NPhardness and Phylogeny Reconstruction

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NP-hardness and Phylogeny Reconstruction

• Tandy Warnow
• Department of Computer Sciences
• University of Texas at Austin

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Phylogeny
From the Tree of the Life Website,University of
Arizona
Orangutan
Human
Gorilla
Chimpanzee
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Phylogeny
From the Tree of the Life Website,University of
Arizona
Orangutan
Human
Gorilla
Chimpanzee
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DNA Sequence Evolution
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Evolution informs about everything in biology

• Big genome sequencing projects just produce data
  -- so what?
• Evolutionary history relates all organisms and
  genes, and helps us understand and predict
• interactions between genes (genetic networks)
• drug design
• predicting functions of genes
• influenza vaccine development
• origins and spread of disease
• origins and migrations of humans

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Molecular Systematics
U
V
W
X
Y
TAGCCCA
TAGACTT
TGCACAA
TGCGCTT
AGGGCAT
X
U
Y
V
W
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Major methods for phylogeny reconstruction

• Biology Polynomial time methods (good enough for
  small datasets), and local search heuristics for
  NP-hard optimization problems
• Linguistics an exact algorithm for an
  NP-hard optimization problem

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Outline for the rest of the talk

• NP-hard and polynomial time problems
• Phylogeny reconstruction in biology the NP-hard
  maximum parsimony problem, and how we can solve
  it better
• Phylogeny reconstruction in linguistics the
  NP-hard perfect phylogeny problem, and how we
  solve it exactly
• An open problem from whole genome phylogeny
• Thoughts about computational biology, and the
  role of mathematics in this field

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Polynomial-time problems

• Shortest path Given edge-weighted graph
  G (V,E) and two vertices, v and w, find
  shortest path from v to w (O(n2) time)
• 2-colorability Given graph G (V,E), determine
  if we can assign two colors to the vertices of G
  so that no edge connects vertices of the same
  color (O(nm) time)
• 3-clique Given graph G (V,E), determine if G
  contains a 3-clique (O(n3) time)
• For all these, nV and mE.

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NP-hard problems

• Some problems seem hard to solve
• Hamilton path Given graph G , determine if G has
  a simple path going through every vertex
• 3-colorability Given graph G, determine if G can
  be properly 3-colored
• Max-clique Given graph G, find a largest clique
  in the graph

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Technical definition of NP-hard

• NP is the class of decision problems for which
  yes instances can be proven in polynomial
  time. (Example I can prove to you that a graph
  has a 3-coloring by presenting that 3-coloring to
  you. So 3-coloring is in NP.)
• Definition A problem X is NP-hard if every
  problem in NP can be reduced to X in polynomial
  time (yes-instances mapped to yes-instances, and
  no-instances mapped to no-instances). So
  2-coloring can be reduced to 3-coloring
• Definition A problem X is in P if it is in NP
  and can be solved in polynomial time.

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NP-hard optimization problems

• Graph-theoretic examples
• Travelling Salesperson (TSP) find minimum cost
  tour visiting every vertex
• Maximum Clique find maximum sized subset of
  vertices which are all pairwise adjacent
• Minimum Vertex Coloring find minimum number of
  colors so that every vertex can be assigned a
  color, and no edge connects vertices of the same
  color.

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NP-hard decision problems

• Each optimization problem has corresponding
  decision problem. For example, the max clique
  optimization problem corresponds to the decision
  problem
• Input Graph G(V,E), positive integer B
• Question Does there exist a subset V of V such
  that VB and V is a clique?

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NP-hard problems and polynomial time problems (P
vs. NP)

• Some decision problems can be solved in
  polynomial time
• Can graph G be 2-colored?
• Does graph G have a 5-clique?
• Some decision problems seem to not be solvable in
  polynomial time
• Can graph G be 3-colored?
• Does graph G have a k-clique?

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P vs. NP, continued

• The big question in theoretical computer
  science is
• Is it possible to solve an NP-hard problem in
  polynomial time?
• If the answer is yes, then all NP-hard problems
  can be solved in polynomial time, so PNP. This
  is generally not believed.

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Coping with NP-hard problems

• Since NP-hard problems may not be solvable in
  polynomial time, the options are
• Solve the problem exactly (but use lots of time
  on some inputs)
• Use heuristics which may not solve the problem
  exactly (and which might be computationally
  expensive, anyway)

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Example Maximum Clique

• Exact solution find largest k so that some
  subset of size k is a clique. Runs in O(nk)
  time.
• Heuristic Pick a vertex at random, and greedily
  assemble a set which is a clique, and stop when
  you cant add any more vertices. Repeat until
  tired (or bored, or running out of time, or ).
  How do we evaluate the running time, or accuracy?

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General comments for NP-hard optimization problems

• Getting exact solutions may not be possible for
  some problems on some inputs, without spending a
  great deal of time.
• You may not know when you have an optimal
  solution, if you use a heuristic.
• Sometimes exact solutions may not be necessary,
  and approximate solutions may suffice. (But this
  may not be true for biology.)

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Major methods for phylogeny reconstruction

• Biology Polynomial time methods (good enough for
  small datasets), and local search heuristics for
  NP-hard optimization problems
• Linguistics an exact algorithm for an
  NP-hard optimization problem

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Polynomial time methods

• Quartet-based methods
• Construct trees on all 4-leaf subsets
• Combine quartet trees into tree on full dataset
• Distance-based methods
• Estimate pairwise distance matrix dij
• Find tree T and edge-weights w(e) so that dTij
  approximates dij
• For both methods, if there are no errors (in
  quartet trees or pairwise distances) then the
  correct tree can be obtained in polynomial time.
  Otherwise, optimization problems are NP-hard.
• Polytime heuristics along these lines are popular.

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Phylogeny reconstruction

• In biology, the most popular approaches for
  reconstructing phylogenetic trees are heuristics
  for Maximum Parsimony (NP-hard) or Maximum
  Likelihood (conjectured to be NP-hard)
• In historical linguistics, a new approach based
  upon exactly solving the NP-hard Perfect
  Phylogeny problem has been useful.

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DNA Sequence Evolution
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Maximum Parsimony

• Given a set S of strings of the same length over
  a fixed alphabet, find a tree T leaf-labelled by
  S and with all internal nodes labelled by strings
  of the same length over the same alphabet which
  minimizes the sum of the edge lengths.
• Motivation seeks to minimize the total number of
  point mutations needed to explain the data
• NP-hard

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Major phylogeny reconstruction methods

• In biology mostly hill-climbing heuristics that
  attempt to solve NP-hard optimization problems
  (maximum parsimony or maximum likelihood)
• In historical linguistics much less is
  established, but an exact solution to an
  NP-hard problem looks very promising.

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Maximum Parsimony
ACT
ACT
ACA
GTA
GTT
GTT
ACA
GTA
GTA
ACA
ACT
GTT
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Maximum Parsimony
ACT
ACT
ACA
GTA
GTT
GTA
ACA
ACT
2
1
1
3
3
2
GTT
GTT
ACA
GTA
MP score 7
MP score 5
GTA
ACA
ACA
GTA
2
1
1
ACT
GTT
MP score 4
Optimal MP tree
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Maximum Parsimony computational complexity
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Exact solutions fixed-parameter approaches

• Fixed-parameter approaches restrict some
  parameter and solve the problem exactly for those
  cases. Examples
• Does graph G(V,E) have a k-clique? Solvable
  in O(nk) time (nV).
• Does graph G(V,E) have a k-coloring? Solvable in
  O(kn) time for general k, and in O(nm) time for
  k2 (nV, and mE).

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Solving MP (maximum parsimony) and ML (maximum
likelihood)

• Why are MP and ML hard? The search space is huge
  -- there are (2n-5)!! trees, it is easy to
  get stuck in local optima, and there can be many
  optimal trees.
• Why try to solve MP or ML? Our experimental
  studies show that polynomial time algorithms
  dont do as well as MP or ML when trees are big
  and have high rates of evolution.
• Why solve MP and ML well? Because trees can
  change in biologically significant ways with
  small changes in objective criterion.

Local optimum
MP score
Global optimum
Phylogenetic trees
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MP/ML heuristics
Fake study
Performance of hill-climbing heuristic
MP score of best trees
Time
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Speeding up MP/ML heuristics
Fake study
Performance of hill-climbing heuristic
MP score of best trees
Desired Performance
Time
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Divide-and-Conquer Approach

• Step 1 Get good starting tree
• 1. Decompose the dataset into smaller,
  overlapping subsets.
• Construct phylogenetic trees on the subsets using
  a base method.
• Merge the subtrees into a single tree on the
  entire dataset.
• Refine the resultant tree to produce a binary
  tree.
• Follow with usual heuristic (hill-climbing or
  other such strategy) to improve tree.

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Divide-and-conquer approaches

• Step 1 Get good starting tree
• Divide dataset into overlapping subsets
• Construct trees on each subset
• Combine subtrees into tree on full dataset
• Refine into binary tree if needed
• Step 2 Apply favored heuristic to improve tree.

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Using divide-and-conquer for MP and ML

• Conjecture better (more accurate) solutions will
  be found in less time, if we analyze a small
  number of smaller subsets and then combine
  solutions
• Need
• 1. techniques for decomposing datasets,
• 2. base methods for subproblems, and
• 3. techniques for combining subtrees

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The DCM3 technique for speeding up MP/ML searches
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DCM Decompositions
Input Set S of sequences, distance matrix d,
threshold value
1. Compute threshold graph
2. Perform minimum weight triangulation
DCM1 decomposition
DCM2 decomposition
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DCM3 Decompositions
Input Set S of sequences, and estimate T of the
true tree
1. Compute short subtree graph G(S,T), based
upon T
2. Find clique separator in the graph G(S,T), and
form subproblems
The graph G(S,T)
DCM3 decomposition
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Strict Consensus Merger (SCM)
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DCM3-boosting a base method

• Decompose the dataset into smaller, overlapping
  subsets, using DCM3
• Construct phylogenetic trees on the subsets using
  a base method
• Merge the subtrees into a single tree using the
  Strict Consensus Merger
• Use PAUP constrained search to refine the
  resultant tree

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Iterative-DCM3 vs Ratchet
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Iterative-DCM3 vs Ratchet
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Comments

• Developing heuristics with good performance takes
  mathematical insights, but may not involve
  proofs. Even so, its really important.
• Extracting information from the set of optimal
  (and near-optimal) solutions is a major open
  problem.
• Other types of data (gene orders, morphology)
  present novel challenges.
• Reticulate evolution detection and reconstruction
  is a major open problem.

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Ringe-Warnow Phylogenetic Tree of Indo-European
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Cognate Classes

• Two words w1 and w2 are in the same cognate
  class, if they evolved from the same word through
  sound changes.
• French champ and Italian champo are both
  descendants of Latin campus thus the two words
  belong to the same cognate class.
• Spanish mucho and English much are not in the
  same cognate class.

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Phylogenies of Languages

• Languages evolve over time, just as biological
  species do (geographic and other separations
  induce changes that over time make different
  dialects incomprehensible -- and new languages
  appear)
• The result can be modelled as a rooted tree
• The interesting thing is that many
  characteristics of languages evolve without back
  mutation or parallel evolution -- so a perfect
  phylogeny is possible!

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Historical Linguistic Data

• A character is a function that maps a set of
  languages, L, to a set of states.
• Three kinds of characters
• Phonological (sound changes)
• Lexical (meanings based on a wordlist)
• Morphological (grammatical features)

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Perfect Phylogeny

• A phylogeny T for a set S of taxa is a perfect
  phylogeny if each state of each character
  occupies a subtree (no character has
  back-mutations or parallel evolution)

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Homoplasy-Free Evolution (perfect phylogenies)

• YES NO

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The Comparative Method(Hoenigswald 1960)

• Used to verify relatedness between languages and
  to infer features of the ancestral languages of a
  group of related languages
• Step 1 establish sound correspondence in a set
  of related languages
• Step 2 establish cognate classes

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The Ringe-Warnow Model of Language Evolution

• The nodes of the tree which contain elements of
  the same cognate class should form a rooted
  connected subgraph of the true tree
• The model is known as the Character Compatibility
  or Perfect Phylogeny.

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Character Compatibility and Perfect Phylogeny

• Ringe and Warnow postulated that all properly
  encoded characters for the Indo-European
  languages should be compatible on the true tree,
  if such a tree existed
• A tree T on which all characters are compatible
  is called a perfect phylogeny

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The Perfect Phylogeny Problem

• Given a set S of taxa (species, languages, etc.)
  determine if a perfect phylogeny T exists for S.
• The problem of determining whether a perfect
  phylogeny exists is NP-hard (McMorris et al.
  1994, Steel 1991).

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Triangulated Graphs

• A graph is triangulated if it has no simple
  cycles of size four or more.

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Triangulating Colored GraphsAn Example

• A graph that can be c-triangulated

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Triangulating Colored GraphsAn Example

• A graph that can be c-triangulated

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Triangulating Colored GraphsAn Example

• A graph that cannot be c-triangulated

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Triangulating Colored Graphs (TCG)

• Triangulating Colored Graphs given a
  vertex-colored graph G, determine if G can be
  c-triangulated.

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The PP and TCG Problems

• Bunemans Theorem
  A perfect phylogeny exists for a set S
  if and only if the associated character state
  intersection graph can be c-triangulated.
• The PP and TCG problems are polynomially
  equivalent and NP-hard.

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Solving the PP Problem Using Bunemans Theorem

• Yes Instance of PP
• c1 c2 c3
• s1 3 2 1
• s2 1 2 2
• s3 1 1 3
• s4 2 1 1

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Solving the PP Problem Using Bunemans Theorem

• Yes Instance of PP
• c1 c2 c3
• s1 3 2 1
• s2 1 2 2
• s3 1 1 3
• s4 2 1 1

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Some special cases are easy

• Binary character perfect phylogeny solvable in
  linear time
• r-state characters solvable in polynomial time
  for each r (combinatorial algorithm)
• Two character perfect phylogeny solvable in
  polynomial time (produces 2-colored graph)
• k-character perfect phylogeny solvable in
  polynomial time for each k (produces k-colored
  graphs -- connections to Robertson-Seymour graph
  minor theory)

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The Indo-European (IE) Dataset

• 24 languages
• 22 phonological characters, 15 morphological
  characters, and 333 lexical characters
• Total number of working characters is 390
  (multiple character coding, and parallel
  development)
• A phylogenetic tree T on the IE dataset (Ringe,
  Taylor and Warnow)
• T is compatible with all but 22 characters 16
  (18) monomorphic and 6 polymorphic
• Resolves most of the significant controversies in
  Indo-European evolution shows however that
  Germanic is a problem (not treelike)

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Phylogenetic Tree of the IE Dataset
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An open problem to take home

• computing the transposition distance
  between two genomes
• (important in whole genome phylogeny
  reconstruction)

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Genomes As Signed Permutations
1 5 3 4 -2 -6or6 2 -4 3 5 1 etc.
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Genomes Evolve by Rearrangements
1 2 3 4 5 6 7 8 9 10

• Inversion (Reversal)

• Transposition

• Inverted Transposition

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An open problem to play with

• Given two permutations on 1,2,n, compute the
  minimum transposition distance (unknown
  computational complexity)
• (The corresponding problem for inversion
  distances involves very beautiful graph theory
  and algorithms.)

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Summary

• NP-hard optimization problems abound in phylogeny
  reconstruction, and in computational biology in
  general, and need very accurate solutions
• Many real problems have beautiful and natural
  combinatorial and graph-theoretic formulations

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Acknowledgements

• NSF and the David and Lucile Packard Foundation
  (funding)
• Collaborators Bernard Moret (UNM CS), Donald
  Ringe (Penn Linguistics)
• Students Usman Roshan and Luay Nakhleh

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Phylolab, U. Texas
Please visit us at http//www.cs.utexas.edu/users/
phylo/