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##### 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**Three-dimensional orthogonal grid drawings of graphs**A drawing of a K4 produced with the Interactive algorithm (Papakostas and Tollis 1997)**“Fork”: two adjacent edges cut by the split**A bad choice of the cuts**A result that is not so nice**dummy node representing a bend final 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 Matching-Cut Problem Instance: A graph Question: Does a set of edges exist, such that it is a cut and a matching?**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)**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**Observation: nodes joined by multiple edges can not be separated by a matching-cut false chain true chain**xi**xi Variable gadget false chain true chain**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!**l m n**Clause gadget For each clause false chain m n l true chain**m**n l m n l m n l Clause gadget matching-cuts (1) l m n falsefalsetrue falsetruefalse falsetruetrue**m**n l m n l m n l Clause gadget matching-cuts (2) l m n truefalsefalse truefalsetrue truetruefalse**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**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:**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**Observation: each node of the construction has even degree replace each star with a “wheel”**Simple graphs**Observation: multiple edges occur only in pairs replace each pair of edges with a triangle**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**A parse tree can be constructed in linear-time describing a sequence of operations producing the series-parallel graph. series edge parallel edge edge**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**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**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**• 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