Search tips
Search criteria 


Logo of bmcbioiBioMed Centralsearchsubmit a manuscriptregisterthis articleBMC Bioinformatics
BMC Bioinformatics. 2012; 13(Suppl 19): S9.
Published online Dec 19, 2012. doi:  10.1186/1471-2105-13-S19-S9
PMCID: PMC3526445
Reconstructing genome mixtures from partial adjacencies
Ahmad Mahmoody,corresponding author1 Crystal L Kahn,1 and Benjamin J Raphaelcorresponding author1
1Department of Computer Science, Brown University, Providence (RI), USA
corresponding authorCorresponding author.
Ahmad Mahmoody: ahmad/at/; Crystal L Kahn: clkahn/at/; Benjamin J Raphael: braphael/at/
Proceedings of the Tenth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics
Mathieu Banchette, Marilia D V Braga and Marie-France Sagot
This supplement has not been supported by sponsorship or other external funding.
Tenth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics
17-19 October 2012
Niteroi, Brazil
Many cancer genome sequencing efforts are underway with the goal of identifying the somatic mutations that drive cancer progression. A major difficulty in these studies is that tumors are typically heterogeneous, with individual cells in a tumor having different complements of somatic mutations. However, nearly all DNA sequencing technologies sequence DNA from multiple cells, thus resulting in measurement of mutations from a mixture of genomes. Genome rearrangements are a major class of somatic mutations in many tumors, and the novel adjacencies (i.e. breakpoints) resulting from these rearrangements are readily detected from DNA sequencing reads. However, the assignment of each rearrangement, or adjacency, to an individual cancer genome in the mixture is not known. Moreover, the quantity of DNA sequence reads may be insufficient to measure all rearrangements in all genomes in the tumor. Motivated by this application, we formulate the k-minimum completion problem (k-MCP). In this problem, we aim to reconstruct k genomes derived from a single reference genome, given partial information about the adjacencies present in the mixture of these genomes. We show that the 1-MCP is solvable in linear time in the cases where: (i) the measured, incomplete genome has a single circular or linear chromosome; (ii) there are no restrictions on the chromosomal content of the measured, incomplete genome. We also show that the k-MCP problem, for k ≥ 3 in general, and the 2-MCP problem with the double-cut-and-join (DCJ) distance are NP-complete, when there are no restriction on the chromosomal structure of the measured, incomplete genome. These results lay the foundation for future algorithmic studies of the k-MCP and the application of these algorithms to real cancer sequencing data.
Nearly all current genome sequencing studies sequence the DNA from a population of cells rather than from single cells. This is because present DNA sequencing technologies cannot sequence the DNA in a single cell without bias-inducing DNA amplification steps. In the majority of applications, sequencing such a population of cells is not problematic because the DNA in every cell is nearly identical. However, there are two notable examples: metagenomics (e.g. environmental sequencing or microbiome studies) and cancer sequencing. In the former case, the genomic differences between cells are due to the presence of mixtures of microorganisms in the sample. In the latter case, the genomic differences between cells are due to somatic mutations that accumulate in individual tumor cells during the progression of cancer [1].
In this paper, we formulate the problem of inferring the organization of each genome present in a mixture in the case where: (1) the individual genomes result from an unknown sequence of genome rearrangements from a known (reference) genome; (2) the adjacencies (breakpoints) of the genomes in the mixture are measured. This situation arises in cancer genome studies where somatic structural aberrations (including inversions, translocations, duplications, deletions, or other rearrangements of large pieces of DNA) induce novel adjacencies, also called breakpoints, that join in the cancer genome two noncontiguous nucleotides from the normal genome. In current cancer sequencing projects, these novel adjacencies are determined from alignments of paired-end reads from cancer DNA to the reference human genome [2,3]. However, these approaches generally do not measure all adjacencies present in the tumor. For example, the quantity of DNA sequence reads (coverage) may be insufficient to measure all adjacencies in all genomes in the tumor, particularly adjacencies that are present in a minority of cancer cells. Moreover, alignment of reads to repetitive regions is challenging, particularly for short reads produced by current sequencing technologies, and thus some adjacencies may not be reliably measured.
We formulate the k-Minimum Completion Problem (k-MCP) of determining the k genomes present in a mixture from a set of measured adjacencies that minimize the total distance between the reference genome and the k measured (i.e. cancer) genomes. The k-MCP is a general problem that encompasses different subproblems that depend on the genomic distance used and the desired chromosomal content of the measured genomes. We show the following results: (1) A linear time algorithm for the 1-MCP in the double cut and join (DCJ) distance [4] when the desired genome has no restrictions on its chromosomal content; (2) A linear time algorithm for the 1-MCP in the DCJ distance when the desired genome has a single circular or linear chromosome; (3) the k-MCP is NP-complete for any distance when k ≥ 3; and (4) the 2-MCP with DCJ distance is NP-complete when the desired genome has no restrictions on its chromosomal content, or when the desired genome has all circular chromosomes.
We emphasize that the k-MCP does not model all the issues arising in cancer sequencing: in particular, we restrict attention to copy-neutral structural variants, and ignore single nucleotide mutations, small indels, or other large copy number aberrations. Single nucleotide mutations and small indels can be addressed separately as they do not produce novel adjacencies of the type studied in k-MCP. Copy number aberrations are common in cancer, but appropriate handling of these mutations when measured in a heterogeneous mixture introduces an entirely different set of challenges: e.g. a deletion of a genomic segment in half of the cells in the mixture with a duplication of the same segment in the other half of the cells will be difficult to distinguish from no copy number change. Finally, we assume that all measured adjacencies are real, while in fact there are likely to be false positive adjacencies. Extending the model to consider these additional complexities is left for future work.
In following sections, we first provide a precise formulation of the k-MCP and describe related work. Then, we provide algorithms and proofs of the complexity of the problem in various cases.
Definitions and problem statement
In this section we present some preliminary definitions and give the formal definition of k-MCP.
A gene g is an oriented sequence of nucleotides, with two extremities: a head gh and a tail gt. An adjacency is an unordered pair of gene extremities. A genome equation M1 on n genes is a set equation M2 of adjacencies such that each of the 2n gene extremities in equation M3 is a member of at most one adjacency in equation M4. The gene extremities which are not members of any adjacency in equation M5 are called telomeres of equation M6, and we denote the set of all telomeres by equation M7 (Figure 1-a). Through this work, we assume that the genes of a genome are distinct.
Figure 1
Figure 1
Genome and genome graph. (a) A genome equation M8 on the set of genes {1, 2, 3, 4, 5} with two chromosomes (one linear and one circular). equation M9. (b) The genome graph (black edges) of equation M10 with additional edges (dotted) connecting the extremities of the same gene. There is (more ...)
The genome graph of a genome equation M11 is a graph whose labeled vertices are the gene extremities in equation M12, and whose edge set is equation M13. We denote the genome graph of equation M14 by equation M15. Because each extremity is in at most one adjacency of equation M16, the graph equation M17 is a matching graph (not necessarily perfect). Note that the genome graph is uniquely determined by the genome, and conversely. For convenience, we also define the augmented genome graph equation M18 to be the genome graph augmented with additional edges connecting extremities of the same gene, i.e., equation M19 is the graph whose labeled vertices are the gene extremities in G, and whose edge set is equation M20.
A chromosome of equation M21 is the set of all adjacencies and telomeres of gene extremities in a connected component of the augmented genome graph (Figure 1-b). A chromosome is linear (resp. circular) if the corresponding connected component is a path (resp. cycle) (Figure 1-b). Note that an adjacency {gh, gt} represents a circular chromosome with the single gene g. A genome is circular or linear if all of its chromosomes are circular or linear, and we say it is mixed if it has both circular and linear chromosomes. A genome is uni-chromosomal if it has only one chromosome, and it is multi-chromosomal, otherwise. A chromosomal condition is a condition on the number or type of chromosomes in a genome. For example we can describe the structure of a genome by two chromosomal conditions: being (i) uni-chromosomal, and (ii) circular.
As described above a paired-end sequencing experiment provides the adjacencies equation M22 of the sequenced genome relative to the genes from a reference genome. However, our knowledge about a genome's adjacencies is typically incomplete. For a set equation M23 of chromosomal conditions, a equation M24 -partial genome equation M25 on n genes is a set of adjacencies equation M26 such that there exists a set equation M27 of pairs of gene extremities such that equation M28 is a genome with chromosomal condition equation M29. When equation M30 is clear in the context we will say partial-genome instead of equation M31 -partial genome. The problems we study below involve adding the missing adjacencies in equation M32 -partial genomes to complete them into genomes with chromosomal condition equation M33. Sometimes we have an idea about the number or the structure of chromosomes in a genome. We define a completion of a partial genome relative to these chromosomal conditions. If equation M34 is a genome, we say equation M35 provided equation M36. A equation M37 of a partial genome equation M38 is a genome equation M39 with equation M40 and satisfying the conditions in equation M41. When equation M42 is clear in the context, we just say completion instead of equation M43.
A multi-genome is a mixture of genomes with the same set of genes. Formally, the multi-genome equation M44 formed from genomes equation M45 is a multiset equation M46 obtained from equation M47, the disjoint union of equation M48 (For a multiset S and an element r, if cS(r) is the number of copies of r in S, the disjoint union of two multisets equation M49 is a multiset in which each element r appears cA(r) + cB(r) times.). Note that the partition of the adjacencies in equation M50 into equation M51 is not known. There is a corresponding genome graph, a multigraph whose vertices are the gene extremities, and whose edge set is the multiset equation M52. We denote the genome graph of a multi-genome equation M53 by equation M54.
The genome graph is related to the breakpoint graph in genome rearrangement studies. The breakpoint graph equation M55 of the genomes equation M56 is an edge-colored multigraph whose labeled vertices are the 2n gene extremities and whose edges are all the adjacencies in equation M57, with each edge assigned a color according to its genome of origin. Thus, the only difference between the breakpoint graph and the genome graph is the lack of edge-coloring in the latter, reflecting our inability to measure the origin of each adjacency.
Our knowledge about a multi-genome can be incomplete. For example a tumor is a mixture of different cancer genomes, and during sequencing process, we obtain a mixture of adjacencies from these genomes. We represent the mixtures of adjacencies by a partial multi-genome. A partial multi-genome is a multi-set equation M58, where each equation M59 is partial genome. We define the genome graph of a partial multi-genome analogously to a multi-genome.
If k is a positive integer and equation M60 is a partial multi-genome, a k-completion of equation M61 is a family of k genomes equation M62, such that equation M63. Note that existence of a completion for a partial (multi-) genome is dependent on the structure of the partial (multi-) genome and the chromosomal conditions. Also, the existence of a completion does not imply its uniqueness.
We use a distance function to distinguish between different completions. A distance function on pairs of genomes (with the same set of genes), is a measure of dissimilarity between the genomes. Having selected a pairwise distance function we must define a distance between the k genomes in a mixture. Motivated by the fact that the different cancer genomes in a tumor are obtained by somatic genome rearrangements from a healthy genome, we model the evolution of the cancer genomes by a rooted tree in which all the cancer genomes are descendants of the healthy one. Suppose equation M64 represents a healthy genome, and equation M65 a mixture of k cancer genomes obtained by rearrangements of the genome equation M66. A mixture tree equation M67 is a rooted tree on equation M68 such that the root vertex is equation M69 and k genomes in equation M70 are (some of) the vertices in equation M71. If ϕ is a distance function on a pair of genomes, then the ϕ-value of equation M72, denoted by equation M73 is defined as follows:
equation M74
where E is the set of edges in equation M75.
We now define the k-Minimum Completion Problem.
k-Minimum Completion Problem (k-MCP) Given a equation M76 -partial multi-genome equation M77, a positive integer k, a reference genome equation M78, and a distance function ϕ, find a k-completion equation M79 and a mixture tree equation M80 such that equation M81 is minimum over all k-completions and mixture trees. If no k-completion exists for equation M82, we say that this k-MCP does not have a valid solution. We say the k-MCP is unrestricted if equation M83, and is restricted, otherwise.
As written, the k-MCP is a general problem that encompasses many subproblems depending on chromosomal condition set equation M84 and the distance ϕ. Common distances in genome rearrangement studies include the breakpoint distance [5], the Hannenhalli-Pevzner distance [6] (which generalizes the reversal distance [7]), and the double-cut-and-join (DCJ) distance [4]. Below we will use the DCJ distance, which approximates the other distances [8].
For two genomes equation M85 and equation M86 on the same set of n genes, their double-cut-and-join (DCJ) distance, denoted by equation M87, is equal to
equation M88
where equation M89 is the number of cycles in equation M90 and equation M91 is the number of paths in B with odd number of vertices [8].
Remark. When at least one of the equation M92 are circular we have equation M93 and dDCJ (G1, G2) = n - c. Thus, having a larger number of cycles in their breakpoint graph is equivalent to having a smaller distance.
Related work
In comparison to other genome rearrangement problems considered in the literature, the k-MCP has three distinguishing features. (1) The input is a mixture of adjacencies from multiple genomes and the genome of origin of each adjacency is unknown. (2) The set of adjacencies is incomplete: not every adjacency from every genome in the mixture is measured. (3) The ancestral relationships between the genomes in the mixture are unknown, and might include both "ancestral" and "present day" genomes. Some of these features have been considered individually in other work, but to our knowledge no previous work has considered all three together. The first feature bears some resemblance to the genome halving problem [9] of finding the doubled ancestor genome by minimizing a rearrangement distance. This problem and further generalizations to polyploidization [10] involves partitioning (or coloring) adjacencies to minimize a rearrangement distance. However, in general no adjacencies are missing and the distance is pairwise (i.e., no tree) in contrast to the 2-MCP.
Regarding the second feature, several authors have considered the problem of inferring missing adjacencies in a manner that optimizes a genome rearrangement distance. Notably, [11] and [12] consider the problem of computing reversal distance between pairs of partially assembled genomes that are provided as unordered sequences of contigs. These problems were motivated by limitations in DNA sequence technologies that result in most whole-genome assemblies being highly fragmented and comprised of contigs whose relative ordering is unknown. These problems are variations of the 1-MCP, where the reference genome equation M94 also has missing adjacencies. In particular, [12] orient sets of contigs from two genomes in such a way that the number of cycles in the breakpoint graph of the resulting genomes is maximized, which they note "has been shown to approximate very well the reversal distance between them." However, there is no work on extending this analysis to more than two genomes.
Regarding the third feature, the genome median problem considers the problem of finding an ancestral genome that minimizes the distance between three given genomes [5,13]. This is different from k-MCP in that the three individual genomes are known (rather than mixed) and the genomes are complete with no missing adjacencies. Also, in the median problem the topology of the phylogenetic tree has been already inferred, while in k-MCP we have to find an optimal topology for the phylogenetic tree as well.
In this section we first consider the 1-MCP problem. We present linear time algorithms that solve 1-MCP in the cases where: (i) the measured, incomplete genome has a single circular or linear chromosome; (ii) there are no restrictions on the chromosomal content of the measured, incomplete genome.
Next we prove that the unrestricted k-MCP is NP-complete when k ≥ 3 for any distance function ϕ. Finally, we show that the unrestricted 2-MCP, and the restricted 2-MCP where all chromosomes are circular (i.e., equation M95), are NP-complete for DCJ distance.
Here, we consider the unrestricted 1-MCP and two restricted versions of 1-MCP problem: (1) the chromosomal condition set is {circular, uni-chromosomal}, which we denote by 1-MCPc; (2) the chromosomal condition set is {linear, uni-chromosomal}, which we denote by 1-MCP[ell]. We first show that unrestricted version is linearly tractable. Then, we show that we can solve the 1-MCPc in linear time. Finally, we prove a relation between 1-MCPc and, 1-MCP[ell] which implies that 1-MCP[ell] is also solvable in linear time.
Note that 1-MCP[ell] is a variation of the Block Ordering Problem (BOP) considered in [12]. In our terminology, the BOP considers two partial genomes, and aims to complete both partial genomes into linear, unichromosomal genomes such that the pairwise distance between the completed genome is minimal. In [12], Gaul and Blanchette provide a linear algorithm for BOP. The algorithm we present for 1-MCP[ell] is simpler than the algorithm for the BOP in [12], and our algorithm is obtained from a straightforward algorithm (Algorithm 1 below) which solves 1-MCPc in linear time.
We begin with the unrestricted 1-MCP, where we have the following result.
Theorem 1. The unrestricted 1-MCP with DCJ distance is linearly tractable.
Proof. In 1-MCP we have a single partial genome equation M96 and a reference genome equation M97 (see Figure 2-a). Since both equation M98 and equation M99 are matchings over the gene extremities, their breakpoint graph equation M100 consists of some paths and cycles. Suppose P1, . . ., Pr are all the paths such that the first and their last edges are adjacencies in equation M101. An optimal completion for equation M102 can be obtained by adding an edge to equation M103 which connects the end points of each Pi, for 1 ≤ i ≤ r (see Figure Figure3),3), since we only can add edges between the vertices which are not incident with any edge in equation M104, i.e., the end vertices of Pi's. Note that adding other possible edges just create longer paths in equation M105. □
Figure 2
Figure 2
Possible mixture trees when k = 1, 2. (a) The only topology in 1-MCP. (b) Branch-tree and (c) path-tree topologies in 2-MCP.
Figure 3
Figure 3
The breakpoint graph equation M106 with possible edges to be added to the adjacencies of equation M107. Breakpoint graph equation M108 consisting of paths and cycles. Thick edges are in equation M109, and thin edges are in equation M110. Dashed edges are the edges that should be added to equation M111 for the paths whose first (more ...)
1-MCPc: circular uni-chromosomal completion
Here we consider 1-MCPc, the restricted 1-MCP for a partial genome equation M113 that we wish to complete to a circular uni-chromosomal genome equation M114. We assume that equation M115 is not already a circular uni-chromosomal genome. Thus equation M116 has a set equation M117 of free extremities, i.e., the extremities that are not in any adjacency in equation M118. Equivalently, equation M119 is the set of vertices of degree 0 in the genome graph equation M120. Finding the completion equation M121 corresponds to finding a partition of equation M122 into pairs of extremities, i.e., into adjacencies. However, this partition cannot be arbitrary as the adjacencies defined by the partition must satisfy two constraints: (1) The resulting genome equation M123 is circular uni-chromosomal, meaning that the augmented genome graph equation M124 has exactly one component, a cycle. Note that equation M125 has only path components, since equation M126 and equation M127. (2) The resulting genome equation M128 must minimize the distance between the reference genome equation M129 and equation M130.
The first constraint on partitioning of equation M131 is that joining extremities at ends of a same path in equation M132 by an edge, which we call an excluded edge, creates a cycle. This cycle must be selected carefully to obtain a uni-chromosomal genome. We define equation M133 to be the set of all excluded edges.
The second constraint on partitioning of equation M134 is provided by our desire to minimize the distance between the reference genome equation M135 and equation M136. For the DCJ distance, we must maximize the number equation M137 of cycles in the breakpoint graph equation M138. Adding an edge to equation M139 increases the number of cycles in B if and only if the edge connects the endpoints of a same path in B. We call such an edge a desired edge and denote by equation M140 the set of all desired edges. Now we combine these two constraints into a graph.
We define the free-extremities graph, equation M141 to be a bicolored graph, whose vertex set is equation M142, and whose edge set is equation M143. The edges from equation M144 are colored blue and the edges from equation M145 are colored red. Note that R is a multi-graph, and R consists of even cycles. This is because both equation M146 and equation M147 are perfect matchings on equation M148: since both equation M149 and {{gh, gt} | g is a gene in equation M150} are perfect matchings on the set of all gene extremities. The restriction of these perfect matchings to equation M151 are equation M152 and equation M153. See Figure 4-b. Thus, we have
Figure 4
Figure 4
Adding adjacencies to a partial genome equation M154 to solve the 1-MCPc. (a) The breakpoint graph equation M155. Gray edges indicate adjacencies of equation M156, black edges indicate adjacencies of equation M157, and the dotted edges connect extremities of the same gene. The set of free vertices is (more ...)
equation M162
To find a completion of the partial genome equation M163 we select pairs {u, v} of free extremities from equation M164 and add them as adjacencies to equation M165. Respecting the constraints encoded in the free-extremities graph R, we define a transformation update(R, {u, v}) that records the effect of adding adjacency {u, v} to equation M166 (Figure (Figure4).4). In particular, since u and v are free vertices of equation M167, there are paths equation M168 and equation M169 in B with an endpoint equal to u and v, respectively. Similarly, there are paths equation M170 and equation M171 in equation M172 having an endpoint equal to u and v, respectively. We may have equation M173 or equation M174. By the definition of equation M175, equation M176 and equation M177 are represented by blue edges bu and bv in R incident to u and v. Similarly by the definition of equation M178, equation M179 and equation M180 are represented by red edges ru and rv in R incident to u and v. Adding the adjacency {u, v} to equation M181 will have the following effects on B and equation M182:
(i) u and v are no longer free vertices.
(ii) If equation M183 then these paths merge into one path in B [union or logical sum] {u, v}. Otherwise these paths merge to create a cycle in B [union or logical sum]{u, v}, and the number of cycles in the breakpoint graph increases by one.
(iii) If equation M184 these paths merge into one path in equation M185. Otherwise these paths merge into a cycle in equation M186. In the latter case, we should add {u, v} as an adjacency if and only if equation M187. This is because adding {u, v} creates a cycle component in equation M188 (i.e., a circular chromosome) and if there are other free vertices any subsequent completion will not be uni-chromosomal.
Therefore, adding the adjacency {u, v} to equation M189 will have three corresponding effects on R: removing the vertices u and v from R based on (i) above, identifying bu and bv based on (ii) above, and identifying ru and rv based on (iii) above. We denote this process of updating R by update(R, {u, v}). Figure Figure44 gives an illustration of this process.
If {u, v} is a blue edge in R, then update(R, {u, v}) increases the number of cycles in the breakpoint graph B by one. Hence, to find a solution to 1-MCPc we want to perform update(R, {u, v}) transformations with as many blue edges as possible. On the other hand, adding new adjacencies has to merge the paths in the graph equation M190 in such a way that we end with a genome with exactly one circular chromosome. Let Mb(R) be the maximum possible number of update transformations using blue edges for the graph R. The following theorem provides the exact value of Mb(R).
Theorem 2. Suppose equation M191 is a partial genome, equation M192 is a reference genome, and equation M193 is their free-extremities graph. We have
equation M194
where Nb(R) is the number of blue edges, and c(R) is the number of cycles in R.
Proof. We prove the theorem by induction on Nb(R). Suppose Nb(R) = 1. Then necessarily R consists of a cycle of length 2 with one blue and one red edge, and c(R) = 1. Thus, we update the graph R with the unique (and the only possible) blue edge obtaining
equation M195
Now suppose Nb(R) >1. Then equation M196, since equation M197. Suppose u, equation M198, and equation M199, i.e., there is no red edge between u and v in R. Then, we have the following three cases for u and v: (i) u and v are from different cycles Cu and Cv in R, (ii) u and v are connected with a blue edge in a cycle C of R, or (iii) u and v are non-neighboring vertices in a cycle C of R.
Let R' = update(R, {u, v}) be the free-extremities graph after the update. Since u and v are incident with blue edges in R, after update(R, {u, v}) the number of blue edges decreases by one, i.e., Nb(R') = Nb(R) - 1.
Thus, by induction hypothesis
equation M200
Considering the above cases we have:
(i) After update(R, {u, v}), Cu and Cv will shrink into one cycle, and c(R') = c(R) - 1. Thus by (2), Mb(R') = Nb(R) - c(R) + 1. By choosing such an edge we can update R with Nb(R) - c(R) + 1 blue edges.
(ii) After update(R, {u, v}), C shrinks into a smaller cycle, and c(R') = c(R). Thus, by (2), Mb(R') = Nb(R) - c(R). Since {u, v} is a blue edge, we can update R with Nb(R) - c(R) + 1 blue edges.
(iii) After update(R, {u, v}), C splits into two smaller cycles. Thus c(R') = c(R) + 1. Thus, by (2), Mb(R') = Nb(R) - c(R) - 1. So by choosing {u, v} we can update R with Nb(R) - c(R) - 1 blue edges.
By calculations above, choosing a pair {u, v} satisfying cases (i) or (ii) will result in a greater number of update moves with blue edges, than choosing a pair satisfies the case (iii). Moreover, considering pairs {u, v} from cases (i) and (ii) gives Mb(R) = Nb(R) - c(R) + 1. □
We call a pair {u, v} (which may or may not be an edge in R) satisfying case (i) or (ii) in the proof of Theorem 2 an optimal adjacency. Optimal adjacencies play an important role in finding a solution of 1-MCPc: updating the free-extremities graph with these adjacencies results in the maximum number of blue edges used in update transformations. We have the following important corollary to this theorem.
Corollary 1. Suppose equation M201 is a partial genome and equation M202 is a reference genome. Adding any optimal adjacency to equation M203 leads to a solution for 1-MCPc. In other words, for any optimal adjacency e, there exists a solution equation M204 for 1-MCPc which includes e as an adjacency.
Proof. By Theorem 2, adding any optimal adjacency to equation M205 will allow the maximum number of blue edges in the update process. Since each update transformation on a blue edge increases the number of cycles in the breakpoint graph by one, a sequence of update transformations on optimal adjacencies gives a solution equation M206 to 1-MCPc. Hence, if equation M207 is the resulting completion of equation M208, we obtain the maximum number of cycles in the breakpoint graph equation M209. □
A linear time (in number of genes) algorithm for solving 1-MCPc adds optimal adjacencies according to cases (i) and (ii) in Theorem 2, and is shown in Algorithm 1. The following corollary is an immediate consequence of Corollary 1 and Algorithm 1.
Corollary 2. The 1-MCPc is solvable in linear time.
Algorithm 1: Solving 1-MCPc
Input : Partial genome equation M210 and reference genome A.
Output: A 1-completion equation M211 that is circular uni-chromosomal and maximizes equation M212.
1  begin
2    Construct the free-extremities graph equation M213;
3    equation M214;
4    while c(R) >1 do
5        u, v ← select two vertices from different cycles in R;
6        equation M215;
7        R ← update (R, {u, v});
8    while the number of blue edges in R >1 do
9        u, v ← select two vertices connected via a blue edge in R;
10        equation M216;
11        R ← update (R, {u, v});
12    Add the single remaining excluded edge in equation M217 to equation M218;
13    Output the resulting circular uni-chromosomal genome equation M219;
14  end
1-MCP[ell]: linear uni-chromosomal completion
In this section we consider the 1-MCP with chromosomal condition of a linear uni-chromosomal genome. We refer to this restricted problem as 1-MCP[ell]. We relate solutions of 1-MCP[ell] to solutions of 1-MCPc. Combined with the results in the previous section, we derive a linear time algorithm for 1-MCP[ell].
Recall that equation M220 is the number of alternating cycles in the breakpoint graph equation M221, for any solution equation M222 of 1-MCPc. Similarly, we define equation M223 to be the number of alternating cycles in equation M224, for any solution equation M225 of 1-MCP[ell]. The following theorem relates the solutions of 1-MCPc to the solutions of 1-MCP[ell].
Theorem 3. Let equation M226 be a partial genome, equation M227 be a circular uni-chromosomal genome, and equation M228 be a linear uni-chromosomal genome obtained from equation M229 by removing an adjacency e. Suppose equation M230 and equation M231 are the reference genomes in 1-MCPc and 1-MCP[ell], respectively. From any solution equation M232 to 1-MCPc we obtain a solution equation M233 for 1-MCP[ell]. Also, from any solution equation M234 to 1-MCP[ell] we obtain a solution equation M235 for 1-MCPc. Moreover, equation M236, where
equation M237
Proof. First, suppose e is not in any cycle in the graph equation M238, and hence θ(e) = 1. Let equation M239 be a solution to 1-MCPc, and let equation M240 be a linear uni-chromosomal genome obtained from equation M241 by removing an adjacency equation M242, such that f and e are in the same cycle in equation M243. Note that such edge f exists, since e is not in any cycle in equation M244 but it is in a cycle of equation M245. See Figure Figure5.5. Both grequation M246 and equation M247 are perfect matchings as equation M248 and equation M249 are both circular. Removing the edges e and f from equation M250 will decrease the number of cycles by exactly one since e and f are in a same cycle in equation M251. Hence equation M252, and we have,
Figure 5
Figure 5
Relating 1-MCPc and 1-MCP[ell]. (a) The breakpoint graph equation M253; black edges are equation M254 and and gray edges are equation M255. The edge e = {1t, 6h} is the only edge in equation M256. Since e is not in a cycle component of B, we have θ(e) = 1. (b) The breakpoint graph equation M257, where (more ...)
equation M261
where the last inequality follows from the definition of equation M262 as the largest number of cycles in any linear chromosomal completion of equation M263.
Now suppose equation M264 is a solution to 1-MCP[ell], so equation M265. Assume equation M266. Let equation M267 be the circular uni-chromosomal genome obtained by adding equation M268 to equation M269. Note that there is at least one path component in equation M270 which becomes a cycle after adding the edges f' to equation M271 and e to equation M272. Hence, equation M273, and we have
equation M274
Thus by (3) and (4) we have equation M275, which implies that equation M276 and equation M277. This means that equation M278 and equation M279 are solutions to 1-MCPc and 1-MCP[ell] that are obtained from equation M280 and equation M281, respectively, which completes the proof for the case θ(e) = 1.
Now suppose e is in a cycle in equation M282, and thus θ(e) = 2. Using the same argument above, we have equation M283 since we cannot find such edge f and the number of cycles in equation M284 decreases by two, when we remove an edge from equation M285 (to obtain a linear genome), and e from equation M286 (to obtain the genome equation M287). Also, equation M288, as adding the excluded edges of equation M289 and equation M290 will increase the number of cycles by 2. Thus, for this case we have equation M291
Notice that the function θ depends only on the partial genome equation M292 and the reference genome equation M293, and not on the completion equation M294. Also, it is easy to see that θ is computable in linear time (in number of genes). We have the following corollary.
Corollary 3. The 1-MCP[ell] is solvable in linear time.
Proof. Suppose equation M295 is a partial genome and equation M296 is a linear chromosomal reference genome. Since equation M297 is linear and uni-chromosomal, equation M298. Assume that equation M299. Let equation M300 be the circular uni-chromosomal genome obtained by adding e to equation M301. Using Algorithm 1 we obtain a solution equation M302 for 1-MCPc with equation M303 as the reference genome. Then by Theorem 3, we can transform the solution equation M304 to a linear uni-chromosomal completion equation M305 in linear time in the following way: If there exists an edge equation M306 such that f and e are in the same cycle of the breakpoint graph equation M307, i.e. θ(e) = 1, remove f from equation M308. Otherwise θ(e) = 2 and we remove an arbitrary edge from equation M309 to make a linear uni-chromosomal genome. Therefore, we obtain a solution to 1-MCP[ell] by viewing equation M310 as a partial genome for a 1-MCPc, solving the problem, and converting the solution equation M311 of 1-MCPc into a solution equation M312 for 1-MCP[ell]. Since all of these steps are done in linear time (in number of genes), the proof is complete. □
(3 ≤ k)-MCP
In the unrestricted case of the k-MCP, the completion of a partial genome is always possible as we can add adjacencies and telomeres arbitrarily to the partial genome, since there is no restriction on the number and type of chromosomes in the resulting genome. The hardness of showing the existence of a k-completion derives from the fact that finding a k-completion for the partial multi-genome results in a proper edge coloring for the genome graph of the partial multi-genome.
Let G = (V, E) be a graph. We define the edge-chromatic number of G, denoted χ'(G), to be the minimum number of colors required to obtain an edge-coloring of G. For each edge-coloring of G a color class is a set of all edges with a specific color. A color class defines a matching in the graph since no two edges of the same color share a vertex.
The following proposition shows the relation between the edge-coloring of a genome graph and the edge color classes of the corresponding breakpoint graph.
Proposition 1. If equation M313 is a multi-genome of k genomes then equation M314.
Proof. Suppose equation M315 is a mixture of k genomes equation M316. Then the breakpoint graph equation M317 can be partitioned into the sets equation M318 of adjacencies, and each equation M319 can be considered as color class. So the edges of B can be colored with k colors. Since B and equation M320 are isomorphic, we have equation M321. □
Using the same argument as in Proposition 1 we have:
Lemma 1. If equation M322 is a partial multi-genome of k partial genomes then equation M323.
Now, in the following theorem we show a relation between the edge-coloring of a genome graph and the k-completion of the corresponding partial multi-genomes.
Theorem 4. Let equation M324 be a partial multi-genome. Then equation M325 has an unrestricted k-completion if and only if equation M326, for any positive integer k.
Proof. ([implies]) If equation M327 has a k completion, then it can be considered as a partial multi-genome of k genomes. Then by Lemma 1 we have equation M328.
([is implied by]) Now assume that equation M329. Hence, we can color the edges of equation M330 with k colors. If C1, . . ., Ck are the color classes of G, we have equation M331. Each Ci is a matching in the graph equation M332, and is a set of adjacencies among the gene extremities. So we can define a partial genome equation M333 by adjacencies equation M334. The color classes partition the edges of equation M335 into k matchings, and we have equation M336. Since there is no restriction on the completions, taking any completion equation M337 for each equation M338 results in a a k-completion equation M339 for equation M340; because equation M341. □
Now, by Theorem 4 and using the following two classic theorems, we show that deciding whether there exists a valid solution to a (k ≥ 3)-MCP is NP-complete. For a graph G let Δ(G) be the maximum degree of G.
Theorem 5 (Vizing [14]). If G is a simple graph, χ'(G) = Δ(G) or Δ(G) + 1.
Theorem 6 (Holyler [15]). For a graph G, deciding whether χ'(G) = Δ(G) or Δ(G) + 1 is NP-complete, if Δ(G) 3.
Corollary 4. If k ≥ 3, deciding whether there exists a valid solution to the unrestricted k-MCP is NP-complete.
Proof. In order to prove this corollary we reduce the edge-coloring problem to k-MCP. Suppose G = (V, E) is a simple graph and k = Δ(G) 3. If |V | is not even, add an isolated vertex so that the number of vertices in G is 2n for some positive integer n. Consider these 2n vertices as gene extremities of a set of n genes. Now, G defines a partial multi-genome equation M342 on these n genes, since the k-MCP is unrestricted and any graph can be considered as a partial multi-genome with no restriction on the chromosomal structure of its partial genomes. If there is a polynomial algorithm for k-MCP, we can input to this algorithm equation M343 as the partial multi-genome, along with an arbitrary distance function ϕ and a healthy reference equation M344. First, suppose the algorithm gives a valid output. Since the algorithm is polynomial, we can find a k-completion for equation M345 in polynomial time, and by Theorem 4, we can find an edge coloring of G with k colors in polynomial time.This implies that the χ'(G) ≤ k. Now if the algorithm does not give a valid output, by Theorem 4 we have χ'(G) > k. This implies that the k-MCP is NP-complete, since the genome graph of a partial multi-genome is always a multigraph and the class of simple graphs is a subset of the class of multigraphs. □
Note that in Corollary 4 we only considered the unrestricted version of k-MCP. This allows us to assume that for each (multi-) graph G there exists a partial multi-genome equation M346 such that G and equation M347 are isomorphic.Thus, if equation M348 | for all partial multi-genomes equation M349} and if equation M350 is the set of all multi-graphs, then equation M351. However, one can restrict the k-MCP by taking a set of chromosomal conditions. Consequently we may have equation M352 such that the new restricted k-MCP is polynomially tractable for all partial multi-genomes (whose genome graph is in equation M353).
Corollary 5. If k ≥ 3, then the unrestricted k-MCP is NP-complete.
Proof. Since in solving a k-MCP we need to find a k-completion for its partial multi-genome, by Corollary 4 the proof is complete. □
In this section, we prove that the unrestricted 2-MCP, and the restricted 2-MCP where all chromosomes are circular (i.e., equation M354), are NP-complete for DCJ distance. The NP-completeness of the unrestricted 2-MCP is done by a reduction from MAX 3-AND problem. The MAX 3-AND is a satisfiability problem, where given a set of conjunctions, each with 3 literals, the goal is to determine an assignment of Boolean value to each variable that maximizes the number of satisfied conjunctions. Note that in 2-MCP there are only two possible topologies for the mixture tree: the branch-tree and path-tree (Figure 2-b, c).
Theorem 7. The unrestricted 2-MCP with DCJ distance is NP-complete.
In order to provide the proof of this theorem, we need the following lemmas.
Lemma 2. Suppose equation M355 is a partial multi-genome whose genome graph, equation M356, consists of m cycles C1, . . ., Cm with even lengths, and equation M357 is a reference genome which consists of [ell] edges (i.e., it has [ell] adjacencies). Assume that there are [ell]' cycles among the cycles in equation M358 such that no edge in A is connected to any of their vertices. If [ell]' >2[ell] then in every solution to the 2-MCP, the optimal mixture tree is a path-tree.
Proof. Note that in 2-MCP there are only two possible topologies for the mixture tree: the branch-tree and path-tree (Figure 2-b, c). Since the degree of each vertex in equation M359 is two, if we partition the edges of equation M360 into two perfect matchings equation M361 and equation M362. Therefore, for any 2-completion equation M363 we have equation M364 and equation M365, since G1 and G2 are maximal (and circular) and we cannot add any edge to them. Also, for each equation M366 we have equation M367, where Mij is a perfect matching on vertices of Cj. Obviously, equation M368. We have equation M369 for i = 1, 2, since equation M370 has [ell] edges and each of them can be in at most one cycle in equation M371. Therefore,
equation M372
which shows that the dDCJ-value of a path tree is smaller than the dDCJ -value of a branch tree, and completes the proof. □
Lemma 3. Any MAX 3-SAT instance is reducible to a MAX 3-AND instance. Moreover, MAX 3-AND is NP-complete.
Proof. Let Δ = [ell]1 V [ell]2 V [ell]3 be a clause (disjunction) of three literals. Define
equation M373
By using basic Boolean rules we have Δ [left and right double arrow ] VS[set membership][ell](Δ) S.
Now, suppose equation M374 is a MAX 3-SAT instance which has m clauses Δ1, . . ., Δm. Let equation M375 be an instance of MAX 3-AND which consists of all the conjunctions in equation M376 Since for every assignment to the variables at most one conjunction in Lj), 1 ≤ j ≤ m, is satisfied and this happens if and only if Δj is satisfied, then every optimal assignment to the variables in equation M377 will be also an optimal assignment to the variables in equation M378. Therefore, MAX 3-SAT is reducible to MAX 3-AND, which implies that MAX 3-AND is NP-complete, as MAX 3-SAT is NP-complete [16]. □
Now, consider an instance equation M379 of the MAX 3-AND problem. We show how to represent equation M380 by a genome graph and a reference genome, to make a reduction from MAX 3-AND to 2-MCP. Suppose we represent a variable x with a cycle C of even length, which we will call a variable-cycle (see Figure 6-a). This cycle has exactly two perfect matchings. We label one of these the true matching, T(x), and the other one the false matching, F(x) (see Figure 6-b, c). We represent an assignment to a variable by choosing one of the matchings T(x) and F(x) and remove the edges in the other matching (see Figure Figure77).
Figure 6
Figure 6
Representing variables with cycles. (a) A variable represented by a cycle, (b) a true matching, and (b) a false matching.
Figure 7
Figure 7
Representing conjunctions with cycles. (a) Three cycles representing the literals equation M381, equation M382, and z, and the conjunction edges (bold) for a conjunction equation M383. (b) For x = y = false and z = true we obtain the conjunction-cycle Δ of length 6. (c) Any other assignment (more ...)
Let [ell](x1), [ell](x2), [ell](x3) be three literals of variables x1, x2, x3, and Δ = ([ell](x1) Λ [ell](x2) Λ [ell](x3)) be a conjunction in equation M384. A conjunction-cycle of Δ is a cycle which is obtained as follows:
1. For each i [set membership] {1, 2, 3} consider an edge in T(xi) if [ell](xi) = xi. If equation M385 take an edge in F(xi).
2. Add three new edges, called conjunction-edges, to the three edges we chose in the previous step, and build a cycle of length 6. This cycle is a conjunction-cycle of Δ.
It is easy to see that an assignment α to xi's satisfy the conjunction Δ if and only if the corresponding matching assignment to α keeps all the edges in the conjunction-cycle of Δ. We call a representation of a MAX 3-AND instance equation M386 with cycles and conjunction-cycles a graphical representation of equation M387.
If the literals of a variable appear in at most t conjunctions, and the variable-cycles have length at least 4t, then by choosing the edges of conjunction-cycles properly, we have a graphical representation of a MAX 3-AND instance, where no edge in a variable-cycle is incident with two conjunction edges from different conjunction-cycles. This implies the following lemma:
Lemma 4. For each MAX 3-AND instance equation M388 there exists a graphical representation equation M389 such that any as-signments to the variables in equation M390 which maximizes the number of satisfied conjunctions, induces a matching assignment that maximizes the number of conjunction-cycles, and vice versa.
Combining Lemmas 2-4 gives the proof of Theorem 7.
Proof of Theorem 7. Since the MAX 3-AND is NP-complete by Lemma 3, it suffices to reduce the MAX 3-AND problem to the 2-MCP. Suppose equation M391 is a MAX 3-AND instance. Assume equation M392 has m conjunctions. We can add 3m + 1 new conjunctions δ1, . . ., δ3m+1 where each δi consists of a new single variable xδi; obviously in any optimal assignment the value of all the xδi's should be true. Now by Lemma 4, there is a graphical representation equation M393 such that finding an optimal assignment in equation M394 is equivalent to finding a matching for each variable-cycle such that the number of preserved conjunction-cycles are maximized. Note that there are 3m conjunction-edges and 3m + 1 variable-cycles which are not connected to any conjunction-edge. Now, consider all the vertices in equation M395 as gene extremities, and all the edges in the variable-cycles as the adjacencies of a partial multi-genome G. Also, consider all the conjunction-edges as the adjacencies of a reference healthy genome equation M396. In the 2-MCP problem with partial multi-genome G and reference healthy genome equation M397, the optimal tree is forced to be a path-tree by Lemma 2 (Figure (Figure2).2). Therefore, in the optimal solution of this 2-MCP, equation M398 should be a genome such that the number of cycles in the breakpoint graph equation M399 is maximized, i.e., the number of conjunction-cycles are maximized. Since equation M400 is a union of perfect matchings of the variable-cycles (see the proof of Lemma 2) it induces an assignment for the variables which maximizes the number satisfied conjunctions, and this completes the proof. □
We end this section by considering the restricted version of k-MCP, where the chromosomal condition set is {circular}, i.e. all genomes have all circular chromosomes. We denote this restricted version by k-MCPc, and the unrestricted version of k-MCP by k-MCP[empty]. If opt(k-MCPc) and opt(k-MCP[empty]) are the dDCJ-value of a solution to k-MCPc and k-MCP[empty], respectively, then:
Theorem 8. For the k-MCPc and k-MCP[empty] versions of k-MCP with DCJ distance we have
equation M401
Proof. First note that each solution to k-MCPc is also a solution of k-MCP[empty], since there is no restriction in k-MCP. Hence, opt(k-MCPc) opt(k-MCP[empty]). Second, for each solution to k-MCP[empty] if the resulting genomes are not circular we can add new edges to the genomes and make them circular. By adding the new edges the number of cycles in the breakpoint graph does not decrease which implies that the dDCJ-value of the newly obtained genomes is not larger than opt(k-MCP[empty]). Therefore, these circular genomes form a solution of k-MCP[empty]. So opt(k-MCPc) opt(k-MCP[empty]) completing the proof. □
Combining this theorem and Theorem 7 we have
Corollary 6. If k ≥ 2, then k-MCPc with DCJ distance is NP-complete.
In this paper we introduced the k-Minimum Completion Problem (k-MCP) motivated by the type of data produced in current cancer genome sequencing studies. We showed the following results. (1) A linear time algorithm for the unrestricted 1-MCP; (2) a linear time algorithm for the restricted versions 1-MCP where the genomes are circular or linear; i.e. the chromosomal condition set equation M402 is {circular, uni-chromosomal} or equation M403 is {linear, uni-chromosomal}; (3) the unrestricted k-MCP is NP-complete for any distance when k ≥ 3; and (4) the 2-MCP with DCJ distance is NP-complete in the unrestricted version and with the condition that all chromosomes are circular, i.e. equation M404. These results lay the foundation for future algorithmic studies of the k-MCP and the application of these algorithms to real cancer sequencing data.
There are numerous further directions to pursue. As noted in the introduction, the model described in this paper does not consider all the complexities of cancer genome sequencing: most importantly copy number aberrations (duplications and deletions) and errors in the measured adjacencies are important features of cancer genome sequencing and should be addressed.
To handle errors, one might consider weighted versions of the k-MCP where adjacencies have a weight corresponding to the confidence in the measurement. Regarding the current model, further work is needed on different chromosomal conditions, genomic distances, or other constraints on the relationships between the genomes in the mixture. For example, the case of linear chromosomes demands further attention, as human chromosomes are linear, although circular chromosomes have been observed in cancer [17]. Similarly, one may impose an upper bound on the number of chromosomes. One may also place restrictions on the structure of the mixture tree.
Another direction is to derive approximation algorithms. In the k-MCP we aim to minimize distance over all possible k-completion and mixture trees simultaneously. However, by separating the completion and distance optimization steps, one may employ techniques that have developed for other problems. For example, one may try to first complete the partial multi-genomes using some clustering techniques, as have been employed in metagenomic studies [18]. With complete genomes, one could then try to find optimal mixture trees rooted at the reference genome. Depending on the allowed structure of the mixture tree, techniques from genome rearrangement phylogeny problems may be employed. For example, in the case of 2-MCP, if the complete genomes are the leaves of the mixture tree, then the problem becomes the median problem (with the reference genome genome as the third genome) [5,13]. Alternatively, if the genomes are the vertices of the mixture tree, then the tree construction problem becomes the problem of finding a minimum spanning tree, which is in generally easier. In between these extremes, where some of the genomes in the mixture are the leaves and some are intermediate nodes (ancestors), the problem becomes a Steiner tree problem. In the cancer application, any of these cases might provide useful approximations, as the process of clonal evolution of cancer [1] might mean that cells at intermediate stages of cancer progression remain present in the tumor.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally to this work.
We thank the anonymous referees for helpful comments on an earlier version of this manuscript. This work was supported by a CAREER Award from the National Science Foundation (#1053753). In addition, BJR is supported by a Career Award from the Scientific Interface from the Burroughs Wellcome Fund and an Alfred P. Sloan Research Fellowship.
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 19, 2012: Proceedings of the Tenth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics. The full contents of the supplement are available online at
  • Nowell PC. The clonal evolution of tumor cell populations. Science. 1976;194(4260):23–28. doi: 10.1126/science.959840. [PubMed] [Cross Ref]
  • Raphael BJ, Volik S, Collins C, Pevzner PA. Reconstructing tumor genome architectures. Bioinformatics. 2003;19(Suppl 2):i162–171. [PubMed]
  • Meyerson M, Gabriel S, Getz G. Advances in understanding cancer genomes through second-generation sequencing. Nat Rev Genet. 2010;11(10):685–696. doi: 10.1038/nrg2841. [PubMed] [Cross Ref]
  • Yancopoulos S, Attie O, Friedberg R. Efficient sorting of genomic permutations by translocation, in-version and block interchange. Bioinformatics. 2005;21(16):3340–3346. doi: 10.1093/bioinformatics/bti535. [PubMed] [Cross Ref]
  • Tannier E, Zheng C, Sankoff D. Multichromosomal median and halving problems under different genomic distances. BMC Bioinformatics. 2009;10 [PMC free article] [PubMed]
  • Hannenhalli S, Pevzner PA. FOCS. IEEE Computer Society; 1995. Transforming Men into Mice (Polynomial Algorithm for Genomic Distance Problem) pp. 581–592.
  • Hannenhalli S, Pevzner PA. Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals. J ACM. 1999;46:1–27. doi: 10.1145/300515.300516. [Cross Ref]
  • Bergeron A, Mixtacki J, Stoye J. A new linear time algorithm to compute the genomic distance via the double cut and join distance. Theor Comput Sci. 2009;410(51):5300–5316. doi: 10.1016/j.tcs.2009.09.008. [Cross Ref]
  • El-Mabrouk N, Sankoff D. The Reconstruction of Doubled Genomes. SIAM J Comput. 2003;32(3):754–792. doi: 10.1137/S0097539700377177. [Cross Ref]
  • Warren R, Sankoff D. In: RECOMB-CG, Volume 6398 of Lecture Notes in Computer Science. Tannier E, editor. Springer; 2010. Genome Aliquoting Revisited; pp. 1–12.
  • Zheng C, Lenert A, Sankoff D. Reversal distance for partially ordered genomes. ISMB (Supplement of Bioinformatics) 2005. pp. 502–508. [PubMed]
  • Gaul É, Blanchette M. In: Comparative Genomics, Volume 4205 of Lecture Notes in Computer Science. Bourque G, El-Mabrouk N, editor. Springer; 2006. Ordering Partially Assembled Genomes Using Gene Arrangements; pp. 113–128.
  • Xu AW. In: RECOMB-CG, Volume 5267 of Lecture Notes in Computer Science. Nelson CE, Vialette S, editor. Springer; 2008. A Fast and Exact Algorithm for the Median of Three Problem-A Graph Decomposition Approach; pp. 184–197.
  • Vizing VG. On an estimate of the chromatic class of a p-graph. (Russian) Diskret Analiz. 1964;3:25–30.
  • Holyer I. The NP-Completeness of Edge-Coloring. SIAM J Comput. 1981;10(4):718–720. doi: 10.1137/0210055. [Cross Ref]
  • Cook SA. In: STOC. Harrison MA, Banerji RB, Ullman JD, editor. ACM; 1971. The Complexity of Theorem-Proving Procedures; pp. 151–158.
  • Raphael BJ, Pevzner PA. Reconstructing tumor amplisomes. ISMB/ECCB (Supplement of Bioinformat-ics) 2004. pp. 265–273. [PubMed]
  • Wu YW, Ye Y. In: RECOMB, Volume 6044 of Lecture Notes in Computer Science. Berger B, editor. Springer; 2010. A Novel Abundance-Based Algorithm for Binning Metagenomic Sequences Using [ell]-Tuples; pp. 535–549.
Articles from BMC Bioinformatics are provided here courtesy of
BioMed Central