The complexity of the matchingcut problem

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The complexity of the matching-cut problem

• Maurizio Patrignani Maurizio Pizzonia

Third University of Rome
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Overview

• Application domain
• Matching-cut problem
• NAE3SAT reduction
• Polynomial-time algorithm for series-parallel
  graphs
• Conclusions

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Three-dimensional orthogonal grid drawings of
graphs
A drawing of a K4 produced with the Interactive
algorithm (Papakostas and Tollis 1997)
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The split push approach
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End of the drawing process
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A simpler example
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A bad choice of the cuts
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A result that is not so nice
dummy node representing a bend
final bend
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Bad VS good cuts
Reducing the number of edges cut by each split
Reducing the forks produced by the cuts
Details in Di Battista, Patrignani, and Vargiu,
"A SplitPush Approach to 3D Orthogonal Drawing",
Journal of Graph Algorithms and Applications, 2000
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The matching-cut problem
A cut
A matching
A matching-cut
Matching-Cut Problem
Instance A graph Question Does a set of edges
exist, such that it is a cut and a matching?
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Previous work

• Recognizing decomposable graphs is NP-complete
  even with graph of maximum degree 4, but it is
  polynomial for graphs of maximum degree 3 (V.
  Chvátal, 1984)
• The problem remains NP-complete even restricting
  to bipartite graphs of minimum degree two (A.M.
  Moshi, 1989)
• The problem remains NP-complete even restricting
  to bipartite graphs with one color class of nodes
  of degree 4 and the other color class of nodes of
  degree 3 (V.B. Le and B. Randerath, 2001)

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The NAE3SAT reduction
Not-All-Equal-3-SAT Problem
Instance A set of clauses, each containing 3
literals from a set of boolean variables Question
Can truth values be assigned to the variables so
that each caluse contains at least one true
literal and at least one false literal?
x1false x2true x3true x4true
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Construction
Observation nodes joined by multiple edges can
not be separated by a matching-cut
false chain
true chain
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Variable gadget
false chain
true chain
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Variable gadget matching-cuts
false chain
false chain
false chain
xi
xi
true chain
true chain
true chain
xi is false (xi is true)
xi is true (xi is false)
Not allowed!
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Clause gadget
For each clause
false chain
m
n
l
true chain
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Clause gadget matching-cuts (1)
l m n
false false true
false true false
false true true
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Clause gadget matching-cuts (2)
l m n
true false false
true false true
true true false
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Connecting to variable gadgets
Each node of the clause gadget that represents a
literal is connected with two edges to the
corresponding literal of the variable gadget
Example
to x1
to x4
x3
x4
x3
x3
x1
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An example of instance
A NAE3SAT instance may be
The corresponding matching-cut instance is
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A solution
x1true x2true x3true
A NAE3SAT solution to
is
The corresponding matching-cut solution is
x2
x3
x2
x1
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Graphs of maximum degree four
Observation each node of the construction has
even degree
replace each star with a wheel
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Simple graphs
Observation multiple edges occur only in pairs
replace each pair of edges with a triangle
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Series-parallel graphs
A series-parallel graph has a source s and a sink
t and can be constructed by recursively applying
the following rules
Serial composition starting from G1(s1,t1) and
G2(s2,t2), obtain G(s1,t2) by identifying t1 and
s2
s
Basic step a single edge between s and t is a
series-parallel graph G(s,t)
t
s1
s1 s2
t1 s2
Parallel composition starting from G1(s1,t1) and
G2(s2,t2), obtain G(s1,t1) by identifying sources
and sinks
t2
t1 t2
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Parse tree construction
A parse tree can be constructed in linear-time
describing a sequence of operations producing the
series-parallel graph.
series
edge
parallel
edge
edge
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Non st-separating matching-cuts
We associate with each node of the parse tree two
labels describing the properties of the
intermediate series-parallel graph with respect
to the existence of a matching-cut Label 1
signals if a non st-separating matching cut
exists in the series-parallel graph
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St-separating matching-cuts
Label 2 signals under which conditions the
series-parallel graph admits an st-separating
matching-cut
s
s
s
label 2
s AND t
t
t
t
s
s
s
label 2
label 2
label 2
t
s OR t
1
t
t
t
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Polynomial-time algorithm
Traverse the parse tree top-down and update the
labels.
series
edge
parallel
edge
edge
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Conclusions and open problems

• We showed an interesting application domain for
  the matching-cut problem in the graph drawing
  field
• We proved that the matching-cut problem is
  NP-complete by using a reduction of the NAE3SAT
  problem
• The result can be extended to graphs of maximum
  degree four and to simple graphs
• We produced a polynomial-time algorithm for
  series-parallel graphs
• It is open whether the problem retains its
  complexity for planar graphs