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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.

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the complexity of the matching cut problem

The complexity of the matching-cut problem

Maurizio Patrignani & Maurizio Pizzonia

Third University of Rome

overview
Overview
  • Application domain
  • Matching-cut problem
  • NAE3SAT reduction
  • Polynomial-time algorithm for series-parallel graphs
  • Conclusions
three dimensional orthogonal grid drawings of graphs
Three-dimensional orthogonal grid drawings of graphs

A drawing of a K4 produced with the

Interactive algorithm (Papakostas and Tollis 1997)

a result that is not so nice
A result that is not so nice

dummy node representing a bend

final bend

bad vs good cuts
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 Split&Push Approach to 3D Orthogonal Drawing", Journal of Graph Algorithms and Applications, 2000

the matching cut problem
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?

previous work
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

x1 x3 x4

x2 x3 x4

x2 x3 x4

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?

x1=false x2=true

x3=true x4=true

construction
Construction

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

false chain

true chain

variable gadget

xi

xi

Variable gadget

false chain

true chain

variable gadget matching cuts

xi

xi

xi

xi

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!

clause gadget

l m n

Clause gadget

For each clause

false chain

m

n

l

true chain

clause gadget matching cuts 1

m

n

l

m

n

l

m

n

l

Clause gadget matching-cuts (1)

l m n

falsefalsetrue

falsetruefalse

falsetruetrue

clause gadget matching cuts 2

m

n

l

m

n

l

m

n

l

Clause gadget matching-cuts (2)

l m n

truefalsefalse

truefalsetrue

truetruefalse

connecting to variable gadgets

x1 x3 x4

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

an example of instance

x1 x2 x3

x2

x1

x3

x3

x3

x2

x2

x1

x1

An example of instance

A NAE3SAT instance may be:

The corresponding matching-cut instance is:

a solution

x1 x2 x3

x1

x3

x3

x2

x1

A solution

x1=true x2=true

x3=true

A NAE3SAT solution to is:

The corresponding matching-cut solution is:

x2

x3

x2

x1

graphs of maximum degree four
Graphs of maximum degree four

Observation: each node of the construction has even degree

replace each star with a “wheel”

simple graphs
Simple graphs

Observation: multiple edges occur only in pairs

replace each pair of edges with a triangle

series parallel graphs
Series-parallelgraphs

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

parse tree construction
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

non st separating matching cuts

s

s

label 1

label 1

false

true

t

t

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

st separating matching cuts

label 2

label 2

0

s

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

polynomial time algorithm

label 2

label 2

label 2

label 2

label 2

s AND t

s AND t

0

s AND t

s

label 1

label 1

label 1

label 1

label 1

false

false

false

false

false

Polynomial-time algorithm

Traverse the parse tree top-down and update the labels.

series

edge

parallel

edge

edge

conclusions and open problems
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