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.
Maurizio Patrignani & Maurizio Pizzonia
Third University of Rome
A drawing of a K4 produced with the
Interactive algorithm (Papakostas and Tollis 1997)
dummy node representing a bend
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
Instance: A graph
Question: Does a set of edges exist, such that it is a cut and a matching?
x2 x3 x4
x2 x3 x4The 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?
Observation: nodes joined by multiple edges can not be separated by a matching-cut
Observation: each node of the construction has even degree
replace each star with a “wheel”
Observation: multiple edges occur only in pairs
replace each pair of edges with a triangle
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
Basic step: a single edge between s and t is a series-parallel graph G(s,t)
s1 = s2
t1 = s2
starting from G1(s1,t1) and G2(s2,t2), obtain G(s1,t1) by identifying sources and sinks
t1 = t2
A parse tree can be constructed in linear-time describing a sequence of operations producing the series-parallel graph.
tNon 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