Thomas C' van Dijk

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Kernelization for Loop Cutset

• Thomas C. van Dijk

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Overview

• Feedback set problems
• the Loop Cutset problem
• A kernelization for Loop Cutset
• Preprocessing with a quality guarantee
• Experimental evaluation

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Feedback Vertex Set

• Instance
• An undirected graph G (V,E)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that G V-S is acyclic?

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Feedback Vertex Set

• Instance
• An undirected graph G (V,E)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all cycles, S contains a vertex on
  that cycle

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Feedback Vertex Set

• Instance
• An undirected graph G (V,E)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all cycles, S contains a vertex on
  that cycle

NP-COMPLETE
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Example
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Example an FVS
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Example an FVS
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Example not an FVS
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Example not an FVS
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Example a lowerbound
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Probabilistic / Bayesian networks

• Probability distribution encode as directed
  acyclic graph with conditional probabilities at
  the arcs
• Recover specific probabilities from this
  representation
• One of the standard algorithms (Pearls) involves
  a loop cutset
• Runtime exponential in size of the cutset

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Feedback Vertex Set

• Instance
• An undirected graph G (V,E)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all cycles, S contains a vertex on
  that cycle

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Loop Cutset

• Instance
• A directed acyclic graph G (V,A)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all simple loops, S contains a vertex
  - on that loop - not a
  sink

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Loop Cutset

• Instance
• A directed acyclic graph G (V,A)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all simple loops, S contains a vertex
  - on that loop - not a
  sink

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Loop Cutset

• Instance
• A directed acyclic graph G (V,A)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all simple loops, S contains a vertex
  - on that loop - not a
  sink

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Loop Cutset

• Instance
• A directed acyclic graph G (V,A)
• An integer k
• Question
• Does there exist an S Vof size at most ksuch
  that for all simple loops, S contains a vertex
  - on that loop - not a
  sink

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Kernelization for Loop Cutset
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Kernelization

• Preprocessing with quality guarantee
• Cf approximation algorithm heuristic with
  quality guarantee.
• Pick a value k. Preprocessing for
• Does the network have a loop cutset of size at
  most k?

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Kernelization for Loop Cutset

• Transform to Blackout FVS
• Kernelize Blackout FVS
• Based on Bodlaenders FVS kernelization
• Transform back

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Blackout Feedback Vertex Set

• Instance
• An undirected graph G(V,E)
• A set X V of blacked out vertices
• An integer k
• Question
• Does there exist an S V-Xof size at most
  ksuch that G V-S is acyclic

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Transform to Blackout FVS

• Split all vertices v into vin and vout
• Blackout all in vertices

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Transform to Blackout FVS

• Split all vertices v into vin and vout
• Blackout all in vertices

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Transform to Blackout FVS

• Split all vertices v into vin and vout
• Blackout all in vertices

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Kernelization overview

• A collection of rules
• A proof if none of the rules apply, thenthe
  graph can have only O( k3 ) vertices
• The algorithmjust keep trying the rules until
  none apply

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Rules simple

• Islet
• Remove degree 0 vertices.
• Twig
• Remove degree 1 vertices.
• Triple edge
• Remove more-than-double edges.

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Degree two

• In regular Feedback Vertex Set
• For each degree-two vertex,exists optimal FVS
  that doesnt use it
• More complicated with blacked out vertices

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Rule degree two

• Suppose v is a vertex with exactly two neighbors
  x and y.
• If v is blacked out, bypass it.

v
y
x
y
x
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Rule degree two

• Suppose v is a vertex with exactly two neighbors
  x and y.
• If v is allowed, and it has an allowed neighbor,
  bypass v.

v
y
x
x
y
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Rule degree two

• Suppose v is a vertex with exactly two neighbors
  x and y.
• If v is allowed, x and y are both blacked out
• Leave it in there.

v
y
x
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Rule self-loop

• Suppose v has a self-loop.
• If v is allowed, select it in the FVS.
• If v is blacked out, conclude No.

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Flower rule example, k 2
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Flower rule example, k 2
v
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Flower rule example, k 2
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Rule flower

• Suppose there are k1 cycles that are
  vertex-disjoint except all include v.
• If v is allowed, select it in the FVS.
• If v is blacked out, conclude No.

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Flower rule example, k 2
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Cascade of degree-2 rule
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Cascade of degree-2 rule
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Cascade of degree-2 rule
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Cascade of degree-2 rule
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Cascade of degree-2 rule
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Flower rule example, before
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Rule 2-approximation

• Suppose a 2-approximate FVS is larger than 2k
  conclude No.

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Rule overview

• Islet, Twig, Triple edge
• Degree two
• Self-loop, Black double edge, Flower
• Blacked out component
• Improvement
• 2-Approximation
• Abdication

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Experimental evaluation
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Implementation

• C
• Library ofEfficient Data structures and
  Algorithms
• By Algorithmic Solutions
• Quite nice

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Runtime on random graphs, k20
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Runtime on random graphs, k20
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Runtime on random graphs
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UAI06 Networks
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UAI06 Networks runtime
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Effectiveness of the kernelization

• After kernelization, one still needs to solve the
  problem on the kernel
• So compare runtime of
• solve, versus
• kernelize and solve kernel
• Following data solve using a simple
  branch-and-bound

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Effectiveness of the kernelization
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Effectiveness on UAI06 Networks

• Unfortunately, those are way too big to find an
  optimal loop cutset for
• Thousands of nodes
• Kernelization improves quality of other heuristics

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Summary

• Loop Cutset also has a kernelwith O( k3 )
  vertices
• Experimental evaluation of the kernelizations
  they are practical
• Questions?