1. Tatsuie Tsukiji (speaker) Tokyo Denki University
Computing Phylogenetic Roots with Bounded Degrees and Errors is NP-complete
Tatsuie Tsukiji (speaker)
Zhi-Zhong Chen
Tokyo Denki University

2. The phylogenetic kth root problem (PRk)
(k ≧ 2 is a fixed constant.)
PR4
Called “phylogeny” k
Output: a tree T such that b
Given: a graph G. j i a d c h f g e f j c b d a e g h i
Leaves of T = vertices of G.
Each vertex represents
an extant specie.
Two vertices are adjacent
in G iff their distance in T
is at most 4.
Each edge corresponds to
similarity in
evolutionary characteristics. k
Degree of each internal
node of T is at least 3.

3. What are known about PRk ?
PRk is solvable in polynomial time for k = 2, 3, 4 .
The complexity of PRk for k > 4 is still known.
ΔPRk
: a natural special case of PRk where the output
phylogeny has maximum degree Δ.
ΔPRk can be solved in linear time

4. An optimization problem
The closest phylogeneitic kth root problem (CPRk)
Given: a graph G = (V, E ).
Output: a phylogeny T that minimizes the number of errors
|T k E |
|T E | =4 T T3
T3－E
G = (V, E )
E－T3

5. An optimization problem
The closest phylogeneitic kth root problem (CPRk)
Given: a graph G = (V, E ).
Output: a phylogeny T that minimizes the number of errors , where
|T k E |
Motivation: G is derived from some similarity data,
which are usually inexact in practice.
CPR2 has been studied extensively.
(See correlation clustering papers in FOCS and STOC.)

6. Results
Known results
PRk is Solvable in polynomial time for k = 2, 3, 4 .
ΔPRk can be solved in linear time
CPRk is NP-hard for any fixed k ≧ 2.
New Result
ΔCPRk is NP-complete, for any fixed k ≧ 3 and Δ≧ 3

7. NP-completeness: CPRk
1. CPR2　＝ Correlation Clustering
Correlation Clustering: Minimize
#(inner nonedges) + #(outer edges) of G
clique of T2 a b c d e f g h
unbounded degree
2. CPR2 ≦CPRk
If ＜the clique size then
|T E(gaget )|
dT(a,b) = 3 G a b
gaget
clique T

8. of graphs with maximum degree 3
NP-completeness: 3CPR3
from Hamiltonian Path
of graphs with maximum degree 3
∃T |T E(G’) | ≦
#(degree-3 vertices)/2
G has HP
T, 3 × G
error = ½ at degree-3 vertices of G
error = 0 at degree-2 vertices of G G’
1/2 1
error ≧ ½　at degree-3 vertices of G

9. NP-completeness: 3CPR3 ≦ 3CPR5
Pad distance 1 at every vertex of G
If |T5 E(7-clique)| ≦　2 then T is
7-clique or
7-clique
∃1 degree-2 internal node
port
Distance( , ) = 1
Distance( , ) ≧ 2
7-clique = (5,1,2)-core graph

10. NP-completeness: 3CPR3 ≦ 3CPR5
: i-port
7-clique
,G’
∃T |T E(G’) |
≦ #(degree-3 vertices)/2　 T
lifted G G
∃T | T 3 at lifted G E(lifted G) |
≦ #(degree-3 vertices)/2 　

11. Core graph: 3CPR3 ≦ 3CPR7 Pad distance 1 ∃1 port Distance( , ) = 1
If |T7 E(11-clique)| ≦　2 then T is
∃1 port
Distance( , ) = 1
Distance( , ) ≧ 2
(7,1,2)-core graph

12. Core graph: 3CPR3 ≦ 3CPR7 ∃1 port Distance( , ) = 2 Distance( , ) ≧ 3
Pad distance 2
Phylogeny of 5-clique
Phylogeny of 11-clique
copies
If |T7 E((the obtained tee)7)| ≦　2 then
∃1 port
Distance( , ) = 2
Distance( , ) ≧ 3
(7,2,2)-core graph

13. Summary and Open Problems
The complexity of PRk for k > 4 ?
ΔPRk ∈P
new
CPRk is NP-hard.
ΔCPRk is NP-hard
TRk,ΔTRk ∈P
CTRk is NP-hard
open
Is ΔCTRk NP-hard ?
Tree 3rd power
Phylogenetic 3rd power