Reconnect 04 ProblemSpecific Preprocessing and a hint at FPT

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Reconnect 04Problem-Specific Preprocessing (and
a hint at FPT)

• Cynthia Phillips
• Sandia National Laboratories

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Preprocessing

• Sets of little tricks that make the problem
  smaller while still allowing you to extract the
  optimal solution
• e.g. There exists an optimal solution for which
  this choice holds, or this choice cannot hold.
• If you know problem structure this can be quite
  powerful

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A Formalization

• Suppose your problem input size is naturally
  described by two values
• n describes most of the problem
• k a single parameter that can be big, but is
  usually small for practical instances
• Examples
• Graphs with bounded degree or bounded diameter
• Nesting level in databases
• Strings with small alphabet size
• Cases where the value of the solution is small
  (cleaning up data)

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Fixed-Parameter Tractability

• A problem with size (n,k) is fixed-parameter
  tractable (FPT) if it can be solved in time
• Where f(k) is an arbitrary function of the
  parameter k
• A problem is FPT if and only if it is
  kernelizable there is a polynomial-time
  transformation from the problem of size (n,k) to
  another instance of the problem (ns,ks) such that
• From here an enumerative method has a better
  chance

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Example Vertex Cover is FPT
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• Find a minimum-size set of vertices such that
  each edge has at least one endpoint in the set.
• k size of the vertex cover

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Kernelizing

• Kernelization usually is a set of (local)
  independent transformation rules
• Rules cascade
• Apply till the problem is irreducible

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Kernelizing Vertex Cover

• Well see 4 rules (not counting trivial degree-0
  one). There are more (even LP-based), but these
  are really simple!
• Make this a decision problem Does graph G(V,E)
  have a vertex cover of size k?
• Each rule transforms to a new question Does
  graph Gs(Vs,Es) have a vertex cover of size ks?
  where

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Rule 1

• If vertex v has degree 1, then put its neighbor u
  in the optimal VC and delete all edges (u,w)

y
x
u
v
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Rule 2

• If vertex v has degree 2, and its two neighbors u
  and w are connected, then put its neighbors u and
  w in the optimal VC

w
u
v
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Rule 3

• If vertex v has degree 2, and its two neighbors u
  and w are not connected, then shrink the (u,v,w)
  triangle to a single node v1.
• Use ks k - 1

v1
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Rule 3 Justification

• If vertex v has degree 2, and its two neighbors u
  and w are not connected, then shrink the (u,v,w)
  triangle to a single node
• If either u or w is necessary for the VC, then
  effectively v becomes degree one and the other is
  selected by rule 1

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Rule 3 Justification

• If vertex v has degree 2, and its two neighbors u
  and w are not connected, then shrink the (u,v,w)
  triangle to a single node v1
• Use ks k - 1
• Find the optimal VC of the graph with v1
• If v1 is not selected, just cover the triple
  independently. Choose v. The total VC has size
  at most ks 1 k

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Rule 3 Justification

• If vertex v has degree 2, and its two neighbors u
  and w are not connected, then shrink the (u,v,w)
  triangle to a single node v1
• Use ks k - 1
• Find the optimal VC of the graph with v1
• If v1 is selected, must take either u or w, so
  have to take both. Size of the VC is at most ks
  - 1 2 k

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Rule 4

• Take all the nodes of degree gt k
• If we didnt take them, wed have to take all (gt
  k) neighbors (too many for a VC of size k)

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Kernelization Results

• If no more rules apply, the graph has

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Proof

• Graph with ns nodes and a VC of size ks
• Satisfies all rules so degree of v (number of
  neighbors) for all v satisfies
• The nodes not in the VC are an independent set of
  size ns-ks

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Proof

• F is the set of edges adjacent to nodes in the VC
• The n-k nodes in the independent set have degree
  at least 3
• The k nodes in the VC have degree at most k

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Proof
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Comments

• These rules are local and independent (any order)
• Could parallelize this