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 K 4 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 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 A cut A matching A matching-cut Instance: A graph Question: 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 x1x3x4x1x3x4 x2x3x4x2x3x4 x2x3x4x2x3x4 x 1 = false x 2 = true x 3 = true x 4 = true

Construction false chain true chain Observation: nodes joined by multiple edges can not be separated by a matching-cut

Variable gadget xixi xixi false chain true chain

Variable gadget matching-cuts Not allowed! x i is true (x i is false ) x i is false (x i is true ) xixi xixi xixi xixi xixi xixi false chain true chain

Clause gadget l m n true chain false chain lmnlmn For each clause

Clause gadget matching-cuts (1) lmn falsefalsetrue falsetruefalse falsetruetrue l m nl m n l m n

Clause gadget matching-cuts (2) lmn truefalsefalse truefalse true true true false l m n l m n l m n

Connecting to variable gadgets x4x4 Each node of the clause gadget that represents a literal is connected with two edges to the corresponding literal of the variable gadget x3x3 x1x3x4x1x3x4 Example: x1x1 x3x3 x3x3 to x 4 to x 1

An example of instance x1x2x3x1x2x3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3 x1x1 x2x2 x3x3 A NAE3SAT instance may be: The corresponding matching-cut instance is:

A solution x1x2x3x1x2x3 x1x1 x2x2 x3x3 x2x2 x1x1 x2x2 x3x3 x1x1 x3x3 A NAE3SAT solution to is: The corresponding matching-cut solution is: x 1 = true x 2 = true x 3 = 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 G 1 (s 1,t 1 ) and G 2 (s 2,t 2 ), obtain G(s 1,t 2 ) by identifying t 1 and s 2 Parallel composition: starting from G 1 (s 1,t 1 ) and G 2 (s 2,t 2 ), obtain G(s 1,t 1 ) by identifying sources and sinks Basic step: a single edge between s and t is a series-parallel graph G(s,t) s t s 1 = s 2 t 1 = t 2 s1s1 t 1 = s 2 t2t2

Parse tree construction A parse tree can be constructed in linear-time describing a sequence of operations producing the series-parallel graph. edge 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 false label 1 true label 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 0 label 2 s

Polynomial-time algorithm Traverse the parse tree top-down and update the labels. edge parallel series edge s AND t label 2 s AND t label 2 s AND t label 2 false label 1 false label 1 false label 1 false label 1 false label 1 0 label 2 s

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