MaxCut on Permutation Graphs is NP-complete

02/28/2022
by   Celina M. H. de Figueiredo, et al.
UFRJ
UFC informa
0

In this paper, we prove that the MaxCut problem is NP-complete on permutation graphs, settling a long-standing open problem that appeared in the 1985 column of the "Ongoing Guide to NP-completeness" by David S. Johnson.

READ FULL TEXT VIEW PDF

Authors

page 1

page 2

page 3

page 4

12/17/2020

Maximum cut on interval graphs of interval count four is NP-complete

The computational complexity of the MaxCut problem restricted to interva...
04/29/2021

Revising Johnson's table for the 21st century

What does it mean today to study a problem from a computational point of...
03/03/2018

Path Puzzles: Discrete Tomography with a Path Constraint is Hard

We prove that path puzzles with complete row and column information--or ...
06/04/2019

Shellings from relative shellings, with an application to NP-completeness

Shellings of simplicial complexes have long been a useful tool in topolo...
07/28/2017

A syntactic tool for proving hardness in the Second Level of the Polynomial-Time Hierarchy

In the nineties Immerman and Medina initiated the search for syn- tactic...
03/06/2019

Proving the NP-completeness of optimal moral graph triangulation

Moral graphs were introduced in the 1980s as an intermediate step when t...
04/27/2022

Insight into Voting Problem Complexity Using Randomized Classes

The first step in classifying the complexity of an NP problem is typical...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

A cut is a partition of the vertex set of a graph into two disjoint parts, and the maximum cut problem (denoted by MaxCut, for short) aims to determine a cut with the maximum number of edges for which each endpoint is in a distinct part. The decision problem MaxCut is known to be NP-complete since the seventies [10], and only recently its restriction to interval graphs has been announced to be hard by Adhikary, Bose, Mukherjee, and Roy [1]. This settles a long-standing open problem from the Ongoing Guide to NP-completeness by David S. Johnson [13].

In his column, David S. Johnson presented a two-page summary table, with a column for each of the ten most famous NP-complete graph problems, and a row for each of thirty selected graph class. Among those graph classes, special emphasis was given to subclasses of perfect graphs and of intersection graphs having broad algorithmic significance. The emphasis was on the restrictions themselves and how they affect the complexity of the considered NP-hard problems. The discussion had focus on the particularly fertile domain of graph theory, where the central open problem at that time was the recognition of perfect graphs.

Many important graph classes are defined or can be characterized by a geometric intersection model. Two particularly well-studied examples are subclasses of perfect graphs: the classes of interval graphs and of permutation graphs [8, 12, 15]. In their respective models, the intersecting objects are line segments in the plane, with different restrictions imposed on their positions. In interval graphs, each line segment must have its endpoints on a single line, while in permutation graphs, their endpoints must lie on two distinct parallel lines.

Besides selecting the recognition of perfect graphs as the famous open problem, in his column, David S. Johnson selected only two others as open and may well be hard problems: Hamiltonian circuit restricted to permutation graphs and edge-coloring restricted to planar graphs. Today, we know that recognition of perfect graphs and Hamiltonian circuit restricted to permutation graphs can both be solved in polynomial time. On the other hand, edge-coloring restricted to planar graphs remains a challenging open problem. Please, refer to [6] for an updated summary table. Surprisingly, after 35 years, the only new resolved entry for permutation graphs is Hamiltonian circuit.

The present paper settles a long-standing open problem proposed by Johnson, by providing the first entry of Johnson’s table for permutation graphs resolved as NP-complete.

Theorem 1.

MaxCut is NP-complete on permutation graphs.

Our proof is based on Adhikary et al.’s construction used to prove the NP-completeness of MaxCut on interval graphs [1]. It is interesting to notice that, among the problems selected by Johnson, MaxCut

is the only one classified as NP-complete for interval graphs and for permutation graphs. Despite that, the interval graph constructed by Adhikary et al. is not a permutation graph, and our constructed permutation graph is not an interval graph. Thus, we leave as an open question the complexity of

MaxCut on permutation interval graphs.

Our paper is organized as follows. In Section 1.1, we present the basic concepts and notations. In Section 2, we present the main gadget in the reduction of Adhikary et al. [1], which also plays an important role in our reduction. In Section 3, we present the construction of Adhikary et al. [1] and show that it does not lead to a permutation graph. The presentation of their construction is also useful in Section 4, where we finally present the proof of Theorem 1. In Section 5, we prove that our constructed permutation graph is not an interval graph, and propose the complexity of MaxCut on permutation interval graphs as an open problem.

1.1 Preliminaries

In this work, all graphs considered are simple. For missing definitions and notation of graph theory, we refer to [3].

Let be a graph. We say that a subset is a clique if every two distinct vertices in are adjacent, and that a subset is a stable set if no two vertices in are adjacent. Let and be two disjoint subsets of . We say that is complete to if every vertex in is adjacent to every vertex in , and that is anti-complete to if there are no edges between and . We let be the subset of with an endpoint in and the other endpoint in . A cut of is a partition of into two parts , denoted by ; the edge set is called the cut-set of associated with . For each two vertices , we say that and are in a same part of if either or ; otherwise, we say that and are in opposite parts of . Denote by the maximum size of a cut-set of . The MaxCut problem has as input a graph and a positive integer , and it asks whether .

Let and be two permutations of a same set, say . A graph is called the intersection graph related to if and, for each two vertices , if and only if and . In this case, we also say that is a permutation model of . A graph is a permutation graph if it is the intersection graph related to a permutation model.

Given two permutations and of disjoint subsets and , respectively, we write to denote the permutation of given by the concatenation of with . Also, we write to denote the reverse of the permutation , that is, if , then . In order to simplify the notation, given a set , we sometimes use the same symbol, , to denote also a chosen permutation of the elements of ; in such cases, represents the reverse of the chosen permutation for .

An interval model is a finite multiset of closed intervals of the real line. Let be a graph and be an interval model. An -representation of is a bijection such that, for every two distinct vertices , we have that if and only if . If such an -representation exists, we say that is an interval model of and that is an interval graph.

We write to mean .

2 Grained gadget

In this section, we present the notion of grained gadgets, which was defined in [5] as a generalization of the so-called -gadgets and -gadgets, these latter introduced by Adhikary et al. [1] in order to prove the NP-completeness of MaxCut on interval graphs.

Let and be positive integers. An -grained gadget is a split graph formed by a clique of size and a stable set of size with being complete to , being complete to , and satisfying and . Figure 1 depicts an interval representation of an -grained gadget. One can readily verify that the intersection graph related to the pair of permutations (see Figure 2) is an -grained gadget. Thus, grained gadgets are interval graphs and permutation graphs.

Figure 1: Interval representation of an -grained gadget c.f.[5].
Figure 2: A permutation model of a grained gadget.

Let be an -grained gadget and be a supergraph of . For each vertex , we say that (see Figure 3): covers if ; weakly intersects if either or ; and that strongly intersects if either or . Moreover, we say that respects the structure of if, for each vertex , either or satisfies one of the previous conditions.

(a) Covering intersection
(b) Weak intersection
(c) Strong intersection
Figure 3: Vertex (subfig:covering) covering , (subfig:weakly) weakly intersecting , and (subfig:strongly) strongly intersecting . The set is a clique and the set is a stable set. A line between sets, or between and some set, means that all the edges occur.

The next lemma establishes the key property of grained gadgets with respect to the MaxCut problem. Intuitively, it states that, for suitable values of and , if is a supergraph that respects the structure of an -grained gadget, then, in any maximum cut of , the vertices belonging to are placed in a same part of , opposite to the part containing the vertices belonging to .

Lemma 1 ([4]).

Let and be positive integers, be an -grained gadget and be a supergraph that respects the structure of . Also, let be a maximum cut of , be the number of vertices in adjacent to some vertex of , be the number of vertices of adjacent to some vertex in , and be the number of vertices of adjacent to some vertex in . If and

are odd,

and , then each of the following holds:

  1. and , or vice versa;

  2. and , or vice versa;

  3. and , or vice versa.

In the remainder of the text, when a grained gadget is not clear in the context, we write , , and to denote the stable sets and and the cliques and of , respectively.

3 Adhikary et al.’s reduction

In this section, we present the construction given by Adhikary et al. [1] of an interval graph that proves NP-completeness of MaxCut in this class. As we see in Section 4, the general idea behind their construction can also be used to obtain a permutation graph instead. Nevertheless, the question of whether their construction is also permutation might arise. We prove here that this is not the case.

Given a cubic graph , let and be arbitrary orderings of and , respectively. Define the values: , , , and . An interval graph is defined through the construction of one of its interval models , described as follows (observe Figure 4 to follow the construction):

  1. Add to a -grained gadget for each vertex . These gadgets should be pairwise disjoint, with appearing completely to the left of for every ;

  2. Add to a -grained gadget for each edge . Likewise, these gadgets should be pairwise disjoint, with appearing completely to the left of for every . Additionally, appears completely to the right of , without intersecting it;

  3. Finally, for each edge , with , add four intervals , called link intervals, such that:

    • and (resp. and ) weakly intersect (resp. ) to the right of (resp. );

    • and (resp. and ) weakly intersect (resp. strongly intersect) to the left of .

Figure 4: Adhikary et al.’s interval model , with , , and .

As claimed, we show that the constructed graph is not a permutation graph. This is because contains the graph depicted in Figure 4(a) as an induced subgraph, and such a graph is a forbidden subgraph for comparability graphs cf. [9, 7], in turn a known superclass of permutation graphs. To see that this claim holds, observe Figure 4(b). Given an edge , with , it shows the intervals in the grained gadgets of , and , as well as some link intervals related to . The adjacencies can be easily checked to be as in the graph of Figure 4(a).

(a)
(b)
Figure 5: (subfig:forbidden_subgraph) Forbidden induced subgraph for comparability graphs cf.[7]. (subfig:forbidden_adhkary)  as an induced subgraph in Adhikary et al.’s construction.

In the next section, we show that a modification of Adhikary et al.’s construction gives us the desired permutation graph.

4 Our reduction

Consider a cubic graph , and orders on its vertex and edge sets, and , respectively. The values of are not the same as in Section 3 and are presented later. Again, for each vertex , create a -grained gadget, , and for each edge , create a -grained gadget . For simplicity, denote the sets , , and by , respectively. Similarly, denote the sets , , and by , respectively.

Recall that for each , the permutation model of consists of the pair of permutations where and . Analogously, for each , the permutation model of consists of the pair of permutations where and . Now, for each edge , with , add four new vertices , called link vertices. In what follows, we modify some of the grained gadget permutations in order to make (resp. ) weakly intersect (resp. ) and strongly intersect (resp. weakly intersect) .

If is incident to edges , with , then modify one of the permutations defining to include the link vertices related to as follows:

where denotes the permutation .

Similarly, for each edge , , we modify one of the permutations defining to include the link vertices related to as follows:

We do not modify and , and keep denoting by the permutation , and by the permutation . Finally, let be the permutation graph related to , where:

Figure 6 illustrates our permutation model , focusing on the vertex grained gadgets and , the edge grained gadget , and the link vertices and related to an edge , with .

Figure 6: Vertex and edge grained gadgets, and link vertices related to an edge , with , in our permutation model .

We remark that the main difference of our permutation graph from the Adhikary et al.’s interval graph is the fact that, in Adhikary et al.’s interval graph, the link vertices form a clique, whereas, as we show in Section 5, some link vertices are not adjacent in our permutation graph. Additionally, for an edge , with , the link vertices (resp. ) weakly intersect (resp. strongly intersect) in Adhikary et al.’s interval graph, whereas in our permutation graph the link vertices (resp. ) strongly intersect (resp. weakly intersect) .

Before our proof, we make some observations about the constructed graph in order to improve the proof’s readability. Note that, for each link vertex and grained gadget , either the relative order between and in is the reverse of their relative order in , in which case is complete to , or the relative order is the same in both and , in which case is anti-complete to , or is related to according to one of the ways described below.

  • and : in this case only the relative orders between and are reversed in and , i.e., is complete to and anti-complete to ;

  • and , with , : in this case the relative orders between and are reversed in and , i.e., is complete to and anti-complete to ; or

  • and , with , : in this case only the relative orders between and are reversed in and , i.e., is complete to and anti-complete to .

Proof of Theorem 1.

Consider the reduction graph and its permutation model as previously defined. For each , let

and for each , let

Also, denote the set of link vertices by , i.e. .

We postpone the assignment of the actual values for and, in addition to the conditions necessary for the application of Lemma 1, we also ask that and .

In what follows, we prove that there exists a bijective relation between the maximum cuts of the input graph and the maximum cuts of our permutation graph . Then, we prove that, for each maximum cut of ,

where and is a well-defined function. Theorem 1 immediately follows.

Let be a maximum cut of . In order to define , we first prove some properties relating the partitioning of vertex and edge grained gadgets of in with the partitioning of the link vertices of in . More specifically, we prove that the two following properties hold:

  1. For each vertex , if , then for each edge , with and ;

  2. For each edge , with and , if , then .

Proof of Property 1. Let and suppose that . For the sake of contradiction, suppose that there exists a link vertex . Then, let be the cut obtained from by setting and . Observe that there is a loss of at most edges between and , and between and the vertices of the edge grained gadget related to , say , since and are always in opposite parts of the cut. On the other hand we gain all the edges between and the vertices in . Therefore, we get an increase of the cut-set of at least edges, and a decrease of less than edges. It follows from the hypothesis that is bigger than , contradicting the maximality of .

Proof of Property 2. Consider an edge , with and , and suppose that . Observe that, because the relative orders among the edge and vertex grained gadgets themselves are the same in and , there are no edges between and any other grained gadgets of , i.e., the only vertices outside of that can be adjacent to the vertices of are those in . Moreover, Lemma 1 tells us that the vertices belonging to are placed in a same part of , opposite to the part containing the vertices belonging to . More formally, either and , or and . As a result, switching the vertices of of part of the cut does not change, and therefore cannot decrease, the number of edges between the vertices of and the vertices belonging to in the cut-set. Consequently, if , then we obtain that there are at least edges in the cut-set that are incident to vertices of ; these are the edges between and the vertices belonging to . On the other hand, if , then we obtain that there are at most edges in the cut-set that are incident to vertices of ; these are the edges between the vertices belonging to and the vertices belonging to . Therefore, since , we get that as we wanted to prove.

We are now ready to prove the existence of the bijective relation . For each maximum cut of , let be the cut of defined as follows:

  • For each vertex , if , then add to and to ; do the opposite otherwise.

  • For each , with and , if , then add to and to ; and do the opposite otherwise.

Based on Properties 1 and 2, one can readily verify that is well-defined and is a bijective relation, as desired.

Now, we count the number of edges in as a function of , , , , , and of the size of the cut-set . First, consider . By construction, we know that there are edges in the cut-set between the vertices of . Additionally, there are exactly link vertices weakly intersecting , while all other link vertices are either complete or anti-complete to . Observe also that the number of link vertices complete to is exactly equal to ; these are the link vertices related to . This gives us a total of edges between the vertices of and the vertices belonging to in the cut-set. Summing up these values for every , we get a total of

edges in the cut-set incident to vertex grained gadgets. Now, let , with and . By construction, we know that there are edges of the cut-set between vertices of , and edges of the cut-set between and the vertices of . Additionally, note that there are exactly link vertices that cover and, therefore, are complete to ; these are the link vertices related to . This gives us a total of edges between the vertices of and the vertices belonging to . Finally, suppose without loss of generality that (the count is analogous if it is in ). If , then we know that and hence there are no edges in the cut-set between vertices and the vertices of . Otherwise, observe that it follows that and , and hence we get additional edges in the cut-set; these additional edges are between the link vertices and the vertices belonging to . Summing up these values for every , we get a total of edges in the cut-set incident to edge grained gadgets, where

Finally, observe that there are at most edges of the cut-set between link vertices. Note also that since each vertex in is related to 6 link vertices, which are all placed in . Similarly, we have . This gives us at most edges in the cut-set between link vertices. Putting everything together, we get:

By setting to , and knowing that will be chosen as functions of and , we want to prove, as stated in the beginning, that if and only if . If , then the first inequality gives us that . On the other hand, if , then the second inequality gives us that . Because we assume that , it follows that and hence .

It only remains to set the values of . Observe that:

  • For every grained gadget , the total number of vertices in adjacent to is at most (these are exactly the link vertices).

  • For every vertex grained gadget , the total number of vertices adjacent to some vertex is exactly , for some positive integer (this is because the number of link vertices adjacent to the vertices in is always even). The same holds for the number of vertices adjacent to the vertices in . We then get that the parity of and in the conditions of Lemma 1 applied to depends only on the parity of .

  • Similarly, if is an edge grained gadget, then the parity of the total number of vertices adjacent to some is equal to the parity of .

Therefore, the necessary conditions of Lemma 1 translate to: and ; and ; and and are odd. Additionally, we need to ensure: and . Hence, consider:

  • ;

  • ;

  • ;

  • .

Since , one can verify that the values described above satisfy all the required conditions. This, therefore, concludes the proof of Theorem 1. ∎

5 MaxCut on permutation interval graphs is an open problem

In this paper, we have presented a proof of NP-completeness for the MaxCut problem when constrained to permutation graphs. Surprisingly enough, we found that the main gadget in the reduction recently presented by Adhikary et al. [1] for interval graphs is also a permutation graph. Additionally, in Section 3, we have seen that being permutation is not a property that holds for the full construction of Adhikary et al. [1]. On the other hand, since the grained gadgets play an important role in our reduction too, one could wonder whether our construction instead is in the intersection between interval and permutation graphs. The answer to that is no as we argue next.

Let be a cubic graph, and consider arbitrary orderings of and , and , respectively. Let be the indices of the edges incident to , with . Also, let be the other endpoint of . We present a in , the graph constructed in Section 4; it thus follows that is not chordal, and hence also not interval [12]. Observe Figure 7 to follow our argument. Let be equal to , be any vertex in , be equal to , and be any vertex in . Since and , we know that the relative order between and in is the same as in ; hence . Also, the relative order in between and any vertex of is reversed in , the same holds between and any vertex belonging to ; hence . Similarly, the relative order between and any vertex belonging to in is reversed in , and the same holds between and any vertex of ; hence . Finally, since , the relative order between and in is the same as in , and therefore , thus finishing our argument.

Figure 7: Existence of a as an induced subgraph in our permutation graph.

The previous paragraph tells us that for any chosen orderings of and , the graph constructed in Section 4 contains a . Since it is known that the class of -free co-comparability graphs is precisely the class of interval graphs [11], and that the class of permutation graphs is equal to the class of comparability co-comparability graphs [14], we get that interval permutation graphs are exactly the class of -free permutation graphs.

A good question is whether there is a construction that produces a permutation graph that is also -free (and hence interval). Up to our knowledge, the largest class in the intersection of permutation and interval graphs for which the complexity is known is the class of the threshold graphs, on which MaxCut is polynomial-time solvable thanks to the algorithm given for cographs, a subclass of permutation graphs that is a superclass of threshold graphs [2].

References

  • [1] R. Adhikary, K. Bose, S. Mukherjee, and B. Roy (2021) Complexity of maximum cut on interval graphs. In 37th International Symposium on Computational Geometry, SoCG 2021, LIPIcs, Vol. 189, pp. 7:1–7:11. External Links: Document Cited by: §1, §1, §1, §2, §3, §5.
  • [2] H. L. Bodlaender and K. Jansen (1994) On the complexity of the maximum cut problem. In Annual Symposium on Theoretical Aspects of Computer Science, pp. 769–780. Cited by: §5.
  • [3] J. A. Bondy and U. S. R. Murty (2008) Graph theory. Graduate Texts in Mathematics, Springer, New York. External Links: Document, ISBN 978-1-84628-970-5 Cited by: §1.1.
  • [4] C. M. H. de Figueiredo, A. A. de Melo, F. de S. Oliveira, and A. Silva (2020) Maximum cut on interval graphs of interval count four is NP-complete. External Links: 2012.09804 Cited by: Lemma 1.
  • [5] C. M. H. de Figueiredo, A. A. de Melo, F. de S. Oliveira, and A. Silva (2021) Maximum cut on interval graphs of interval count four is NP-complete. In 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, LIPIcs, Vol. 202, pp. 38:1–38:15. External Links: Document Cited by: Figure 1, §2.
  • [6] C. M.H. de Figueiredo, A. A. de Melo, D. Sasaki, and A. Silva (2021) Revising Johnson’s table for the 21st century. Discret. Appl. Math.. External Links: ISSN 0166-218X, Document Cited by: §1.
  • [7] H. N. de Ridder et al. Graphclass: comparability graphs. information system on graph classes and their inclusions (isgci). Note: https://www.graphclasses.org/classes/gc_72.htmlAccessed: 2022-02-17 Cited by: Figure 5, §3.
  • [8] P. C. Fishburn (1985) Interval graphs and interval orders. Discret. Math. 55 (2), pp. 135–149. External Links: Document Cited by: §1.
  • [9] T. Gallai (1967) Transitiv orientierbare graphen. Acta Mathematica Hungarica 18 (1-2), pp. 25–66. Cited by: §3.
  • [10] M. R. Garey, D. S. Johnson, and L. J. Stockmeyer (1976) Some simplified NP-complete graph problems. Theor. Comput. Sci. 1 (3), pp. 237–267. External Links: Document Cited by: §1.
  • [11] P. C. Gilmore and A. J. Hoffman (1964) A characterization of comparability graphs and of interval graphs. Canadian Journal of Mathematics 16, pp. 539–548. Cited by: §5.
  • [12] M. C. Golumbic (2004) Algorithmic graph theory and perfect graphs (annals of discrete mathematics, vol 57). North-Holland Publishing Co., NLD. External Links: ISBN 0444515305 Cited by: §1, §5.
  • [13] D. S. Johnson (1985) The NP-completeness column: an ongoing guide. J. Algorithms 6 (3), pp. 434–451. External Links: Document Cited by: §1.
  • [14] A. Pnueli, A. Lempel, and S. Even (1971) Transitive orientation of graphs and identification of permutation graphs. Canadian Journal of Mathematics 23, pp. 160–175. Cited by: §5.
  • [15] J. P. Spinrad (2003) Efficient Graph Representations