the complexity of the matching-cut problem
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The complexity of the matching-cut problem. Maurizio Patrignani & Maurizio Pizzonia. Third University of Rome. Overview. Application domain Matching-cut problem NAE3SAT reduction Polynomial-time algorithm for series-parallel graphs Conclusions. - PowerPoint PPT PresentationTRANSCRIPT
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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
“Fork”: two adjacent edges cut by the split
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A result that is not so nice
final bend
dummy node representing a bend
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Bad VS good cutsReducing the number of edges
cut by each splitReducing the forks produced by the cuts
Details in: Di Battista, Patrignani, and Vargiu, "A Split&Push 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
Instance: A graphQuestion: Does a set of edges exist, such that it is a cut
and a matching?
Matching-Cut Problem
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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
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?
Not-All-Equal-3-SAT Problem
x1 x3 x4
x2 x3 x4
x2 x3 x4
x1=false x2=truex3=true x4=true
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Construction
false chain
true chain
Observation: nodes joined by multiple edges can not be separated by a matching-cut
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Variable gadget
xi xi
false chain
true chain
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Variable gadget matching-cuts
Not allowed! xi is true(xi is false)
xi is false(xi is true)
xi xixi xi xi xi
false chainfalse chain false chain
true chain true chain true chain
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Clause gadget
l
mn
true chain
false chain
l m nFor each clause
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Clause gadget matching-cuts (1) l m n
false false true
false true false
false true true
lm
n
lm
n
l
mn
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Clause gadget matching-cuts (2) l m n
true false false
true false true
true true false
lm
n
l
m
n
lm
n
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Connecting to variable gadgets
x4
Each node of the clause gadget that represents a literal is connected with two edges to the corresponding literal of the variable gadget
x3
x1 x3 x4Example:
x1
x3
x3
to x4 to x1
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An example of instancex1 x2 x3
x1 x1x2 x2
x3 x3x1
x2
x3
A NAE3SAT instance may be:
The corresponding matching-cut instance is:
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A solutionx1 x2 x3
x1x2
x3
x2
x1 x2x3 x1 x3
A NAE3SAT solution to is:
The corresponding matching-cut solution is:
x1=true x2=truex3=true
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Graphs of maximum degree four
replace each star
with a “wheel”
Observation: each node of the construction has even degree
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Simple graphs
replace each pair of edges
with a triangle
Observation: multiple edges occur only in pairs
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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
Parallel composition:starting from G1(s1,t1) and G2(s2,t2), obtain G(s1,t1) by identifying sources and sinks
Basic step: a single edge between s and t is a series-parallel graph G(s,t)
s
t
s1 = s2
t1 = t2
s1
t1 = s2
t2
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Parse tree constructionA parse tree can be constructed in linear-time describing a sequence of operations producing the series-parallel graph.
edgeedge
parallel
series
edge
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Non st-separating matching-cuts
s
t
s
t
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
falselabel 1
truelabel 1
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St-separating matching-cuts
s
t
s
t
s AND t
s
t
s
t
t
s
t
s OR t
s
t
1
Label 2 signals under which conditions the series-parallel graph admits an st-separating matching-cut
0label 2 label 2
slabel 2
label 2 label 2 label 2
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Polynomial-time algorithmTraverse the parse tree top-down and update the labels.
edgeedge
parallel
series
edge
s AND tlabel 2s AND t
label 2
s AND tlabel 2
falselabel 1
falselabel 1
falselabel 1
falselabel 1
falselabel 1
0label 2
slabel 2
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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