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PLoS One. 2010; 5(10): e13055.
Published online Oct 18, 2010. doi:  10.1371/journal.pone.0013055
PMCID: PMC2956629

A Kernelisation Approach for Multiple d-Hitting Set and Its Application in Optimal Multi-Drug Therapeutic Combinations

Maria A. Deli, Editor

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 Odkd) 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.

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 (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e013.jpg-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e014.jpg-Feature Set [15] problem.

The decision version of the Hitting Set problem is defined as follows:

Hitting Set

Instance: A set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e015.jpg and a collection An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e016.jpg and an integer An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e017.jpg.

Question: Is there a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e018.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e019.jpg such that for every An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e020.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e021.jpg?

The set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e022.jpg is called a hitting set for An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e023.jpg, or simply a hitting set. For an element An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e024.jpg and an element An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e025.jpg if An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e026.jpg we say that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e027.jpg hits An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e028.jpg. This problem is An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e029.jpg-complete even when the maximum size of each element of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e030.jpg is two (by equivalence with Vertex Cover [13]) and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e031.jpg-complete for parameter An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e032.jpg; Cotta and Moscato [16] give a parameterized proof via An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e033.jpg-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e034.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e035.jpg the problem, while remaining An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e036.jpg-complete, becomes fixed-parameter tractable where An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e037.jpg is a constant and the parameter is An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e038.jpg [18]. In this case the problem is known as the Hitting Set for Sets of Size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e039.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e040.jpg-Hitting Set problem. We note that Hitting Set has several equivalent formulations, in particular we choose to use the bipartite graph representation where An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e041.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e042.jpg form the two partite vertex sets of the graph and an edge An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e043.jpg corresponds to the element An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e044.jpg being an element of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e045.jpg. 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e046.jpg to be hit more than once. In particular this includes the case where we ask if all the sets of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e047.jpg can be hit An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e048.jpg times, but extends to the case where the elements of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e049.jpg can be hit up to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e050.jpg times. We encode this by the use of a hitting function An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e051.jpg. Our problem then becomes the An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e052.jpg-Multiple An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e053.jpg-Hitting Set (or (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e054.jpg)-Hitting Set):

An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e055.jpg-Hitting Set

Instance: A bipartite graph An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e056.jpg where for all An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e057.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e058.jpg, a hitting function An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e059.jpg and an integer An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e060.jpg.

Question: Is there a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e061.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e062.jpg such that for every An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e063.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e064.jpg?

When An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e065.jpg for all An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e066.jpg, (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e067.jpg)-Hitting Set can be An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e068.jpg-approximated in time An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e069.jpg [19], but cannot be approximated with a factor of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e070.jpg for any An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e071.jpg unless An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e072.jpg [20].

Results and Discussion

The Fixed-Parameter Tractability of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e073.jpg)-Hitting Set

As we prove in the Materials and Methods section, the (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e074.jpg)-Hitting Set problem is fixed-parameter tractable, and indeed a more general variant the (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e075.jpg)-Hitting Set problem is also fixed parameter tractable when we take the maximum degree An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e076.jpg of the class vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e077.jpg as a constant and the size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e078.jpg of the hitting set and the maximum desired coverage An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e079.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e080.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e081.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e082.jpg, 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e083.jpg drugs against the An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e084.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e085.jpg seconds, where most of the time is spent loading and reducing the data, with CPLEX solving the integer programming instance in approximately An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e086.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e087.jpg compounds. We note that employing the new GI50 response data we are able to uncover An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e088.jpg 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.

Figure 1
Minimal hitting set hitting for the NCI60 dataset.
Table 1
Minimal hitting set using 2009 NCI60 data.

Employing the (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e089.jpg)-Hitting Set model gives more flexibility in what kind of therapy we would like to pursue. For instance, by choosing An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e090.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e091.jpg, 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 (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e092.jpg)-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e093.jpg and all other cell lines to have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e094.jpg. 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e095.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e096.jpg 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].

Figure 2
Minimal hitting set hitting only breast cancer cell lines.
Figure 3
Minimal hitting set hitting only breast cancer cell lines.
Figure 4
Minimal hitting set hitting only breast cancer cell lines.
Figure 5
Minimal hitting set hitting breast cancer cell lines twice.
Figure 6
Minimal hitting set hitting breast cancer cell lines twice.
Table 2
Minimal double hitting set.
Table 3
Minimal hitting set targeting only breast cancer.
Table 4
Minimal hitting set targeting only breast cancer without MDA-N.
Table 5
Minimal hitting set targeting breast cancer but allowing other cell lines to be hit.
Table 6
Minimal hitting set hitting breast cancer twice, and no others, with MDA-N.
Table 7
Minimal hitting set hitting breast cancer twice, and no others, without MDA-N.

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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e097.jpg for melanoma cell lines and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e098.jpg for all others) the minimal hitting set size is An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e099.jpg (or An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e100.jpg if MDA-N is included as a melanoma cell line – Table 8 and Figures 7 & 8). Considering that a therapeutic cocktail involving An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e101.jpg compounds may have excessive side effects, we can relax the requirements, and allow An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e102.jpg for non-melanoma cell lines. In this case we find that the smallest hitting set is of size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e103.jpg. By altering the focus when solving the kernel by fixing the hitting set size (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e104.jpg) at An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e105.jpg and maximizing the total degree of the vertices in the hitting set, subject to the An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e106.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e107.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e108.jpg anti-cancer drug trials [35] and has been identified as a potent kinase inhibitor [36], [37].

Figure 7
Minimal hitting set hitting melanoma cell lines at least 2 and no other cell lines.
Figure 8
Minimal hitting set hitting melanoma cell lines at least 2 and no other cell lines.
Figure 9
Minimal hitting set hitting melanoma cell lines at least 2 and all other cell lines at most once.
Figure 10
Minimal hitting set hitting melanoma cell lines at least 2 and all other cell lines at most once.
Table 8
Minimal hitting set targeting melanoma twice, without MDA-N.
Table 9
Minimal hitting set targeting melanoma, without MDA-N.
Table 10
Minimal hitting set targeting melanoma, with MDA-N.

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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e111.jpg-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e112.jpg-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e113.jpg-complete, it would be relatively quick to solve when these parameters are small. However we demonstrate that the much more flexible variant (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e114.jpg)-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 (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e115.jpg)-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e116.jpg-Hitting Set problem, with a sound theoretical background as to why and how the problem can be solved efficiently, despite its An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e117.jpg-completeness. Moreover the existence of a kernelization for (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e118.jpg)-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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e119.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e120.jpg, and anything below, including “NA” values, to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e121.jpg. 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e122.jpg is of importance, for the graph constructed as stated, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e123.jpg (as we allow the natural value, rather than imposing an external limit). In practice a An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e124.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e125.jpg value). The first objective function simply minimizes the size of the hitting set (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e126.jpg), 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e127.jpg 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e128.jpg (the vertices), and a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e129.jpg of two element subsets of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e130.jpg (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., An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e131.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e132.jpg.

Given a graph An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e133.jpg and two vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e134.jpg, we denote the edge between An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e135.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e136.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e137.jpg or equivalently An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e138.jpg. Given two vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e139.jpg in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e140.jpg, if there is an edge An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e141.jpg we say that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e142.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e143.jpg are adjacent and the An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e144.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e145.jpg are incident on An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e146.jpg. Given a vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e147.jpg, the set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e148.jpg is the (open) neighborhood of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e149.jpg and consists off all vertices adjacent to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e150.jpg in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e151.jpg, 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e152.jpg is fixed-parameter tractable if there is an algorithm An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e153.jpg such that for every instance An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e154.jpg where An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e155.jpg is the input, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e156.jpg is the parameter and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e157.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e158.jpg correctly answers Yes or No in time bounded by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e159.jpg where An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e160.jpg is a polynomial and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e161.jpg is a computable function.

A polynomial time kernelization (or just kernelization) is a polynomial time mapping that given an instance An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e162.jpg of a parameterized problem produces a new instance An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e163.jpg of the problem such that:

  1. An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e164.jpg is a Yes-instance if and only if An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e165.jpg is a Yes-instance,
  2. An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e166.jpg and
  3. An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e167.jpg for some computable function An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e168.jpg.

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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e169.jpg-hard or An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e170.jpg-complete for any An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e171.jpg, then the problem is not fixed-parameter tractable unless certain complexity theoretic assumptions are false, which seems unlikely.

The Fixed-Parameter Tractability of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e172.jpg)-Hitting Set

Our kernelization for (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e173.jpg)-Hitting Set follows the basic format of Abu-Khzam's kernelization for An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e174.jpg-Hitting Set [18].

Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e175.jpg be an instance of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e176.jpg)-Hitting Set which we assume to have been preprocessed for nonsense input such as vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e177.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e178.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e179.jpg. Therefore we may assume that for all An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e180.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e181.jpg and that for all vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e182.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e183.jpg.

We first apply Reduction Rules 1 to 3 exhaustively, before applying Rules 4 and 5.:

Reduction Rule 1: If there is a vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e184.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e185.jpg then for every vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e186.jpg for every vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e187.jpg reduce An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e188.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e189.jpg, delete An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e190.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e191.jpg and reduce An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e192.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e193.jpg. Finally, delete An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e194.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e195.jpg.

Lemma 1 Reduction Rule 1 is sound.

Proof. If such a vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e196.jpg exists, then all its neighbors in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e197.jpg must be in the hitting set, and we can remove them from the graph after suitably noting the effect for the vertices of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e198.jpg.

Note in particular that this rule effectively allows us to assume that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e200.jpg is at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e201.jpg. This will be used implicitly in Reduction Rule 4.

Reduction Rule 2: If there is a vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e202.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e203.jpg, delete An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e204.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e205.jpg.

Lemma 2 Reduction Rule 2 is sound.

Proof. Clearly An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e206.jpg requires no vertices to hit it, so may be ignored.

Reduction Rule 3: If there are two vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e208.jpg such that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e209.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e210.jpg, delete An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e211.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e212.jpg.

Lemma 3 Reduction Rule 3 is sound.

Proof. If two such vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e213.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e214.jpg exist, then any hitting set that hits An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e215.jpg at least An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e216.jpg times will hit An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e217.jpg at least An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e218.jpg times.

Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e220.jpg be a set of size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e221.jpg vertices such that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e222.jpg is the pairwise intersection of the neighborhoods of a vertex set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e223.jpg. Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e224.jpg.

Reduction Rule 4: Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e225.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e226.jpg be vertex sets as described. For each An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e227.jpg such that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e228.jpg add a vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e229.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e230.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e231.jpg and edges such that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e232.jpg and delete An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e233.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e234.jpg.

Lemma 4 Reduction Rule 4 is sound.

Proof. Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e235.jpg be a Yes-instance of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e236.jpg)-Hitting Set. Then there is a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e237.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e238.jpg that hits each element An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e239.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e240.jpg at least An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e241.jpg times. Assume that there are sets An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e242.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e243.jpg as described in the reduction rule and that for some An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e244.jpg we have that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e245.jpg. Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e246.jpg be the subset of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e247.jpg that hits An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e248.jpg. Assume further that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e249.jpg, then for each An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e250.jpg there is at least one other vertex in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e251.jpg, but then An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e252.jpg, which contradicts the assumption that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e253.jpg is a Yes-instance.

Therefore the set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e254.jpg must be hit by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e255.jpg, 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e257.jpg for some An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e258.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e259.jpg, we construct an auxiliary graph An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e260.jpg by taking for each An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e261.jpg the subgraph of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e262.jpg induced by the vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e263.jpg. If there is a maximum matching in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e264.jpg of size greater than An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e265.jpg, then the matched vertices from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e266.jpg form the required set with pairwise neighbohood intersection An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e267.jpg.

As An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e268.jpg is a constant, we can iterate over all sets of vertices of size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e269.jpg in time An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e270.jpg. The matchings can be computed in time An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e271.jpg.

Definition 6 (Weakly Related Vertices) Given two vertices An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e273.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e274.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e275.jpg are weakly related if An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e276.jpg, and both An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e277.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e278.jpg.

Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e279.jpg be a maximal set of pairwise weakly related vertices. Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e280.jpg be a set of vertices, and denote by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e281.jpg the set of vertices of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e282.jpg whose neighborhood is a superset of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e283.jpg. Further denote by An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e284.jpg the subset of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e285.jpg where for each An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e286.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e287.jpg.

Reduction Rule 5: Compute a maximal collection An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e288.jpg of pairwise weakly related vertices. If An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e289.jpg apply the following algorithm:

for An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e290.jpg downto An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e291.jpg do

 for An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e292.jpg downto An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e293.jpg do

  for each set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e294.jpg where An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e295.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e296.jpg do

   if An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e297.jpg then

    Add a vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e298.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e299.jpg, edges such that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e300.jpg and set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e301.jpg.

    Delete An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e302.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e303.jpg.

Lemma 7 Reduction Rule 5 is sound.

Proof. We defer the proof of the bound on the size of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e304.jpg until the proof of Lemma 8.

Let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e305.jpg be a Yes-instance of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e306.jpg)-Hitting Set. Then there is a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e307.jpg that hits An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e308.jpg sufficiently. For sets of size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e309.jpg, 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 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e310.jpg holds, then let An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e311.jpg be set of size An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e312.jpg where An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e313.jpg for some An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e314.jpg. If An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e315.jpg then by the pigeon hole principle there is some vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e316.jpg that is in at least An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e317.jpg neighborhoods of vertices in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e318.jpg, but then An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e319.jpg is a set that is the intersection of at least An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e320.jpg neighborhoods of vertices in some subset of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e321.jpg, contradicting the correctness of the previous iteration. Therefore the entire set of vertices hitting each An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e322.jpg vertex is contained within An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e323.jpg if An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e324.jpg, so we may replace An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e325.jpg with a single vertex.

Note also that for each element of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e326.jpg there is at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e327.jpg sets An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e328.jpg, so we may iterate through all sets in time An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e329.jpg, so we can perform the replacements in polynomial time.

Lemma 8 If An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e331.jpg is a Yes-instance of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e332.jpg)-Hitting Set, reduced under Reduction Rules 1 to 5, then An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e333.jpg.

Proof. If An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e334.jpg is a Yes-instance of (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e335.jpg)-Hitting Set, then there is a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e336.jpg such that for every An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e337.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e338.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e339.jpg.

Claim 9 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e340.jpg.

By construction, every vertex in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e341.jpg with degree at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e342.jpg is in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e343.jpg. Assume there is some An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e344.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e345.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e346.jpg, then there must be some vertex An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e347.jpg such that An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e348.jpg, but then as the degree of any vertex in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e349.jpg is at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e350.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e351.jpg, and Reduction Rule 3 would apply. Therefore there are no vertices from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e352.jpg not in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e353.jpg.

Claim 10 An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e354.jpg.

As An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e355.jpg hits each vertex of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e356.jpg at least once, by Reduction Rule 5 each element of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e357.jpg as a singleton is in the neighborhood of at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e358.jpg vertices from An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e359.jpg. Therefore An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e360.jpg.

Combining Claims 9 and 10 we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e361.jpg. As each vertex of An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e362.jpg has degree at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e363.jpg, there are at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e364.jpg vertices in An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e365.jpg, and the bound follows.

Theorem 11 (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e367.jpg)-Hitting Set is fixed-parameter tractable with parameter An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e368.jpg and has a kernel of size at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e369.jpg.

We note that although An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e370.jpg must be a constant to obtain a polynomial time kernelization, An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e371.jpg 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:

An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e372.jpg-Hitting Set

Instance: A bipartite graph An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e373.jpg where for all An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e374.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e375.jpg, two hitting functions An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e376.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e377.jpg and an integer An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e378.jpg.

Question: Is there a set An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e379.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e380.jpg such that for every An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e381.jpg we have An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e382.jpg?

Corollary 12 (An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e383.jpg)-Hitting Set is fixed-parameter tractable with parameter An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e384.jpg and has a kernel of size at most An external file that holds a picture, illustration, etc.
Object name is pone.0013055.e385.jpg.

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

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.

References

1. Albain KS, Crowley JJ, LeBlanc M, Livingston RB. Determinants of improved outcome in small-cell lung cancer: an analysis of the 2,580-patient southwest oncology group data base. Journal of Clinical Oncology. 1990;8:1563–1574. [PubMed]
2. Flamant F, Schwartz L, Delons E, Caillaud JM, Hartmann O, et al. Nonseminomatous malignant germ cell tumors in children. Multidrug therapy in stages III and IV. Cancer. 1984;54:1687–1691. [PubMed]
3. Fu KK, Silverberg IJ, Phillips TL, Friedman MA. Combined radiotherapy and multidrug chemotherapy for advanced head and neck cancer: results of a radiation therapy oncology group pilot study. Cancer Treatment Reports. 1979;63:351–357. [PubMed]
4. Vazquez A. Optimal drug combinations and minimal hitting sets. BMC Systems Biology. 2009;3:81–86. [PMC free article] [PubMed]
5. Berman P, DasGupta B, Sontag ED. Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks. Discrete Applied Mathematics. 2007;155:733–749.
6. Haus UU, Klamt S, Stephen T. Computing knock-out strategies in metabolic networks. Journal of Computational Biology. 2008;15:259–268. [PubMed]
7. de Kleer J, Mackworth AK, Reiter R. Characterizing diagnoses and systems. Artificial Intelligence. 1992;56:197–222.
8. Leipins GE, Potter WD. A genetic algorithm approach to multiple-fault diagnosis. In: Davis L, editor. Handbook of Genetic Algorithms, Van Nostrand Reinhold Company; 1991. pp. 237–250.
9. Reiter R. A theory of diagnosis from first principles. Artificial Intelligence. 1987;32:57–95.
10. Hvidsten TR, Lægreid A, Komorowski HJ. Learning rule-based models of biological process from gene expression time profiles using gene ontology. Bioinformatics. 2003;19:1116–1123. [PubMed]
11. Ruchkys D, Song S. A parallel approximation hitting set algorithm for gene expression analysis. 2002. pp. 75–81. In: 14th Symposium on Computer Architecture and High Performance Computing (SBAC-PAD 2002). Vitoria, Espirito Santo, Brazil.
12. Vinterbo SA, Kim EY, Ohno-Machado L. Small, fuzzy and interpretable gene expression based classifiers. Bioinformatics. 2005;21:1964–1970. [PubMed]
13. Garey MR, Johnson DS. Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman & Co; 1979.
14. Fernau H. Parameterized Algorithmics: A Graph-Theoretic Approach. Germany: Habilitationsschrift, Universität Tübingen; 2005.
15. Davies S, Russell S. NP-completeness of searches for smallest possible feature sets. In: Greiner R, Subramanian D, editors. AAAI Symposium on Intelligent Relevance. New Orleans: 1994. pp. 41–43.
16. Cotta C, Moscato P. The k-feature set problem is W[2]-complete. Journal of Computer and System Sciences. 2003;67:686–690.
17. Paz A, Moran S. Non deterministic polynomial optimization problems and their approximations. Theoretical Computer Science. 1981;15:251–277.
18. Abu-Khzam FN. A kernelization algorithm for d-hitting set. Journal of Computer and Systems Sciences. 2010;76:524–531.
19. Vazirani V. Approximation Algorithms. Berlin: Springer-Verlag; 2001.
20. Feige U. A threshold of ln n for approximating set cover. Journal of the ACM. 1998;45:634–652.
21. Shoemaker RH. The NCI60 human tumour cell line anticancer drug screen. Nature Reviews Cancer. 2006;6:813–823. [PubMed]
22. NCI/NIH Website (Accessed 2010). Developmental Theraputics Program. http://dtp.nci.nih.gov/
23. IBM Website (Accessed 2010). ILOG CPLEX. http://www-01.ibm.com/software/integration/optimization/cplex/
24. Rao RD, Windschitl HE, Allred JB, Lowe VJ, Maples WJ, et al. Phase II trial of the mTOR inhibitor everolimus (RAD-001) in metastatic melanoma. Journal of Clinical Oncology. 2006;24:8043.
25. Peyton JD, Spigel DR, Burris HA, Lane C, Rubin M, et al. Phase II trial of bevacizumab and everolimus in the treatment of patients with metastatic melanoma: Preliminary results. Journal of Clinical Oncology. 2009;27:9027.
26. Lorenzi PL, Reinhold WC, Varma S, Hutchinson AA, Pommier Y, et al. DNA fingerprinting of the NCI-60 cell line panel. Molecular Cancer Therapeutics. 2009;8:713–724. [PMC free article] [PubMed]
27. Chambers AF. MDA-MB-435 and M14 cell lines: Identical but not M14 melanoma? Cancer Research. 2009;69:5292–5293. [PubMed]
28. Shin SY, Yong Y, Kim CG, Lee YH, Lim Y. Deoxypodophyllotoxin induces G2/M cell cycle arrest and apoptosis in HeLa cells. Cancer Letters. 2010;287:231–239. [PubMed]
29. Lyseng-Williamson KA, Fenton C. Docetaxel: A review of its use in metastatic breast cancer. Drugs. 2005;65:2513–2531. [PubMed]
30. Slamon D, Eiermann W, Robert N, Pienkowski T, Martin M, et al. Phase III randomized trial comparing doxorubicin and cyclophosphamide followed by docetaxel (AC-¿T) with doxorubicin and cyclophosphamide followed by docetaxel and trastuzumab (AC-¿TH) with docetaxel, carboplatin and trastuzumab (TCH) in Her2neu positive early breast cancer patients: BCIRG 006 study. Cancer Research. 2009;69:62.
31. Perez EA, Hillman DW, Dentchev T, Le-Lindqwister NA, Geeraerts LH, et al. North central cancer treatment group (NCCTG) N0432: phase II trial of docetaxel with capecitabine and bevacizumab as first-line chemotherapy for patients with metastatic breast cancer. Annals of Oncology. 2010;21:269–274. [PMC free article] [PubMed]
32. Polyzos A, Malamos N, Boukovinas I, Adamou A, Ziras N, et al. FEC versus sequential docetaxel followed by epirubicin/cyclophosphamide as adjuvant chemotherapy in women with axillary node-positive early breast cancer: a randomized study of the Hellenic Oncology Research Group (HORG). Breast Cancer Research and Treatment. 2010;119:95–104. [PubMed]
33. Joensuu H, Bono P, Kataja V, Alanko T, Kokko R, et al. Fluorouracil, Epirubicin, and Cyclophosphamide With Either Docetaxel or Vinorelbine, With or Without Trastuzumab, As Adjuvant Treatments of Breast Cancer: Final Results of the FinHer Trial. Journal of Clinical Oncology. 2009;27:5685–5692. [PubMed]
34. Sparano JA, Makhson AN, Semiglazov VF, Tjulandin SA, Balashova OI, et al. Pegylated Liposomal Doxorubicin Plus Docetaxel Significantly Improves Time to Progression Without Additive Cardiotoxicity Compared With Docetaxel Monotherapy in Patients With Advanced Breast Cancer Previously Treated With Neoadjuvant-Adjuvant Anthracycline Therapy: Results From a Randomized Phase III Study. J Clin Oncol. 2009;27:4522–4529. [PubMed]
35. ClinicalTrialsgov Website (Accessed 2010). U.S. clinical trial registry. http://clinicaltrials.gov/ct2/home.
36. Davies B, Logie A, McKay JS, Martin P, Steele S, et al. AZD6244 (ARRY-142886), a potent inhibitor of mitogen-activated protein kinase/extracellular signal-regulated kinase kinase 1/2 kinases: mechanism of action in vivo, pharmacokinetic/pharmacodynamic relationship, and potential for combination in preclinical models. Molecular Cancer Theraputics. 2007;6:2209–2219. [PubMed]
37. Yeh TC, Marsh V, Bernat BA, Ballard J, Colwell H, et al. Biological characterization of ARRY-142886 (AZD6244), a potent, highly selective mitogen-activated protein kinase kinase 1/2 inhibitor. Clinical Cancer Research. 2007;13:1576. [PubMed]
38. Java Universal Network/Graph Framework Website (Accessed 2010). JUNG. http://jung.sourceforge.net/
39. yWorks Website (Accessed 2010). yEd. http://www.yworks.com/en/products_yed_about.html.
40. Flum J, Grohe M. Parameterized Complexity Theory. Berlin: Springer; 2006.
41. Niedermeier R. Invitation to Fixed-Parameter Algorithms. Oxford: Oxford University Press; 2006.
42. Downey RG, Fellows MR. Parameterized Complexity. Berlin: Springer; 1999.

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