r
-Simple
k
-Path and Related Problems Parameterized by
k
/
r
Authors: Gutin Gregory , Wahlström Magnus , Zehavi Meirav 
Published: January 2021

Abasi et al. (2014) introduced the following two problems. In the
r
-S
imple
k
-P
ath
problem, given a digraph
G
on
n
vertices and positive 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
)-M
onomial
D
etection
problem, given an arithmetic circuit that succinctly encodes some polynomial
P
on
n
variables and positive integers
k
,
r
, decide whether
P
has a monomial of total degree
k
where the degree of each variable is at most
r
. Abasi et al. obtained randomized algorithms of running time 4

(
k
/
r
)log
r

⋅
n

O
(1)

for both problems. Gabizon et al. (2015) designed deterministic 2

O
((
k
/
r
)log
r
)

⋅
n

O
(1)

-time algorithms for both problems (however, for the (
r
,
k
)-M
onomial
D
etection
problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the
P
-S
et
(
r
,
q
)-P
acking
P
roblem
, given a universe
V
, positive integers (
p
,
q
,
r
), and a collection H of sets of size
P
whose elements belong to
V
, decide whether there exists a subcollection H
′
of H of size
q
where each element occurs in at most
r
sets of H
′
. Gabizon et al. obtained a deterministic 2

O
((
pq
/
r
)log
r
)

⋅
n

O
(1)

-time algorithm for
P
-S
et
(
r
,
q
)-P
acking
.

The above results prove that the three problems are
single-exponentially
fixed-parameter tractable (FPT) parameterized by the product of
two
parameters, that is,
k
/
r
and log
r
, where
k
=
pq
for
P
-S
et
(
r
,
q
)-P
acking
. Abasi et al. and Gabizon et al. asked whether the log
r
factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (
r
,
k
)-M
onomial
D
etection
by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2

o
((
k
/
r
) log
r
)

⋅ (
n
+ log
k
)

O
(1)

-time algorithm for (
r
,
k
)-M
onomial
D
etection
, i.e., (
r
,
k
)-M
onomial
D
etection
is unlikely to be single-exponentially FPT when parameterized by
k
/
r
alone. The question remains open for
r
-S
imple
k
-P
ath
and
P
-S
et
(
r
,
q
)-P
acking
.

We consider the question from a wider perspective: are the above problems FPT when parameterized by
k
/
r
only, i.e., whether there exists a computable function
f
such that the problems admit a
f
(
k
/
r
)(
n
+log
k
)

O
(1)

-time algorithm? Since
r
can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by
k
/
r
alone. 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
-S
imple
k
-P
ath
on digraphs and a 2

O
(
k
/
r
)

&sdot (
n
+ log
k
)

O
(1)

-time algorithm for
r
-S
imple
k
-P
ath
on undirected graphs (i.e., for undirected graphs, we answer the original question in affirmative), (b) showing that
P
-S
et
(
r
,
q
)-P
acking
is FPT (in contrast, we prove that
P
-M
ultiset
(
r
,
q
)-P
acking
is W[1]-hard), and (c) proving that (
r
,
k
)-M
onomial
D
etection
is para-NP-hard even if only two distinct variables are in polynomial
P
and the circuit is non-canceling. For the special case of (
r
,
k
)-M
onomial
D
etection
where
k
is polynomially bounded by the input size (which is in XP), we show W[1]-hardness. Along the way to solve
P
-S
et
(
r
,
q
)-P
acking
, 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. All our algorithms are deterministic.