether $P$ has a monomial of degree $k$ where the
degree of each variable is at most~$r$. In the $p$-Set $(r,q)$-Packing problem,
given a universe $V$, positive integers $p,q,r$, and a collection $\cal H$ of
sets of size $p$ whose elements belong to $V$, decide whether there exists a
subcollection ${\cal H}'$ of $\cal H$ of size $q$ where each element occurs in
at most $r$ sets of ${\cal H}'$. Abasi et al. and Gabizon et al. proved that
the three problems are single-exponentially fixed-parameter tractable (FPT)
when parameterized by $(k/r)\log r$, where $k=pq$ for $p$-Set $(r,q)$-Packing
and asked whether the $\log r$ factor in the exponent can be avoided.
  We consider their question from a wider perspective: are the above problems
FPT when parameterized by $k/r$ only? We resolve the wider question by (a)
obtaining a $2^{O((k/r)^2\log(k/r))} (n+\log k)^{O(1)}$-time algorithm for
$r$-Simple $k$-Path on digraphs and a $2^{O(k/r)} (n+\log k)^{O(1)}$-time
algorithm for $r$-Simple $k$-Path on undirected graphs (i.e., for undirected
graphs we answer the original question in affirmative), (b) showing that
$p$-Set $(r,q)$-Packing is FPT, and (c) proving that $(r,k)$-Monomial Detection
is para-NP-hard. For $p$-Set $(r,q)$-Packing, we obtain a polynomial kernel for
any fixed $p$, which resolves a question posed by Gabizon et al. regarding the
existence of polynomial kernels for problems with relaxed disjointness
constraints.

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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph