Figures
Abstract
Therapies consisting of a combination of agents are an attractive proposition, especially in the context of diseases such as cancer, which can manifest with a variety of tumor types in a single case. However uncovering usable drug combinations is expensive both financially and temporally. By employing computational methods to identify candidate combinations with a greater likelihood of success we can avoid these problems, even when the amount of data is prohibitively large. Hitting Set is a combinatorial problem that has useful application across many fields, however as it is NP-complete it is traditionally considered hard to solve exactly. We introduce a more general version of the problem (α,β,d)-Hitting Set, which allows more precise control over how and what the hitting set targets. Employing the framework of Parameterized Complexity we show that despite being NP-complete, the (α,β,d)-Hitting Set problem is fixed-parameter tractable with a kernel of size O(αdkd) when we parameterize by the size k of the hitting set and the maximum number α of the minimum number of hits, and taking the maximum degree d of the target sets as a constant. We demonstrate the application of this problem to multiple drug selection for cancer therapy, showing the flexibility of the problem in tailoring such drug sets. The fixed-parameter tractability result indicates that for low values of the parameters the problem can be solved quickly using exact methods. We also demonstrate that the problem is indeed practical, with computation times on the order of 5 seconds, as compared to previous Hitting Set applications using the same dataset which exhibited times on the order of 1 day, even with relatively relaxed notions for what constitutes a low value for the parameters. Furthermore the existence of a kernelization for (α,β,d)-Hitting Set indicates that the problem is readily scalable to large datasets.
Citation: Mellor D, Prieto E, Mathieson L, Moscato P (2010) A Kernelisation Approach for Multiple d-Hitting Set and Its Application in Optimal Multi-Drug Therapeutic Combinations. PLoS ONE 5(10): e13055. https://doi.org/10.1371/journal.pone.0013055
Editor: Maria A. Deli, Hungarian Academy of Sciences, Hungary
Received: June 15, 2010; Accepted: August 19, 2010; Published: October 18, 2010
Copyright: © 2010 Mellor et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The authors acknowledge the support of the Hunter Medical Research Institute, The University of Newcastle, and ARC Discovery Project DP0773279 (Application of novel exact combinatorial optimisation techniques and metaheuristic methods for problems in cancer research). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Typically the selection of a drug therapy for a disease is limited to a single drug, however diseases such as cancer may present as a heterogeneous mix of subtypes of the general disease. In cases such as these multi-drug therapies may prove more effective than single drug therapies, and many trials have been conducted to this end [1]–[3]. Furthermore combinations of drugs may allow a more targeted approach for a selection of subtypes of a disease, while minimizing effects on unaffected cells. Unfortunately with the abundance of compounds available for the treatment of many conditions of interest, the time and expense in testing even all two drug combinations may be prohibitive. Therefore a smarter approach is needed. Vazquez [4] introduces the Hitting Set problem for this task in the context of oncological drug therapy. The Hitting Set problem is a combinatorial problem that proves extremely useful in modeling a large variety of problems in many domains including protein network discovery [5], metabolic network analysis [6], diagnostics [7]–[9], gene ontology [10] and gene expression analysis [11], [12].
The Hitting Set Problem
Hitting Set is a combinatorial problem that models the problem
of selecting a small group of elements to represent or cover a collection of
sets. Such a group that covers every set in the collection is called a hitting
set. Finding such a set without any constraint is simple, however if we required
that the size of the hitting set be relatively small, the problem becomes
computationally challenging (-complete in a formal sense). This difficulty in obtaining
solutions with desirable qualities thus requires more thoughtful approaches.
We now give some technical details and formal definitions of the problems of interest.
Hitting Set is equivalent to the Set Cover
problem [13], and when otherwise unrestricted, is equivalent
to the Red/Blue Dominating Set
[14]
problem and is related to the -Feature Set
[15]
problem.
The decision version of the Hitting Set problem is defined as follows:
Hitting Set
Instance: A set
and a collection
and an integer
.
Question: Is there a set
with
such that for every
we have
?
The set is called a hitting set for
, or simply a hitting set. For an element
and an element
if
we say that
hits
. This problem is
-complete even when the maximum size of each element of
is two (by equivalence with Vertex Cover
[13])
and
-complete for parameter
; Cotta and Moscato [16] give a parameterized
proof via
-Feature Set and Paz and Moran [17] give a
proof which along with the equivalence of Hitting Set and
Set Cover leads to the same result, though predates the
parameterized complexity framework. However if we restrict the cardinality of
the elements of
to
the problem, while remaining
-complete, becomes fixed-parameter tractable where
is a constant and the parameter is
[18]. In this case the problem is known as the
Hitting Set for Sets of Size
or
-Hitting Set problem. We note that
Hitting Set has several equivalent formulations, in
particular we choose to use the bipartite graph representation where
and
form the two partite vertex sets of the graph and an edge
corresponds to the element
being an element of
. This allows us to employ some simplifying graph theoretic
terminology and techniques. We generalize this problem to include the case where
we may want the elements of
to be hit more than once. In particular this includes the case
where we ask if all the sets of
can be hit
times, but extends to the case where the elements of
can be hit up to
times. We encode this by the use of a hitting function
. Our problem then becomes the
-Multiple
-Hitting Set (or (
)-Hitting Set):
-Hitting Set
Instance: A bipartite graph
where for all
we have
, a hitting function
and an integer
.
Question: Is there a set
with
such that for every
we have
?
When for all
, (
)-Hitting Set can be
-approximated in time
[19], but cannot be approximated with a factor of
for any
unless
[20].
Results and Discussion
The Fixed-Parameter Tractability of (
)-Hitting Set
As we prove in the
Materials and Methods
section, the ()-Hitting Set problem is fixed-parameter
tractable, and indeed a more general variant the (
)-Hitting Set problem is also fixed parameter
tractable when we take the maximum degree
of the class vertices
as a constant and the size
of the hitting set and the maximum desired coverage
as a joint parameter. Though the problem is formally hard -
which would normally give the intuition that an exact solution would be too
expensive to compute - the fixed-parameter tractability indicates that it is
likely that we can obtain an exact solution efficiently. Armed with this
knowledge we proceed with the experiments of the following section, where we use
the drug response data of the NCI60 anti-tumor drug screening program to
determine a sets of drugs that hit cancerous cell lines multiple times. These
drug sets are than mathematically supportable candidates for combination
chemotherapies. Moreover we are able to tune the nature of the hitting sets via
the numbers
,
and
, which allows us to control which cell lines are targetted
(and which are specifically not) and how much each cell line is hit in the
solution.
A Comparative Application
The NCI60 human tumor anti-cancer drug screen dataset [21] was established
in the 1980s as an enabling tool for anti-cancer drug development. Included in
this dataset is response data for over drugs against the
cell lines of the dataset. Vazquez [4] highlights the
utility of a hitting set approach in developing multi-drug therapies for
heterogeneous malignancies; given the plethora of available compounds, testing
multi-drug combinations exhaustively is prohibitive if not impossible. Applying
hitting set to efficacy data measured on an individual basis for each compound
allows us to determine possible drug combinations that would provide the best
chance of efficacy against many cancer types. Using the GI50 response NCI60
dataset (available from the DTP website [22]) Vazquez uncovers a
minimum hitting set with three compounds that cumulatively gives a good response
with all cell lines in the dataset, where a response is considered good if it is
more than two standard deviations above the mean of the z-transformed response
data. Vazquez uses first a greedy highest-degree-first approach to give an
estimate of the maximum size of a minimum hitting set, followed by either an
exhaustive search or simulated annealing, depending on the size of the hitting
set. Vazquez reports times for such approaches on the order of one day on a
desktop computer.
We revisit Vasquez's experiment, using data reduction (though it is not
necessary to employ the more complex rules given in the kernelization proof)
with IBM ILOG CPLEX [23] as the kernel solver by framing the problem
as a integer programming problem. We use the same threshold for the
z-transformation to identify significant response levels. Using this approach we
reduce the time to solve the instance to less than seconds, where most of the time is spent loading and reducing
the data, with CPLEX solving the integer programming instance in approximately
milliseconds. Furthermore this approach guarantees optimality
in the size of the hitting set.
From here we employ more a more recent version of the NCI60 dataset (2009 as
compared to Vazquez's 2006). At the time of writing, the latest NCI60
dataset includes 14 additional cell lines, however we remove these, as there is
insufficient response data in the dataset, leading to inflated hitting set
sizes. The latest data also includes a further compounds. We note that employing the new GI50 response data
we are able to uncover
element hitting sets involving compounds not available in the
earlier dataset (an example is given in Table 1 and Figure 1), in particular Everolimus (NSC
733504) a drug now used for the treatment of advanced renal cancer which is also
giving positive results in phase II trials for metastatic melanoma [24], [25].
However there have recently been some concerns over the provenance of some of
the cell lines in the NCI60 dataset. In particular Lorenzi et al.
[26]
suggested that the MDA-N cell line, nominally a breast cancer cell line is in
fact similar the M14 and MDA-MB-435 cell lines, and thus should be is in fact a
melanoma cell line. Chambers [27] however suggests that although M14 and
MDA-MB-435 are identical cell lines, they may not in fact be melanoma cell
lines. We do not attempt to resolve this dispute, however with regard to this,
and as a indication of the flexibility of the method we employ we consider both
the case where MDA-N is a breast cancer cell line and the the case where MDA-N
is a melanoma cell line.
This hitting set hits all cell lines at least once, but is further optimized to hit all target cell lines the maximal number of times. Of particular note are NSC 174121, a methotrexate derivative and NSC733504, Everolimus/Afinitor, both known anti-cancer agents.
Employing the ()-Hitting Set model gives more flexibility in
what kind of therapy we would like to pursue. For instance, by choosing
for all vertices, we are able to find a hitting set that hits
every cell line at least twice (see Table 2). However the size of this hitting
set is
, which is likely to be beyond the point where the trade off
between anti-cancer efficacy and side effects is acceptable. Fortunately we can
exploit (
)-Hitting Set more intelligently. For example
we may wish to find a hitting set that specifically targets breast cancer cell
lines – for which we set all breast cancer cell line vertices to have
and all other cell lines to have
. This gives a hitting set that hits only
breast cancer cell lines, which may be useful in minimizing unwanted peripheral
damage to non-breast cancer cells. This gives a hitting set with three elements.
In the case where we considered MDA-N to be a breast cancer cell line (see Table 3 and Figure 2) this set includes
the compound deoxypodophyllotoxin, which is known to induce apoptosis [28]. If
we consider MDA-N as a melanoma cell line we obtain a different hitting set (see
Table 4 and Figure 3). If we relax our
requirements an allow other cell lines to be hit at most once we can obtain a
hitting set that hits the breast cancer cell lines more (Table 5 and Figure 4). The results when we set
to
for all breast cancer lines are given in Table 6 and Figure 5 (including MDA-N) and
Table 7 and Figure 6 (excluding MDA-N). We
note particularly that in the case where MDA-N is included, the optimal hitting
set uncovered includes Docetaxel, a well known anti-cancer agent [29] for several cancer types including breast cancer.
Interestingly Docetaxel is also currently included in several clinical trials
examining its potential as part of a multi-drug therapy [30]–[34].
Including the disputed MDA-N cell line. This hitting set also reveals additional structure with each drug targeting a specific, disjoint subset of the breast cancer cell lines. Only cell lines with at least one adjacent compound are shown.
Excluding the disputed MDA-N cell line. In this case the hitting set is much less clearly separated, though two of the cell lines are now hit twice. Only cell lines with at least one adjacent compound are shown.
Excluding the disputed MDA-N cell line. In this case we allow non-breast cancer cell lines to be hit at most once. By relaxing the restriction on hitting non-breast cancer cell lines, we obtain a hitting set which hits more of the breast cancer cell lines repeatedly. The trade-off being that other cell lines are also affected, increasingly the likelihood that non-cancerous cells are also affected by the treatment, as the compounds are less specific to a particular genetic signature. Only cell lines with at least one adjacent compound are shown.
Including the disputed MDA-N cell line. In this case the breast cancer cell lines separate neatly into two groups, with the first group forming a cycle and the second group forming a complete bipartite graph. Only cell lines with at least one adjacent compound are shown.
Excluding the disputed MDA-N cell line. Without the MDA-N cell line, the breast cancer cell lines do not separate, although the complete bipartite component is a subgraph of this graph, however we gain a greater number of hits per cell line in this case. Only cell lines with at least one adjacent compound are shown.
In another example, we may wish to target melanoma cell lines exclusively, and
furthermore, we may wish to attack each cell line with at least two drugs at
once. However in this case (where for melanoma cell lines and
for all others) the minimal hitting set size is
(or
if MDA-N is included as a melanoma cell line – Table 8 and Figures 7 & 8). Considering that a
therapeutic cocktail involving
compounds may have excessive side effects, we can relax the
requirements, and allow
for non-melanoma cell lines. In this case we find that the
smallest hitting set is of size
. By altering the focus when solving the kernel by fixing the
hitting set size (
) at
and maximizing the total degree of the vertices in the hitting
set, subject to the
and
constraints, we can obtain the minimal size hitting set that
hits our targets as much as possible, within the bounds given by the
constraints. This results in the hitting sets in Tables 9 & 10 and Figures 9 & 10. Of note is AZD6244, which is currently
involved in
anti-cancer drug trials [35] and has
been identified as a potent kinase inhibitor [36], [37].
This hitting set also maximizes the number of hits on the melanoma cell lines. Only cell lines with at least one adjacent compound are shown.
Including the disputed MDA-N cell line. It is interesting to note that
including MDA-N as a melanoma cell line rather than a breast cancer cell
line reduces the size of the minimal hitting set from to
. This hitting set also maximizes the number of hits on
the melanoma cell lines. Only cell lines with at least one adjacent
compound are shown.
For this we consider MDA-N as a non-melanoma cell line, however it is also hit by the hitting set, though only once. This hitting set also maximizes the number of hits on the melanoma cell lines. Only cell lines with at least one adjacent compound are shown.
Including MDA-N as a melanoma cell line. The key difference with the case where we consider MDA-N to be a non-melanoma cell line is that in this case we obtain a hitting set that hits the melanoma cell lines slightly more. Only cell lines with at least one adjacent compound are shown.
Conclusion
Given the size of modern datasets, and the expectation that they will only get
larger, it is clear that we require efficient approaches to solving important
computational biology problems. The first phase of any such approach is simply
defining the problem at hand. Unfortunately once clearly stated, many such
problems are -hard or worse. However this need not mean that we must resort
to inexact or approximate approaches, which could be undesirable in a field such
as drug selection. Parameterized Complexity provides a toolkit for dealing with
nominally hard problems, and identifying cases where despite super-polynomial
running times, we may still expect good performance.
The drug selection problem as examined here is one such problem. It is modeled
well by the -Hitting Set problem, which is
fixed-parameter tractable when parameterized by the maximum size of the hitting
set. Therefore we can expect that despite being
-complete, it would be relatively quick to solve when these
parameters are small. However we demonstrate that the much more flexible variant (
)-Hitting Set is also fixed-parameter
tractable, with only the addition of a single parameter - the maximum of the
minimum number of times any vertex should be hit. With (
)-Hitting Set we are able to better control
the nature of the hitting set uncovered, and thus tailor any such hitting set to
a useful set of constraints, such as limits on which cell lines are to be hit,
the maximum any of these can be hit and of course the minimum number of times
any cell line should be hit. Moreover we can solve this problem quickly, and
guarantee optimality - without any notable restrictions on the parameters and
constants. This allows the quick generation of possible drug combinations for
testing, with guarantees of a certain baseline performance, eliminating the need
to exhaustively test all possible combinations, which would be financially and
temporally prohibitive.
In brief this paper provides a robust and flexible methodology for multiple drug
selection, which can easily be applied to other domains that are modeled by the -Hitting Set problem, with a sound
theoretical background as to why and how the problem can be solved efficiently,
despite its
-completeness. Moreover the existence of a kernelization for (
)-Hitting Set indicates that even without
using a specialized commercial solver such as CPLEX, the problem is readily
scalable to large datasets. Given the speed at which we are able to solve
instances with on the order of
vertices, we can expect that much larger datasets are also
solvable in a reasonable time.
A future extension that may be of interest would be to somehow encode in the problem the notion that some hitting vertices are incompatible, e.g., two compound may have severe adverse interactions, and thus can never be used together as a therapy, regardless of their individual usefulness.
Materials and Methods
Dataset and Computational Method
The dataset primarily employed is the NCI60 DTP Human Tumor Cell Line Screen, available from [22]. We use the version released in October 2009, and downloaded in April 2010. The raw dataset is presented as a series of cell line and compound pairs, along with the GI50 response measurement (the method for producing the measurements is also detailed by [22]) for that pair plus concentration information and statistical information. Where there are multiple entries for the same compound-cell line pair, we select the entry resulting from the experiment using the highest concentration of the compound. We extract this data into a matrix cross indexed by the NSC number of the compound and the name of the cell line. Where an entry does not exist for a given compound-cell line pair, we enter “NA” for that entry in the matrix.
Once the data is in this matrix format we threshold the data according to the
method used by Vazquez [4] whereby the raw data is subject to a
z-transformation over a logarithmic scale and then any value above a certain
threshold expressed in terms of the standard deviation to , and anything below, including “NA”
values, to
. In line with Vazquez we choose two standard deviations as our
particular threshold for this paper, though this is adjustable.
We then construct a graph for the hitting set instance using the Java Universal
Network/Graph Framework (JUNG) [38] with the SetHypergraph class, representing
each compound with a vertex and each cell line with a (hyper)edge which carries
a weight indicating the number of times that edge is to be hit. This graph is
then reduced to remove vertices of zero degree, edges with no incident vertices
(which are noted as technically this would indicate a no instance unless that
edge does not require hitting) and vertices that are only adjacent to edges that
require zero hits. This basic reduction alone typically reduces the number of
vertices significantly, bringing the graph within a reasonable size for
immediate processing. From a theoretical standpoint the constant is of importance, for the graph constructed as stated,
(as we allow the natural value, rather than imposing an
external limit). In practice a
value of this magnitude proves perfectly workable, and
returning to the theoretical viewpoint indicates that the instance is in a sense
already kernelized.
Once the graph is reduced, we construct an integer programming instance
equivalent of the problem given the graph, and pass this instance to CPLEX [23]
(version 11.200) and search for an optimal solution to one of two objective
functions, given the constraints of the number of hits for each cell line (given
by the value). The first objective function simply minimizes the size
of the hitting set (
), for the second objective function we fix the size of the
hitting set, and maximize the number of hits on vertices where no maximum number
of hits has been set (the
value). As part of this search CPLEX may apply some
unspecified proprietary reduction process.
The figures were created using yEd Graph Editor [39].
The computer hardware employed is a Dell PowerEdge III Dual Xeon 5550 server with 32Gb of RAM, operating Red Hat Linux 64 bit EL 4 Server.
Theoretical Background and Kernelization Proof
Graph Theory and Notation.
A (simple undirected) graph consists of a set (the vertices), and a set
of two element subsets of
(the edges). A bipartite graph is a graph
where the vertices are partitioned into two partite sets, where all edges
have one endpoint in one set and the other endpoint in the other set, i.e.,
and
.
Given a graph and two vertices
, we denote the edge between
and
by
or equivalently
. Given two vertices
in
, if there is an edge
we say that
and
are adjacent and the
and
are incident on
. Given a vertex
, the set
is the (open) neighborhood of
and consists off all vertices adjacent to
in
, we extend this notion in the natural way to sets of
vertices.
Parameterized Complexity.
A parameterized (decision) problem is a formally defined
computational problem consisting of three components; the input, a special
part of the input called the parameter, and the question. Following Flum and
Grohe's [40] definition we may assume that the
parameter is derived from a polynomial time computable mapping from the
input to the natural numbers. A parameterized problem is fixed-parameter tractable if there is
an algorithm
such that for every instance
where
is the input,
is the parameter and
,
correctly answers Yes or No in time
bounded by
where
is a polynomial and
is a computable function.
A polynomial time kernelization (or just
kernelization) is a polynomial time mapping that given
an instance of a parameterized problem produces a new instance
of the problem such that:
is a Yes-instance if and only if
is a Yes-instance,
and
for some computable function
.
It is easy to see that if a problem has kernelization, then it is fixed-parameter tractable. It is also easy to prove that if a problem is fixed-parameter tractable, then it has a kernelization [41].
Parameterized complexity has a fully developed theory for determining when a
problem is unlikely to be fixed-parameter tractable, but as this is not
necessary for this work, we refer the reader to the monographs of Flum and
Grohe [40] and Downey and Fellows [42] for full
discussion, and simply state that if a problem is -hard or
-complete for any
, then the problem is not fixed-parameter tractable unless
certain complexity theoretic assumptions are false, which seems
unlikely.
The Fixed-Parameter Tractability of (
)-Hitting Set
Our kernelization for ()-Hitting Set follows the basic format of
Abu-Khzam's kernelization for
-Hitting Set
[18].
Let be an instance of (
)-Hitting Set which we assume to have been
preprocessed for nonsense input such as vertices
with
or
. Therefore we may assume that for all
we have
and that for all vertices
we have
.
We first apply Reduction Rules 1 to 3 exhaustively, before applying Rules 4 and 5.:
Reduction Rule 1: If there is a vertex with
then for every vertex
for every vertex
reduce
by
, delete
from
and reduce
by
. Finally, delete
from
.
Lemma 1 Reduction Rule 1 is sound.
Proof. If such a vertex exists, then all its neighbors in
must be in the hitting set, and we can remove them from the
graph after suitably noting the effect for the vertices of
.
Note in particular that this rule effectively allows us to assume that is at most
. This will be used implicitly in Reduction Rule 4.
Reduction Rule 2: If there is a vertex with
, delete
from
.
Lemma 2 Reduction Rule 2 is sound.
Proof. Clearly requires no vertices to hit it, so may be ignored.
Reduction Rule 3: If there are two vertices such that
and
, delete
from
.
Lemma 3 Reduction Rule 3 is sound.
Proof. If two such vertices and
exist, then any hitting set that hits
at least
times will hit
at least
times.
Let be a set of size
vertices such that
is the pairwise intersection of the neighborhoods of a vertex
set
. Let
.
Reduction Rule 4: Let and
be vertex sets as described. For each
such that
add a vertex
to
with
and edges such that
and delete
from
.
Lemma 4 Reduction Rule 4 is sound.
Proof. Let be a Yes-instance of (
)-Hitting Set. Then there is a set
with
that hits each element
of
at least
times. Assume that there are sets
and
as described in the reduction rule and that for some
we have that
. Let
be the subset of
that hits
. Assume further that
, then for each
there is at least one other vertex in
, but then
, which contradicts the assumption that
is a Yes-instance.
Therefore the set must be hit by
, so we may restrict our search to the intersection.
Lemma 5 Reduction Rule 4 can be computed in polynomial time.
Proof. Given a set of vertices for some
with
, we construct an auxiliary graph
by taking for each
the subgraph of
induced by the vertices
. If there is a maximum matching in
of size greater than
, then the matched vertices from
form the required set with pairwise neighbohood intersection
.
As is a constant, we can iterate over all sets of vertices of
size
in time
. The matchings can be computed in time
.
Definition 6 (Weakly Related Vertices) Given two vertices ,
and
are weakly related if
, and both
and
.
Let be a maximal set of pairwise weakly related vertices. Let
be a set of vertices, and denote by
the set of vertices of
whose neighborhood is a superset of
. Further denote by
the subset of
where for each
we have
.
Reduction Rule 5: Compute a maximal collection of pairwise weakly related vertices. If
apply the following algorithm:
for
downto
do
for
downto
do
for each set where
and
do
if
then
Add a vertex to
, edges such that
and set
.
Delete from
.
Lemma 7 Reduction Rule 5 is sound.
Proof. We defer the proof of the bound on the size of until the proof of Lemma 8.
Let be a Yes-instance of (
)-Hitting Set. Then there is a set
that hits
sufficiently. For sets of size
, Reduction Rule 4 proves the soundness of the first iteration
of the outer loop.
For each other iteration, assume that the iteration for sets of size holds, then let
be set of size
where
for some
. If
then by the pigeon hole principle there is some vertex
that is in at least
neighborhoods of vertices in
, but then
is a set that is the intersection of at least
neighborhoods of vertices in some subset of
, contradicting the correctness of the previous iteration.
Therefore the entire set of vertices hitting each
vertex is contained within
if
, so we may replace
with a single vertex.
Note also that for each element of there is at most
sets
, so we may iterate through all sets in time
, so we can perform the replacements in polynomial time.
Lemma 8
If is a Yes-instance
of (
)-Hitting Set, reduced under
Reduction Rules 1 to 5, then
.
Proof. If is a Yes-instance of (
)-Hitting Set, then there is a set
such that for every
we have
with
.
Claim 9
.
By construction, every vertex in with degree at most
is in
. Assume there is some
with
and
, then there must be some vertex
such that
, but then as the degree of any vertex in
is at most
,
, and Reduction Rule 3 would apply. Therefore there are no
vertices from
not in
.
Claim 10
.
As hits each vertex of
at least once, by Reduction Rule 5 each element of
as a singleton is in the neighborhood of at most
vertices from
. Therefore
.
Combining Claims 9 and 10 we have . As each vertex of
has degree at most
, there are at most
vertices in
, and the bound follows.
Theorem 11 ()-Hitting Set
is fixed-parameter tractable with parameter
and has a kernel of size at most
.
We note that although must be a constant to obtain a polynomial time kernelization,
may be alternatively given as an additional parameter, without
change to the kernelization.
This kernelization may be extended to an even more general version of the problem, where we not only specify lower bounds for the number of hits, but also upper bounds:
-Hitting Set
Instance: A bipartite graph
where for all
we have
, two hitting functions
and
and an integer
.
Question: Is there a set
with
such that for every
we have
?
Corollary 12 ()-Hitting Set
is fixed-parameter tractable with parameter
and has a kernel of size at most
.
Author Contributions
Conceived and designed the experiments: DM LM PM. Performed the experiments: DM LM PM. Analyzed the data: LM PM. Contributed reagents/materials/analysis tools: DM EP LM PM. Wrote the paper: DM EP LM PM.
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