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Level- k phylogenetic networks: uniqueness and complexity

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### Level-k phylogenetic networks: uniqueness and complexity

Matthias Mnich, Steven Kelk, Leo van Iersel

Phylogenetic networks

root

A level-k phylogenetic network is a directed acyclic graph where every biconnected component contains at most k recombination vertices.

split vertex

- Split vertex:indegree 1, outdegree 2
- Recombination vertex: indegree 2, outdegree 1

recombination vertex

leaf

Example: a level-2 network

A level-k phylogenetic network is a directed acyclic graph where every biconnected component contains at most k recombination vertices.

- blue = biconnected component
- red = recombination vertex

Triplets

ab|c

N

A triplet ab|c is consistent with a phylogenetic network Nif N contains a subdivision of ab|c.

Triplets

ab|c

N

A triplet ab|c is consistent with a phylogenetic network Nif N contains a subdivision of ab|c.

Constructing networks from triplets

- ab|c, ac|b, ad|b, bd|a, bd|c, ad|c and ac|d are consistent with this network

- bc|a, ab|d, cd|b, bc|d and cd|a are not

Consistent level-k network (CL-k)

Input: set of triplets T

Output: level-k network consistent with T, if one exists.

Maximum consistent level-k network (MCL-k) on dense triplet sets

Input:dense set of triplets T

Output: level-k network N that maximises the number of triplets in T consistent with N.

A triplet set is dense if it contains at least one triplet for each combination of three leaves

Previous work

Remember: k is the level of the networks.

Note: a level-0 network is a phylogenetic tree.

New results

- A level-k network (Nk) that is uniquely defined by its triplets
- CL-k is NP-hard for all k
- MCL-k is NP-hard for all k, even for dense triplet sets

2. CL-k is NP-hard for all k

- Problem:Set Splitting
- Input: set S = {s1, …, sn} and collection C = {C1, …, Cm} of cardinality-3 subsets of S.
- Question: can S be partitioned into S1 and S2 (a set splitting) such that Cj is not a subset of S1 and not of S2 for all j?

REDUCTION FROM:

TO:

- Problem:Consistent level-k network (CL-k)
- Input: set of triplets T.
- Question: does there exist a level-k network consistent with T?

Start with all triplets consistent with Nk.

- For each set Cj={sa,sb,sc} add leavessaj, sbj and scj.
- Add triplets that force these leaves togo to one of the red sides.
- Add triplets saj r1|sbj, sbj r1|scj and scj r1|saj,which make sure that each set is split.

Nk

Suppose that set Cj={sa,sb,sc} is not split,

i.e. saj, sbj and scj are on the same side.

- Then the triplet saj r1|sbj means that saj is below sbj
- And the triplet sbj r1|scj means that sbj is below scj
- And the triplet scj r1|saj means that scj is below saj

So saj r1|sbj, sbj r1|scj and scj r1|saj make it

impossible that saj, sbj and scj are all on the same side.

contradiction

Example

- Instance of Set Splitting:C1 = {s1,s3,s4},C2 = {s2,s3,s4},C3 = {s1,s2,s4}.
- For C1 we add leaves For C2 we add leavesFor C3 we add leaves
- The tripletsenforce that goes to one of the red sides.

Nk

Example

- The tripletsmake sure that C1 is split overthe two red sides.
- The tripletsenforce that areon the same side.
- In this example there exists aset splitting S1={s1,s3}, S2={s2,s4}.

Nk

3. MCL-k is NP-hard for all k, even for dense triplet sets

REDUCTION FROM:

- Problem:Feedback Arc Set
- Input: directed graph G=(V,A)
- Output: smallest set A’ of arcs such that G’=(V, A \ A’) is acyclic.

TO:

- Problem: Maximum consistent level-k network (MCL-k) on dense triplet sets
- Input: dense set of triplets T
- Output: level-k network N that maximises the number of triplets in T consistent with N.

Example for level-2

BigN2

Example for level-2

BigN2

Start with all triplets consistent with BigN2.

- Add triplets that enforce the encoding leaves (u,v,w and q) to go to the red side.
- For each arc (v,u) of G add a triplet xu|v, requesting that“v is above u”

The arcs corresponding to triplets that are not satisfied form a feedback arc set of G.

- Triplet corresponding to arc (w,v) is not satisfied sincew is not “above” v.
- Removing (w,v) makes G acyclic because all arcs go to a vertex that is lower on the red path.

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