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Level- k phylogenetic networks: uniqueness and complexity Matthias Mnich, Steven Kelk, Leo van Iersel Phylogenetic trees Time Phylogenetic networks root

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level k phylogenetic networks uniqueness and complexity

Level-k phylogenetic networks: uniqueness and complexity

Matthias Mnich, Steven Kelk, Leo van Iersel

phylogenetic networks
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
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
Triplets

ab|c

N

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

triplets6
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
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
slide8
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
Previous work

Remember: k is the level of the networks.

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

new results
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
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?
slide14
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

slide15
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
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

example17
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
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.
slide21
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”
slide22
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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