Level- k phylogenetic networks: uniqueness and complexity

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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 trees**Time**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**1. A level-k network (Nk) that is uniquely defined by its**triplets**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.