1 research outputs found
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 -Simple -Path problem, given a digraph on vertices and
integers , decide whether has an -simple -path, which is a walk
where every vertex occurs at most times and the total number of vertex
occurrences is . In the -Monomial Detection problem, given an
arithmetic circuit that encodes some polynomial on variables and
integers , decide whether has a monomial of degree where the
degree of each variable is at most~. In the -Set -Packing problem,
given a universe , positive integers , and a collection of
sets of size whose elements belong to , decide whether there exists a
subcollection of of size where each element occurs in
at most sets of . Abasi et al. and Gabizon et al. proved that
the three problems are single-exponentially fixed-parameter tractable (FPT)
when parameterized by , where for -Set -Packing
and asked whether the factor in the exponent can be avoided.
We consider their question from a wider perspective: are the above problems
FPT when parameterized by only? We resolve the wider question by (a)
obtaining a -time algorithm for
-Simple -Path on digraphs and a -time
algorithm for -Simple -Path on undirected graphs (i.e., for undirected
graphs we answer the original question in affirmative), (b) showing that
-Set -Packing is FPT, and (c) proving that -Monomial Detection
is para-NP-hard. For -Set -Packing, we obtain a polynomial kernel for
any fixed , which resolves a question posed by Gabizon et al. regarding the
existence of polynomial kernels for problems with relaxed disjointness
constraints