Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an
additive form
defined as follows. Given an instance (
I,k
) of some (parameterized) problem π with a
guarantee
g(I)
, decide whether
I
admits a solution of size at least (or at most)
k
+
g(I)
. Here,
g(I)
is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M
AX
SAT and M
AX
C
UT
(with
g(I)
being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a
multiplicative
form of parameterization above (or, rather, times) a guarantee: Given an instance (
I,k
) of some (parameterized) problem π with a guarantee
g(I)
, decide whether
I
admits a solution of size at least (or at most)
k
·
g(I)
. In particular, we study the
Long Cycle
problem with a multiplicative parameterization above the girth
g(I)
of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above
g(I)
1+ε
of
Long Cycle
yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.
Towards fixed-parameter tractable algorithms for abstract argumentation