From RS Posets to NPCompleteness: Adventures in Fixed Parameter Tractability

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From RS Posets to NP-Completeness Adventures in
Fixed Parameter Tractability
W. Henry Suters Department of Computer
Science University of Tennessee Research Lead by
Mike Langston Contributions by Faisal Abu-Khzam,
Pushkar Shanbhag and Chris Symons Advances in
Graph and Matroid Theory Ohio State University
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Complexity Theory

• The Classic View

easy
P


NP
?
P
PSPACE
2
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Complexity Theory

• The Classic View

easy
P


NP
?
P
PSPACE
2
hard
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Complexity Theory

• The Classic View

impossible
easy
P


NP
?
P
PSPACE
2
hard
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Complexity Theory

• But, consider time bounds such as
• O(2kn)
• versus
• O(2kn)
• when k is fixed.

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Complexity Theory

• Hence, the Parameterized View

solvable (even if NP-hard!)


W1
W2
XP
FPT
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Complexity Theory

• The Parameterized View

solvable (even if NP-hard!)


W1
W2
XP
FPT
heuristics only
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Complexity Theory

• The Parameterized View

Impossible
solvable (even if NP-hard!)


W1
W2
XP
FPT
heuristics only
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Graph Algorithms

• Clique is a good example. But it is not FPT
• (unless the fixed parameter hierarchy
  collapses).
• Fortunately, Vertex Cover is FPT.
• And Vertex Cover is a dual to Clique

_
G
G
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Vertex Cover
Given a graph G (V,E) a vertex cover is a
subset C of V so that for any edge (u,v) in E
either u or v is in C.
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Vertex Cover
Given a graph G (V,E) a vertex cover is a
subset C of V so that for any edge (u,v) in E
either u or v is in C.
Optimization problem what is the size of the
smallest vertex cover?
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Vertex Cover
Given a graph G (V,E) a vertex cover is a
subset C of V so that for any edge (u,v) in E
either u or v is in C.
Search Problem what is the smallest vertex
cover?
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Vertex Cover
Given a graph G (V,E) a vertex cover is a
subset C of V so that for any edge (u,v) in E
either u or v is in C.
Decision Problem Is there a vertex cover with
size k or smaller?
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Showing Vertex Cover is FPT

• Decision Problem Is there a vertex cover with
  size k or smaller?
• Algorithm
• Produce a binary search tree using the following
  rules.

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Showing Vertex Cover is FPT

• Decision Problem Is there a vertex cover with
  size k or smaller?
• Algorithm
• Produce a binary search tree using the following
  rules.
• Select an arbitrary uncovered edge (u,v) .

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Showing Vertex Cover is FPT

• Decision Problem Is there a vertex cover with
  size k or smaller?
• Algorithm
• Produce a binary search tree using the following
  rules.
• Select an arbitrary uncovered edge (u,v) .
• One of its endpoints (u or v) must be in any
  vertex cover.

or
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Showing Vertex Cover is FPT

• Decision Problem Is there a vertex cover with
  size k or smaller?
• Algorithm
• Produce a binary search tree using the following
  rules.
• Select an arbitrary uncovered edge (u,v) .
• One of its endpoints (u or v) must be in any
  vertex cover.
• Form a new branch of the binary search tree
  based on the vertex
• selected.

Sets containing u
Sets containing v
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Showing Vertex Cover is FPT

• Decision Problem Is there a vertex cover with
  size k or smaller?
• Algorithm
• Produce a binary search tree using the following
  rules.
• Select an arbitrary uncovered edge (u,v) .
• One of its endpoints (u or v) must be in any
  vertex cover.
• Form a new branch of the binary search tree
  based on the vertex
• selected.
• Repeatedly apply these rules, constructing the
  tree, until either
• a vertex cover is found (we are done and the
  answer is yes)

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Showing Vertex Cover is FPT

• Decision Problem Is there a vertex cover with
  size k or smaller?
• Algorithm
• Produce a binary search tree using the following
  rules.
• Select an arbitrary uncovered edge (u,v) .
• One of its endpoints (u or v) must be in any
  vertex cover.
• Form a new branch of the binary search tree
  based on the vertex
• selected.
• Repeatedly apply these rules, constructing the
  tree, until either
• a vertex cover is found (we are done and the
  answer is yes) or
• all the candidate sets with k or fewer vertices
  are eliminated
• (we are done and the answer is no).

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Showing Vertex Cover is FPT

• Observations
• The depth of the binary search tree is at most
  k.

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Showing Vertex Cover is FPT

• Observations
• The depth of the binary search tree is at most
  k.
• The binary search tree has at most 2k1-1 nodes.

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Showing Vertex Cover is FPT

• Observations
• The depth of the binary search tree is at most
  k.
• The binary search tree has at most 2k1-1 nodes.
• At each node we check to see if we have a vertex
  cover.
• Time complexity of O(n2).

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Showing Vertex Cover is FPT

• Observations
• The depth of the binary search tree is at most
  k.
• The binary search tree has at most 2k1-1 nodes.
• At each node we check to see if we have a vertex
  cover.
• Time complexity of O(n2).
• The time complexity of the algorithm is O(n22k).

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Showing Vertex Cover is FPT

• Observations
• The depth of the binary search tree is at most
  k.
• The binary search tree has at most 2k1-1 nodes.
• At each node we check to see if we have a vertex
  cover.
• Time complexity of O(n2).
• The time complexity of the algorithm is O(n22k).
• If k is small, the problem is easy, no matter
  the size of n.

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Showing Vertex Cover is FPT

• Observations
• The depth of the binary search tree is at most
  k.
• The binary search tree has at most 2k1-1 nodes.
• At each node we check to see if we have a vertex
  cover.
• Time complexity of O(n2).
• The time complexity of the algorithm is O(n22k).
• If k is small, the problem is easy, no matter
  the size of n.
• Vertex cover is FPT.

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Showing Vertex Cover is FPT

• More Observations
• Better branching choices can result in a time
  complexity
• of O(1.2852k kn).

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Showing Vertex Cover is FPT

• More Observations
• Better branching choices can result in a time
  complexity
• of O(1.2852k kn).
• Simplifying the graph (and reducing k) can
  reduce run times.

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The Vertex Cover Project

• Preprocessing via degree structure

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The Vertex Cover Project

• Preprocessing via degree structure
• Kernelize to computational core

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The Vertex Cover Project

• Preprocessing via degree structure
• Kernelize to computational core
• Branching explores core

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The Vertex Cover Project

• Preprocessing via degree structure
• Kernelize to computational core
• Branching explores core
• Interleave all three

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Preprocessing

• Low degree rules (e.g., degree one)
• Observation There is an optimal vertex cover
  that contains v and not u.

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Preprocessing
High degree rule Observation if
a vertex has degree greater than k, then it must
be in any vertex cover of size lt k.
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Preprocessing
Combining low and high degree rules
Resultant graph has size O(k2) at
most k(1k/3) vertices
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LP - Kernelization

• Based on linear programming (Bill
  Cooks code)
• minimize
• subject to
• where

• Resultant graph has size O(k)
• at most 2k vertices

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Crown Reduction

• An independent set of vertices C

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Crown Reduction

• An independent set of vertices C
• Whose neighborhood H is no larger than C

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Crown Reduction

• An independent set of vertices C
• Whose neighborhood H is no larger than C
• With a matching on the edges between C and H so
  that every vertex in H is matched.

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Crown Reduction
Observation There is an optimal vertex cover
containing no vertices from C and all vertices
from H. Resultant graph has size O(k) at most 3k
vertices.
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Representative Results on Large Non-Synthetic
Graphs
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An Example of Cliques Utility Microarray Data
Analysis
Complexity Analysis Algorithm Development Extensiv
e Parallelism Hardware Acceleration
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Variety of Applications

• Gene co-regulation

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Variety of Applications

• Phylogeny

A phylogenetic tree organizes life forms
according to their putative evolutionary
relationships.
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Variety of Applications

• SELDI

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Variety of Applications

• Gene co-regulation
• Phylogeny
• SELDI
• Upstream sequence analysis (motifs, modules,
  other cis-regulatory elements)

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Variety of Applications

• Gene co-regulation
• Phylogeny
• SELDI
• Upstream sequence analysis (motifs, modules,
  other cis-regulatory elements)
• Protein interaction networks

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Other Amenable Problems

• Hitting Set
• Dominating Set
• Cutwidth, others

Recent Codes Released

• Clustal XP (High Performance, Parallel
• Clustal W)
• CAMDA (Disease Screening)