- Email: [email protected]
Maximum Cuts in Edge-colored Graphs
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 62 (2017) 87–92
www.elsevier.com/locate/endm
Maximum Cuts in Edge-colored Graphs 1 Rubens Sucupira a , Luerbio Faria a , Sulamita Klein b Ignasi Sau c,d , and U´everton S. Souza e,2 a b
IME, Universidade Estadual do Rio de Janeiro, Rio de Janeiro, Brasil
IM, COPPE, Universidade Federal do Rio de janeiro, Rio de Janeiro, Brasil c
d
CNRS, LIRMM, Universit´e de Montpellier, Montpellier, France
Departamento de Matem´ atica, Universidade Federal do Cear´ a, Fortaleza, Brasil e
IC, Universidade Federal Fluminense, Niter´ oi, Brasil
Abstract The input of the Maximum Colored Cut problem consists of a graph G = (V, E) with an edge-coloring c : E → {1, 2, 3, . . . , p} and a positive integer k > 0, and the question is whether G has a nontrivial edge cut using at least k colors. The Colorful Cut problem has the same input but asks for a nontrivial edge cut using all colors. Unlike what happens for the classical Maximum Cut problem, we prove that both problems are NP-complete even on complete, planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is NP-complete even when each color class induces a clique of size at most 3, but is trivially solvable when each color induces a K2 . On the positive side, we prove that Maximum Colored Cut is fixed-parameter tractable when parameterized by either k or p, and that it admits a cubic kernel in both cases. Keywords: colored cuts, edge cuts, max cut, planar graph, polynomial kernel.
1 2
Supported by CNPQ, CAPES, and FAPERJ. Email: [email protected]
https://doi.org/10.1016/j.endm.2017.10.016 1571-0653/© 2017 Elsevier B.V. All rights reserved.
88
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Introduction
Let G = (V, E) be a simple graph with an edge coloring c : E → {1, 2, . . . , p}, not necessarily proper. Given a proper subset S ⊂ V , we define the edge cut ∂S as the subset of E where the edges have one endpoint in S and the other in V \ S. We represent by c(∂S) the set of colors that are in ∂S, i.e., c(∂S) = {c(e) | e ∈ ∂S}. We are interested in the problem of finding a subset S ⊂ V such that |c(∂S)| ≥ |c(∂T )| for every T ⊂ V . This problem is called Maximum Colored Cut and its decision form is stated next. Maximum Colored Cut instance: A graph G = (V, E) with an edge coloring c : E → {1, 2, . . . , p} and an integer k > 0. question: Is there a proper subset S ⊂ V such that |c(∂S)| ≥ k? The classical (simple) Maximum Cut problem [5] is the particular case of Maximum Colored Cut when c : E → N is an injective function. Therefore, we are interested in analyzing the complexity of Maximum Colored Cut on graph classes C for which Maximum Cut is solvable in polynomial time. In addition, we are also interested in the complexity of determining if the input graph has a subset S ⊂ V such that |c(∂S)| = p, i.e., if there is an edge cut ∂S using all the colors; we call this problem Colorful Cut. Colorful Cut instance: A graph G = (V, E) with an edge coloring c : E → {1, 2, . . . , p}. question: Is there a proper subset S ⊂ V such that c(∂S) = p? Although the complexity of Minimum Colored Cut, which is defined analogously but with ‘|c(∂S)| ≤ k?’, has been explored in recent years (cf., for instance, [2]), to the best of our knowledge a study on the complexity of Maximum Colored Cut is novel. As Colorful Cut is a particular case of Maximum Colored Cut, our hardness results deal with Colorful Cut while the tractable cases will be presented for Maximum Colored Cut.
2
NP-completeness
Hadlock [6] proved that (simple) Maximum Cut is polynomial-time solvable on planar graphs. In this section we prove, among other results, the NPcompleteness of Colorful Cut on a particular subclass of planar graphs.
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Theorem 2.1 Colorful Cut is NP-complete on planar graphs. Proof. Let I = (U, C) be an instance of 3-sat. We construct in polynomial time a planar instance G = (V, E) with an edge coloring c such that I = (U, C) is satisfiable if and only if (G, c) has an edge cut with all colors of c. For each clause cj = (x ∨ y ∨ z) ∈ C we associate a K3 where each edge is labeled by a literal of cj together with its occurrence in C (see Fig. 1(a)), obtaining a graph G with m connected components, each of them associated with a clause of C. Starting with G we construct a multigraph Gc where each literal in a clause of C is associated with a colored edge as follows: •
•
For each pair {xji , xi k } where the integers j and k represent the occurrence of the literals xi and xi in the clauses of C, respectively, give the color Sij,k . The edge labeled with xji in G is replaced with parallel edges colored with Sij,k in Gc for all k. Analogously, the edge labeled by xi k in G is replaced with parallel edges in Gc colored with Sij,k for all j.
(a)
(b)
Fig. 1. Graphs G and Gc associated to (x1 ∨ x2 ∨ x3 ) ∧(x1 ∨ x2 ∨ x3 ) ∧ (x1 ∨ x2 ∨ x3 ).
Without loss of generality, we may assume that all variables have both positive and negative literals in I. From a truth assignment AI of I, we can construct a colorful cut of Gc as follows. For each clause cj of C, arbitrarily pick one edge {v, w} corresponding to a true literal of C. Then, put v and w in the same part of the partition, leaving the remaining vertex of the clause in the other part. This procedure gives a cut with all colors. Indeed, for each true literal xji , there is at least one false literal xi k that places the color Sij,k in the cut. Conversely, suppose that Gc has a colorful cut. Without loss of generality, we may assume that each K3 has a cut edge. Beginning with this cut, we construct a truth assignment AI that satisfies I, putting xi = T if at least one of the edges associated with some xji is inside a part of the partition, and xi = F otherwise. Note that this assignment is well-defined: there is no pair of literals {xji , xi k } such that the edges corresponding to both literals are inside a part of the partition, otherwise the color Sij,k is missing and the cut does not have all colors. Besides that, each K3 has the edges corresponding to some
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literal inside a part of the partition, which defines a truth assignment for I. Finally, we can transform Gc into a simple graph replacing each edge {v, w} colored with Sij,k by a path {v, x, y, w} such that c({x, y}) = Sij,k and the remaining edges of this path receive new different colors. Also, if c({v , w }) = c({v, w}) in Gc , we replace it by a path {v , x , y , w } such that c({x , y }) = c({x, y}) = Sij,k , c({v, x}) = c({v , x }) and c({y, w}) = c({y , w }). 2 Several NP-hard problems, such as (simple) Maximum Cut, are polynomialtime solvable on bounded treewidth graphs [1]. As each clause gadget has bounded size, using a similar argument we can modify the graph obtained in the previous construction in order to obtain the following corollary. Corollary 2.2 Colorful Cut remains NP-complete even when the input graph G satisfies simultaneously the following properties: (i) G is planar and connected.
(iii) G has maximum degree three.
(ii) G has bounded treewidth.
(iv) Each color class of G contains at most two edges.
Any bipartite graph has a colorful cut. Thus, it is natural to ask about the complexity of the problem on graphs with small odd cycle transversal. Corollary 2.3 Colorful Cut remains NP-complete even when the input graph G has odd cycle transversal number one. Proof. It is enough to pick a vertex of each gadget of Gc , constructed as described in the proof of Theorem 2.1, and identify them into a single vertex.2 Note that Maximum Cut is trivial on complete graphs, and that it is polynomial time solvable on cographs [1]. By adding a new vertex and edges colored with a new color, we can construct a hard instance in order to show the NP-completeness of Colorful Cut on complete graphs. Theorem 2.4 Colorful Cut is NP-complete on complete graphs. Note that if each color class of a graph G induces a K2 , then G has a colorful cut if and only if G is bipartite. The next result shows the NP-completeness of Colorful Cut when each color class induces either a K2 or a K3 . Theorem 2.5 Colorful Cut is NP-complete when each color class induces a clique of size at most three. Proof. For this proof we use a reduction from Not All Equal 3-sat (nae 3-sat), which is NP-complete [7]. Let I = (U, C) be an instance of nae 3-sat
R. Sucupira et al. / Electronic Notes in Discrete Mathematics 62 (2017) 87–92
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such that U = {u1 , u2 , . . . , un } and C = {c1 , c2 , . . . , cm }. The construction of a graph Gc = (V, E, f ) is given by the following procedure, illustrated in Fig. 2: •
For each clause cj = (x, y, z) ∈ C construct a clique {xj , yj , zj } with all the edges colored with the color j.
•
For each variable ui ∈ U create the vertices ai and bi in V such that ai is only adjacent to all positive occurrences of ui , and bi is only adjacent to all negative occurrences of the same variable. Add an edge joining the first positive occurrence and the first negative occurrence of ui .
•
Excluding the edges of the clause cliques, all other edges must be colored with different colors.
Fig. 2. Graph Gc associated to the formula (u1 ∨u2 ∨u3 )∧(u1 ∨u2 ∨u3 )∧(u1 ∨u2 ∨u3 ).
At this point, it is not difficult to see that I = (U, C) is a satisfiable instance of nae 3-sat if and only if Gc has a colorful cut. 2
3
Fixed-parameter tractability
From the previous results it follows that Maximum Colored Cut is paraNP-complete [4] parameterized by any of these parameters: treewidth, neighborhood diversity, genus, degeneracy, odd cycle transversal number, p − k, and several combinations of such parameters. The next result shows its fixedparameter tractability when parameterized by the number of colors. Theorem 3.1 Maximum Colored Cut admits a cubic kernel parameterized by the number of colors. Proof. First recall that a cut of a graph G is a bipartite subgraph of G. Claim 1. Let H = (V1 , V2 , E) be a bipartite graph having λ edges and no isolated vertices. The maximum number λ of edges having endpoints in the same part that can be added to H is 2 2 .
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Suppose that λ is the maximum number of colors in a cut of G and let S ⊂ V be a set such that |c(∂S)| = λ. Forming a bipartite graph H by selecting exactly one edge of each color class in [S, V \ S], we can observe that any color class that is not in H has at most 2 λ2 edges, otherwise λ would not be maximum. Let Ei ⊂ E be the set of edges colored with color class i. As λ ≤ p, if |Ei | > 2 p−1 , then the color class i is in any maximum colored cut. 2 Such a property us the following reduction rule: () If some color class gives i has |Ei | > 2 p−1 replace (G, p) by (G[E \ Ei ], p − 1). After the application 2 of this rule we obtain a kernel of size O(p3 ). 2 Lemma 3.2 Any simple graph G = (V, E) with an edge coloring c : E → {1, 2, . . . , p} has an edge cut ∂S such that |c(∂S)| ≥ p2 . Proof. Follows from the well-known property that any graph with at least m 2 edges contains a bipartite subgraph with at least m2 edges [3]. Corollary 3.3 Maximum Colored Cut admits a cubic kernel parameterized by the cost of the solution. Proof. If p ≥ 2k, by Lemma 3.2 we have an Yes-instance. Otherwise, applying a reduction similar to Rule () we obtain a kernel of size O(k 3 ). 2
References [1] H. L. Bodlaender and K. Jansen. On the complexity of the maximum cut problem. Nordic Journal on Computing, 7(1):14–31, 2000. [2] D. Coudert, P. Datta, S. Perennes, H. Rivano, and M. E. Voge. Shared risk resource group complexity and approximability issues. Parallel Processing Letters, 17(2):169–184, 2007. [3] P. Erd˝os. On some extremal problems in graph theory. Mathematics, 3:113–116, 1965.
Israel Journal of
[4] J. Flum and M. Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. Springer, 2006. [5] M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical computer science, 1(3):237–267, 1976. [6] F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing, 4(3):221–225, 1975. [7] T. J. Schaefer. The complexity of satisfiability problems. In the 10th Annual ACM Symposium on Theory of Computing (STOC), pages 216–226, 1978.
Electronic Notes in Discrete Mathematics 62 (2017) 87–92
www.elsevier.com/locate/endm
Maximum Cuts in Edge-colored Graphs 1 Rubens Sucupira a , Luerbio Faria a , Sulamita Klein b Ignasi Sau c,d , and U´everton S. Souza e,2 a b
IME, Universidade Estadual do Rio de Janeiro, Rio de Janeiro, Brasil
IM, COPPE, Universidade Federal do Rio de janeiro, Rio de Janeiro, Brasil c
d
CNRS, LIRMM, Universit´e de Montpellier, Montpellier, France
Departamento de Matem´ atica, Universidade Federal do Cear´ a, Fortaleza, Brasil e
IC, Universidade Federal Fluminense, Niter´ oi, Brasil
Abstract The input of the Maximum Colored Cut problem consists of a graph G = (V, E) with an edge-coloring c : E → {1, 2, 3, . . . , p} and a positive integer k > 0, and the question is whether G has a nontrivial edge cut using at least k colors. The Colorful Cut problem has the same input but asks for a nontrivial edge cut using all colors. Unlike what happens for the classical Maximum Cut problem, we prove that both problems are NP-complete even on complete, planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is NP-complete even when each color class induces a clique of size at most 3, but is trivially solvable when each color induces a K2 . On the positive side, we prove that Maximum Colored Cut is fixed-parameter tractable when parameterized by either k or p, and that it admits a cubic kernel in both cases. Keywords: colored cuts, edge cuts, max cut, planar graph, polynomial kernel.
1 2
Supported by CNPQ, CAPES, and FAPERJ. Email: [email protected]
https://doi.org/10.1016/j.endm.2017.10.016 1571-0653/© 2017 Elsevier B.V. All rights reserved.
88
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R. Sucupira et al. / Electronic Notes in Discrete Mathematics 62 (2017) 87–92
Introduction
Let G = (V, E) be a simple graph with an edge coloring c : E → {1, 2, . . . , p}, not necessarily proper. Given a proper subset S ⊂ V , we define the edge cut ∂S as the subset of E where the edges have one endpoint in S and the other in V \ S. We represent by c(∂S) the set of colors that are in ∂S, i.e., c(∂S) = {c(e) | e ∈ ∂S}. We are interested in the problem of finding a subset S ⊂ V such that |c(∂S)| ≥ |c(∂T )| for every T ⊂ V . This problem is called Maximum Colored Cut and its decision form is stated next. Maximum Colored Cut instance: A graph G = (V, E) with an edge coloring c : E → {1, 2, . . . , p} and an integer k > 0. question: Is there a proper subset S ⊂ V such that |c(∂S)| ≥ k? The classical (simple) Maximum Cut problem [5] is the particular case of Maximum Colored Cut when c : E → N is an injective function. Therefore, we are interested in analyzing the complexity of Maximum Colored Cut on graph classes C for which Maximum Cut is solvable in polynomial time. In addition, we are also interested in the complexity of determining if the input graph has a subset S ⊂ V such that |c(∂S)| = p, i.e., if there is an edge cut ∂S using all the colors; we call this problem Colorful Cut. Colorful Cut instance: A graph G = (V, E) with an edge coloring c : E → {1, 2, . . . , p}. question: Is there a proper subset S ⊂ V such that c(∂S) = p? Although the complexity of Minimum Colored Cut, which is defined analogously but with ‘|c(∂S)| ≤ k?’, has been explored in recent years (cf., for instance, [2]), to the best of our knowledge a study on the complexity of Maximum Colored Cut is novel. As Colorful Cut is a particular case of Maximum Colored Cut, our hardness results deal with Colorful Cut while the tractable cases will be presented for Maximum Colored Cut.
2
NP-completeness
Hadlock [6] proved that (simple) Maximum Cut is polynomial-time solvable on planar graphs. In this section we prove, among other results, the NPcompleteness of Colorful Cut on a particular subclass of planar graphs.
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Theorem 2.1 Colorful Cut is NP-complete on planar graphs. Proof. Let I = (U, C) be an instance of 3-sat. We construct in polynomial time a planar instance G = (V, E) with an edge coloring c such that I = (U, C) is satisfiable if and only if (G, c) has an edge cut with all colors of c. For each clause cj = (x ∨ y ∨ z) ∈ C we associate a K3 where each edge is labeled by a literal of cj together with its occurrence in C (see Fig. 1(a)), obtaining a graph G with m connected components, each of them associated with a clause of C. Starting with G we construct a multigraph Gc where each literal in a clause of C is associated with a colored edge as follows: •
•
For each pair {xji , xi k } where the integers j and k represent the occurrence of the literals xi and xi in the clauses of C, respectively, give the color Sij,k . The edge labeled with xji in G is replaced with parallel edges colored with Sij,k in Gc for all k. Analogously, the edge labeled by xi k in G is replaced with parallel edges in Gc colored with Sij,k for all j.
(a)
(b)
Fig. 1. Graphs G and Gc associated to (x1 ∨ x2 ∨ x3 ) ∧(x1 ∨ x2 ∨ x3 ) ∧ (x1 ∨ x2 ∨ x3 ).
Without loss of generality, we may assume that all variables have both positive and negative literals in I. From a truth assignment AI of I, we can construct a colorful cut of Gc as follows. For each clause cj of C, arbitrarily pick one edge {v, w} corresponding to a true literal of C. Then, put v and w in the same part of the partition, leaving the remaining vertex of the clause in the other part. This procedure gives a cut with all colors. Indeed, for each true literal xji , there is at least one false literal xi k that places the color Sij,k in the cut. Conversely, suppose that Gc has a colorful cut. Without loss of generality, we may assume that each K3 has a cut edge. Beginning with this cut, we construct a truth assignment AI that satisfies I, putting xi = T if at least one of the edges associated with some xji is inside a part of the partition, and xi = F otherwise. Note that this assignment is well-defined: there is no pair of literals {xji , xi k } such that the edges corresponding to both literals are inside a part of the partition, otherwise the color Sij,k is missing and the cut does not have all colors. Besides that, each K3 has the edges corresponding to some
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literal inside a part of the partition, which defines a truth assignment for I. Finally, we can transform Gc into a simple graph replacing each edge {v, w} colored with Sij,k by a path {v, x, y, w} such that c({x, y}) = Sij,k and the remaining edges of this path receive new different colors. Also, if c({v , w }) = c({v, w}) in Gc , we replace it by a path {v , x , y , w } such that c({x , y }) = c({x, y}) = Sij,k , c({v, x}) = c({v , x }) and c({y, w}) = c({y , w }). 2 Several NP-hard problems, such as (simple) Maximum Cut, are polynomialtime solvable on bounded treewidth graphs [1]. As each clause gadget has bounded size, using a similar argument we can modify the graph obtained in the previous construction in order to obtain the following corollary. Corollary 2.2 Colorful Cut remains NP-complete even when the input graph G satisfies simultaneously the following properties: (i) G is planar and connected.
(iii) G has maximum degree three.
(ii) G has bounded treewidth.
(iv) Each color class of G contains at most two edges.
Any bipartite graph has a colorful cut. Thus, it is natural to ask about the complexity of the problem on graphs with small odd cycle transversal. Corollary 2.3 Colorful Cut remains NP-complete even when the input graph G has odd cycle transversal number one. Proof. It is enough to pick a vertex of each gadget of Gc , constructed as described in the proof of Theorem 2.1, and identify them into a single vertex.2 Note that Maximum Cut is trivial on complete graphs, and that it is polynomial time solvable on cographs [1]. By adding a new vertex and edges colored with a new color, we can construct a hard instance in order to show the NP-completeness of Colorful Cut on complete graphs. Theorem 2.4 Colorful Cut is NP-complete on complete graphs. Note that if each color class of a graph G induces a K2 , then G has a colorful cut if and only if G is bipartite. The next result shows the NP-completeness of Colorful Cut when each color class induces either a K2 or a K3 . Theorem 2.5 Colorful Cut is NP-complete when each color class induces a clique of size at most three. Proof. For this proof we use a reduction from Not All Equal 3-sat (nae 3-sat), which is NP-complete [7]. Let I = (U, C) be an instance of nae 3-sat
R. Sucupira et al. / Electronic Notes in Discrete Mathematics 62 (2017) 87–92
91
such that U = {u1 , u2 , . . . , un } and C = {c1 , c2 , . . . , cm }. The construction of a graph Gc = (V, E, f ) is given by the following procedure, illustrated in Fig. 2: •
For each clause cj = (x, y, z) ∈ C construct a clique {xj , yj , zj } with all the edges colored with the color j.
•
For each variable ui ∈ U create the vertices ai and bi in V such that ai is only adjacent to all positive occurrences of ui , and bi is only adjacent to all negative occurrences of the same variable. Add an edge joining the first positive occurrence and the first negative occurrence of ui .
•
Excluding the edges of the clause cliques, all other edges must be colored with different colors.
Fig. 2. Graph Gc associated to the formula (u1 ∨u2 ∨u3 )∧(u1 ∨u2 ∨u3 )∧(u1 ∨u2 ∨u3 ).
At this point, it is not difficult to see that I = (U, C) is a satisfiable instance of nae 3-sat if and only if Gc has a colorful cut. 2
3
Fixed-parameter tractability
From the previous results it follows that Maximum Colored Cut is paraNP-complete [4] parameterized by any of these parameters: treewidth, neighborhood diversity, genus, degeneracy, odd cycle transversal number, p − k, and several combinations of such parameters. The next result shows its fixedparameter tractability when parameterized by the number of colors. Theorem 3.1 Maximum Colored Cut admits a cubic kernel parameterized by the number of colors. Proof. First recall that a cut of a graph G is a bipartite subgraph of G. Claim 1. Let H = (V1 , V2 , E) be a bipartite graph having λ edges and no isolated vertices. The maximum number λ of edges having endpoints in the same part that can be added to H is 2 2 .
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Suppose that λ is the maximum number of colors in a cut of G and let S ⊂ V be a set such that |c(∂S)| = λ. Forming a bipartite graph H by selecting exactly one edge of each color class in [S, V \ S], we can observe that any color class that is not in H has at most 2 λ2 edges, otherwise λ would not be maximum. Let Ei ⊂ E be the set of edges colored with color class i. As λ ≤ p, if |Ei | > 2 p−1 , then the color class i is in any maximum colored cut. 2 Such a property us the following reduction rule: () If some color class gives i has |Ei | > 2 p−1 replace (G, p) by (G[E \ Ei ], p − 1). After the application 2 of this rule we obtain a kernel of size O(p3 ). 2 Lemma 3.2 Any simple graph G = (V, E) with an edge coloring c : E → {1, 2, . . . , p} has an edge cut ∂S such that |c(∂S)| ≥ p2 . Proof. Follows from the well-known property that any graph with at least m 2 edges contains a bipartite subgraph with at least m2 edges [3]. Corollary 3.3 Maximum Colored Cut admits a cubic kernel parameterized by the cost of the solution. Proof. If p ≥ 2k, by Lemma 3.2 we have an Yes-instance. Otherwise, applying a reduction similar to Rule () we obtain a kernel of size O(k 3 ). 2
References [1] H. L. Bodlaender and K. Jansen. On the complexity of the maximum cut problem. Nordic Journal on Computing, 7(1):14–31, 2000. [2] D. Coudert, P. Datta, S. Perennes, H. Rivano, and M. E. Voge. Shared risk resource group complexity and approximability issues. Parallel Processing Letters, 17(2):169–184, 2007. [3] P. Erd˝os. On some extremal problems in graph theory. Mathematics, 3:113–116, 1965.
Israel Journal of
[4] J. Flum and M. Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. Springer, 2006. [5] M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical computer science, 1(3):237–267, 1976. [6] F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing, 4(3):221–225, 1975. [7] T. J. Schaefer. The complexity of satisfiability problems. In the 10th Annual ACM Symposium on Theory of Computing (STOC), pages 216–226, 1978.