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

Three-dimensional orthogonal grid drawings of graphs

A drawing of a K4 produced with the Interactive algorithm (Papakostas and Tollis 1997)

The “split & push” approach

End of the drawing process

A simpler example

A bad choice of the cuts

“Fork”: two adjacent edges cut by the split

A result that is not so nice

final bend

dummy node representing a bend

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

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

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)

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

Construction

false chain

true chain

Observation: nodes joined by multiple edges can not be separated by a matching-cut

Variable gadget

xi xi

false chain

true chain

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

Clause gadget

l

mn

true chain

false chain

l m nFor each clause

Clause gadget matching-cuts (1) l m n

false false true

false true false

false true true

lm

n

lm

n

l

mn

Clause gadget matching-cuts (2) l m n

true false false

true false true

true true false

lm

n

l

m

n

lm

n

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

An example of instancex1 x2 x3

x1 x1x2 x2

x3 x3x1

x2

x3

A NAE3SAT instance may be:

The corresponding matching-cut instance is:

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

Graphs of maximum degree four

replace each star

with a “wheel”

Observation: each node of the construction has even degree

Simple graphs

replace each pair of edges

with a triangle

Observation: multiple edges occur only in pairs

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

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

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

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

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

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