The problem of finding a maximum cut in a weighted graph, called MaxCut
problem, is well-known in combinatorial optimization, and one of Karp’s original 21 NP-complete problems[Kar72]. The research on MaxCut is driven by a variety of applications ranging from mathematical problems like -embeddability [DL94a] over quantum mechanics [Bar82, DL94b] to design of electronic circuits [BR88]. An overview of applications is given in [DL94a, DL94b].
Formally, considering a graph with edge weights , MaxCut is the problem of finding a node subset that maximizes , where . The cut polytope is defined as the convex hull of the indicator vectors of cuts , for all , given by
Although MaxCut is NP-complete on general graphs, there are some classes of graphs on which polynomial algorithms are known. In [OD72, Had75] it was shown that MaxCut can be solved in polynomial time for unweighted planar graphs. This result can be extended to the weighted case [LP12, SWK90].
By Kuratowski’s Theorem [Kur30], a graph is planar if and only if it contains no - or -subdivision. As an extension of this, Wagner [Wag37] proved that a graph is planar if and only if it contains no - or -minor.
Using Wagner’s result, Barahona [Bar83] introduced a polynomial-time algorithm solving MaxCut on -minor-free graphs in time. This was generalized by Kaminski [Kam12] by proving that MaxCut can be solved in time on -minor-free graphs, for an arbitrary graph that admits a drawing with exactly one crossing. An extension of the class of -minor-free graphs was given by Grötschel and Pulleyblank by introducing weakly bipartite graphs [GP81, FMU92]. By definition these are the graphs, whose bipartite subgraph polytope is completely described by certain cycle- and edge-inequalities (see Section 2). Moreover, they proved that for positive edge-weights, MaxCut
can be solved in polynomial time one these graphs by using linear programming. In contrast to these results,MaxCut is NP-complete on -minor-free graphs [Bar83].
Considering cut polytopes, it is particularly interesting to find their linear description, i.e., their facet-defining inequalities. If there is a linear description of polynomial size in the input, this gives a polynomial algorithm for MaxCut. Even though it is unlikely to find such a description for arbitrary graphs, a better understanding of cut polytopes is expected to improve algorithmic results.
Although cut polytopes of complete graphs have been intensively studied (see, e.g., [DL10]), we are far from a good understanding of these objects, especially for , . Even much less is known for cut polytopes of arbitrary graphs. The latter were considered, e.g., by Barahona and Mahjoub [BM86]. As an additional result to the polynomial algorithm on -minor-free graphs, they determined all facets of cut polytopes of those graphs.
Not too long ago, Sturmfels and Sullivant [SS08] established a new connection between the study of cut polytopes and commutative algebra, as well as algebraic geometry, by considering related toric varieties. In particular, they conjectured that the cut polytope of a graph is normal if and only if the graph is -minor-free. Among others, the research on these toric varieties and associated cut algebras has been pursued by Engström [Eng11], Ohsugi [Ohs10, Ohs14], and Römer and Saeedi Madani [RS18].
It turns out that not much is known about the polyhedral structure of cut polytopes as objects in discrete geometry. We expect new insights in the study of MaxCut by considering cut polytopes of graphs not containing a specific minor.
Our contribution and organization of this paper
In Section 2, we recall basic definitions on graphs and polytopes, and summarize known results on cut polytopes.
In Section 3, we consider -minor-free graphs. Complementing the results on -minor-free graphs, we provide the full linear description of cut polytopes of -minor-free graphs.
Moreover, we give an algorithm solving MaxCut on -minor-free graphs, requiring only the running time for MaxCut on planar graphs. This is somewhat surprising, as -minor-free graphs admit an easier linear description, while we achieve a better running time for MaxCut on -minor-free graphs.
Starting the investigation of geometric properties of cut polytopes, in Section 4 we completely characterize graphs that provide a simple or simplicial cut polytope. In particular, it turns out that graphs providing a simple cut polytope are precisely the -minor-free graphs. The simplicial case can only occur for finitely many graphs.
In this section we provide some basic background on graphs and polytopes. Then, we recapitulate some known results on cut polytopes. For notation and results related to graphs we refer to [Die18], for those related to polytopes to [BG09, Zie12].
We only consider undirected graphs. A graph is simple, if it does neither have parallel edges connecting the same two nodes, nor self-loops. Unless specified otherwise, we only consider simple graphs that contain no isolated nodes in the following. For , let . Given a graph we also write and for its set of nodes and its set of edges , respectively. For , let be the edge between and . Two nodes and are adjacent if .
A path of length is a sequence of edges with such that for . Such a sequence with is a cycle; a cycle of length 3 is a triangle. A graph is a subgraph of , denoted by , if contains (a subgraph isomorphic to) . Given a subset , the subgraph induced by is the graph . If an induced subgraph forms a cycle, this is an induced cycle and thus chordless. A graph is chordal, if every induced cycle in has length . Maximal planar graphs are triangulations. We fix the following notations for some special classes of graphs: for the cycle of length for the complete graph on nodes; for the complete bipartite graph on and nodes per partition set.
is a -subdivision, if is obtained from by replacing edges by internally node-disjoint paths. The graph is obtained from by deleting the edge . The graph is obtained from by contracting edge , i.e., the nodes and are identified, and we delete the arising self-loop and merge parallel edges. contains an -minor, if can be obtained from by contracting and deleting edges. Otherwise is -minor-free.
is -connected if for each pair of nodes there exist internally node-disjoint paths from to . In particular, -connected graphs are called connected. If is connected but not -connected, there exists some cut-node such that is disconnected.
For two graphs and , their union is disjoint if their node sets are; in this case we write . Assume two graphs , contain as a subgraph, for some . The -sum (or clique-sum) is obtained by taking the union of and , identifying the subgraphs and possibly also removing edges contained in this specific . A -sum is strict, if no edges are removed. We denote the strict -sum of and by .
A polytope is the convex hull of finitely many points in . The dimension of is the dimension of its affine hull. A linear inequality is a valid inequality for if it is satisfied by all points . It is homogenous if . A (proper) face of is a (non-empty) set of the form for some valid inequality with . Each face is itself a polytope. The faces of dimension and dimension are vertices and facets, respectively. For polytopes and we define the product . It is a polytope with , and the proper faces of are given by products of proper faces of and proper faces of .
If is a facet of , the inequality is facet-defining. Each polytope can be represented as the bounded intersection of finitely many closed half-spaces, i.e., admits a linear description for some matrix and some vector . This is given, e.g., by taking the system of all facet-defining inequalities. A simplex of dimension is the convex hull of affinely independent points. A -dimensional polytope is simple if each vertex of is contained in exactly facets; the polytope is simplicial if each facet of is a simplex.
Given a graph and a subset , the set is a cut in . If is connected, this gives pairwise different cuts. To each cut in we associate its indicator vector given by
The cut polytope of is defined as their convex hull
and has dimension , see, e.g., [BGM85, p.344]. For disconnected graphs we have
Similarly, we may consider clique sums. As many classes of graphs can be described in terms of these, it is reasonable to study their effect on cut polytopes.
Theorem 2.1 (see [Bar83, Theorem 3.1.]).
Let be a strict -sum with . Then the facet-defining inequalities of are given by taking all facet-defining inequalities of and and identifying the variables of common edges. In particular, it holds that
Any automorphism of a graph gives rise to a map on cuts. Thus, induces a permutation on the vertices of by mapping to , which yields a symmetry of . Another symmetry of cut polytopes is given by switching:
Lemma 2.2 (Switching Lemma, see [Bm86, Corollary 2.9.]).
Let be a graph and be a facet-defining inequality for . Let , and define , and for all . Then defines a facet of .
On the level of cuts, switching in is induced by the map . Switching a facet-defining inequality by a cut corresponding to a vertex of this facet gives a homogeneous facet-defining inequality. Thus, all symmetry classes of facets of contain facets of the cut cone . Hence, it suffices to understand the facets of cut cones to understand the facets of cut polytopes.
Since is contained in the unit cube, the inequalities are valid. Given a cut and a cycle in , the number of edges in clearly is even. These observations give rise to the following edge- and cycle-inequalities:
Theorem 2.3 (see [Bm86, Section 3]).
The valid inequalities define facets of if and only if does not belong to a triangle. The valid inequalities
define facets if and only if is chordless.
Moreover, a graph is -minor-free if and only if is defined completely by the cycle- and edge inequalities.
In particular, for each triangle with the following metric inequalities (up to permuting the edges) are facet-defining for :
As a generalization of metric inequalities we get hypermetric inequalities by considering the complete graph instead of triangles, see [BM86, Theorem 2.4.]. An example for these is given by the hypermetric inequality of in Inequality (3.1). All above facets correspond to complete subgraphs or, in the case of cycle inequalities, subdivisions of these. There also exist facet-defining inequalities whose support graph is not complete, see, e.g., [BM86, Theorem 2.3].
3. -minor-free Graphs
In this section, we consider -minor-free graphs and provide the complete linear description of their cut polytopes. We also show that this yields an efficient algorithm for MaxCut on -minor-free graphs. This complements the known facts on -minor-free graphs. Moreover, since is maximal -minor-free but not weakly bipartite, we obtain the first full polyhedral description of a general minor-closed graph class apart from weakly bipartite graphs.
We first characterize maximal -minor-free graphs. Per se, this is not new: it is sometimes referenced to (different papers by) Wagner; a complete proof in modern terminology was given in [Tho99]. Here, we propose a slightly different approach, using -connectivity components. This provides a simpler, more basic proof and turns out to be directly usable for our polyhedral and our algorithmic results.
Let be a -connected, not necessarily simple graph and let be a split pair in , i.e., is disconnected or there are parallel edges connecting and . The split classes of are given by a partition of such that two edges are in a common split class if and only if there is a path between them neither containing nor as an internal node. As is -connected, it is easy to see that and are both incident to each split class. For a split class let . A Tutte split replaces by the two graphs and , provided that or remains -connected. Thereby, is a new virtual edge connecting and ; the other edges are called original. Observe that this operation may yield parallel edges. Iteratively splitting the graphs via Tutte splits gives the unique -connectivity decomposition of . Its components can be partitioned into the following sets: a set of cycles, a set of edge bundles (two nodes joined by at least edges), and a set of -connected graphs, see, e.g., [Tut66, HT73].
Any maximal -minor-free graph is -connected.
Clearly, is connected, as otherwise we could join two connected components via an edge without obtaining a -minor. Assume that is not -connected and let be a cut-node separating into and . Choose and adjacent to and obtain the graph from by adding the edge . As a sidenote, this operation retains planarity for planar . Since contains only two paths between and but is -connected, is still -minor-free. This contradiction concludes the proof. ∎
Let be a maximal -minor-free graph. Then, can be decomposed as a strict clique-sum , where each is either a planar triangulation or a copy of .
Let be a maximal -minor-free graph. By creftypecap 3.1, is -connected, so we may consider its -connectivity decomposition. Whenever a virtual edge was introduced, both parts of the Tutte split contain a path between and . Furthermore, contains a -minor if and only if one of the components of its decomposition does. But then, if would not contain an original edge connecting and , we could introduce it without creating a -minor. Thus, each virtual edge corresponds to an edge by maximality of , and is the strict -sum of cycles and -connected graphs. By maximality of , the cycles are triangles, a trivial form of a planar triangulations.
Let be a -connected graph from this sum. If is planar, then – by maximality – it is a triangulation. Otherwise, by Kuratowski’s Theorem, contains a -subdivision. Assume that . If contains as a subgraph, then it contains the graph shown in Figure 1(K33minor_a) as a minor, and thus a -minor, which yields a contradiction.
Assume that contains a proper -subdivision with Kuratowski nodes and let be a node of this subdivision. Since is 3-connected, there are disjoint paths from to three pairwise distinct Kuratowski nodes, say . But then contains the graph of Figure 1(K33minor_b) as a minor, which itself contains a -minor. This concludes the proof. ∎
creftypecap 3.2 allows us to classify all facets of cut polytopes of maximal -minor-free graphs:
Let be a maximal -minor-free graph. Then all facets of are given by the metric inequalities for each triangle contained in and switchings of the facet-defining inequality
We know from creftypecap 2.1 that the facets of the cut polytope of a -sum of graphs are given by taking all facets of the cut polytopes of both graphs and identifying common variables. Moreover, by creftypecap 2.3 all facets of a planar triangulation are given by metric inequalities; the facets of are given by metric inequalities and switchings of (3.1) [DL10, Chapter 30.6]. Since maximal -minor-free graphs are -sums of copies of and planar triangulations, this yields the claimed result. ∎
We can use creftypecap 3.3 to classify the facets of the cut polytope of any -minor-free graph.
Let be a -minor-free graph. Then, can be decomposed as a (not necessarily strict) k-sum of planar graphs and/or copies of , with .
Let be a maximal -minor-free graph containing . Then, the facets of are obtained by projecting onto .
The decomposition claim follows from creftypecap 3.2. Alternatively, we can obtain from by deleting edges. On the level of cut polytopes, the effect of an edge deletion corresponds to a projection onto . ∎
On the level of facets, a projection of a polytope to a coordinate hyperplane is given by eliminating variables. This can be done by Fourier-Motzkin elimination[Zie12, Chapter 1.2], which is made more precise in the following example.
Consider the non-maximal -minor-free graph shown in Figure 2. It is obtained by taking the non-strict -sum of two copies of . Let these copies of be and with and .
Both and are planar and each edge is contained in a triangle.
Thus, all facets of their cut polytopes are given by metric inequalities, and those are also facets of .
All other facets of are obtained by taking a pair of facets of and of and eliminating the variable by summing the corresponding inequalities.
In the following we focus on the latter class of facets.
Choosing one representative for each class of facet-defining inequalities of and we get:
one metric inequality of : 7cm,
one hypermetric inequality of : 7cm,
one metric inequality of : 7cm,
one hypermetric inequality of : 7cm.
Using Fourier-Motzkin elimination we have to sum each pair of inequalities such that there is one facet of each graph:
(1+3) is a cycle inequality. Switching (1+4) at shows that this inequality is equivalent to (2+3). These inequalities correspond to copies of with one subdivided edge contained in . The support graph of facet (2+4) is : This type of inequality is neither facet-defining for complete graphs nor does it belong to one of the mentioned classes of facet-defining inequalities in Section 2.
As demonstrated in the above example, Fourier-Motzkin elimination yields all facet-defining inequalities of a non-maximal -minor-free graph as sums of metric inequalities and hypermetric -inequalities. The support graph of a valid inequality for is the graph induced by edges with non-zero coefficients in . From creftypecap 2.3 we can thus deduce that the support graph of a facet-defining inequality is an edge, a cycle or contains a -minor. Considering the sum of two facets and used to eliminate the variable we observe the following: If is a cycle-inequality, summing it to acts on the support graph of as subdividing ; the effect of subdividing an edge in the support graph of a facet is described in [BM86, Corollary 2.10]. If is a hypermetric -inequality, summing it to acts on the support graph of as replacing by ; all non-zero coefficients of the obtained inequality are . Although possible, it is tedious to determine the exact signs and thus the constant term of the inequalities. However, we can concisely describe the facets’ support graphs.
Let be a facet of the cut polytope of a -minor-free graph . All its non-zero coefficients are and its support graph is an induced subgraph of that is either
an edge that is not contained in a triangle, or
obtained from a triangle or a by repeatedly (possibly zero times) subdividing edges and/or replacing an edge by .
Barahona [Bar83, Section 4] gave an algorithm for MaxCut on -minor-free graphs. Complementing this result an algorithm for MaxCut on -minor-free graphs whose running time is identical to that of planar MaxCut is given. Currently, the best known running time for this is [LP12, SWK90]. We use a data structure to efficiently consider the components of the -connectivity decomposition of . Recall that they are cycles , edge bundles , and -connected graphs . The SPR-tree has a node for each element of , , and [dBT96, CH17]111The data structure is also known as SPQR-tree. However, the originally proposed nodes of type (as well as the tree’s orientation) have often turned out to be superfluous.. For a node , let denote its corresponding component. Two nodes are adjacent if and only if and share a virtual edge. can be reconstructed from by taking the non-strict -sum of components whenever their corresponding nodes are adjacent in . Following this interpretation, -nodes containing a non-virtual edge represent strict -sums of their adjacent components of the decomposition. has only linear size and can be computed in time [HT73, Lemma 15].
The MaxCut problem on -minor-free graphs can be solved in the same time complexity as MaxCut on planar graphs.
Let be a -minor-free graph with edge weights , . Let be the best known running time for MaxCut on planar graphs with nodes. For we denote by the maximum weight over cuts with and . If is not -connected, we apply the algorithm to its -connected components (which can be identified in linear time). Assume in the following that is -connected.
We want to insert “original” edges of weight into between split pairs corresponding to Tutte splits. This will allow us to only consider strict -sums. To this end compute the SPR-tree . For any -node whose contains only virtual edges, introduce a new original edge of weight into , and therefore also into . For any adjacent non--nodes , let be the virtual edge shared between their components. We introduce a new original edge into . This yields a new -node subdividing the edge in . The edge bundle contains the new original edge together with two virtual edges, one shared with , the other with . By this construction, for every virtual edge there is an original edge with the same end nodes. Throughout the following, we always consider the weight of a virtual edge to be identical to the weight of the original edge . We continue to denote the resulting graph and tree by and , respectively.
Let be a leaf in and be the virtual edge contained in . Note that is either an - or an -node and thus, is either a copy of or planar. We compute and . If , this requires only constant time. Thus the needed work is bounded by . Let be the gain/loss by having in the cut, respectively. Removing from and therefore all edges of from yields a graph . is obtained from by removing the potential -node-leaf (and considering the “dangling” virtual edge as original, retaining its current cost). Setting the cost of the original edge to (after the computation of and ) yields that the maximum cut on is exactly , where is the maximum cut in (after updating the edge weight).
In this way, we can iteratively compute a maximum cut on by eliminating all nodes of its SPR-tree. The SPR-tree of can be built in time. Let be the components corresponding to - and -nodes in , . By planarity (or constant size of ), we have , and hence . For each , , we require only time. Since we have . The claim follows. ∎
4. Simple and Simplicial Cut Polytopes
In this section, we completely characterize graphs whose cut polytopes are simple or simplicial.
In [Gan13], it was claimed that is simple if and only if contains no -minor. Unfortunately, the given proof has some gaps. For example, [Gan13, Proposition 3.2.4.] claims that a --polytope is simple if and only if it is smooth. The proof mistakenly assumes that is always the polytope corresponding to the cut-variety in the sense of toric geometry. It is then used that a toric variety is smooth if and only if the corresponding polytope is, see [CLS11, Theorem 2.4.3]. However, the cut polytope is simple but not smooth, since the edges , and do not form a basis of . Contrarily the cut variety of is smooth, see [SS08, Corollary 2.4].
Nevertheless, in the following we show that the claimed characterization of graphs whose cut polytopes are simple is true. Our proof only requires basic tools from graph theory and discrete geometry.
An ear in a graph is a maximal path whose internal nodes have degree in . An ear decomposition of a -connected graph is a decomposition such that is a cycle and is an ear of for all .
A graph is -connected if and only if it admits an ear decomposition, see, e.g., [Die18, Proposition 3.1.2]. This allows us to prove the following:
Let be a connected graph. Then the following are equivalent:
with or for each .
Since and are -minor-free and -sums create cut-nodes, it is easy to see that implies . To show the reverse direction, let be a -minor-free graph. Considering its -connected components gives a decomposition , where or is -connected.
It is left to show that the only -connected -minor-free graph is . Assume that is a -connected -minor-free graph and consider its ear-decomposition . Since is -minor-free, is a copy of . Attaching an ear to two of its nodes would yield a -minor. Hence . ∎
Given this characterization, we are able to show that -minor-free graphs are exactly those graphs whose cut polytopes are simple.
The following are equivalent:
If is not connected, then is the product of the cut polytopes of the connected components of . Since the product of polytopes is simple if and only if each of the polytopes is simple, it suffices to show the equivalence for connected graphs.
If is not -connected, it can be decomposed as such that is either -connected or a copy of . Hence is simple if and only if is simple for all .
We show that for -connected graphs, the cut polytope is simple if and only if the graph equals . As is simple this fact together with creftypecap 4.2 yields the claim.
Observe that is a simplex. It hence remains to show that is not simple in case that is -connected and . In this case, each edge is contained in a cycle and thus in particular in a chordless cycle . By creftypecap 2.3 the inequalities
define many different facets of that contain the origin.
If , , no edge is contained in a triangle and defines a facet of for all . Hence, is contained in at least many different facets and as , the cut polytope is not simple.
Similarly, if , then there has to exist a chord in some cycle. In particular, lies in two chordless cycles. Thus, the origin is contained in at least facets and hence is not simple. ∎
Next we study graphs whose cut polytopes are simplicial. It was shown in [DL10] that the cut polytope of is not simplicial for . We generalize this result by giving a complete characterization of graphs with simplicial cut polytopes. We start by proving that the cut polytope of an arbitrary graph on at least nodes is not simplicial. In addition to this, Table 1 lists all graphs on at most four non-isolated nodes and indicates whether their cut polytopes are simplicial or not.
Let with without isolated nodes. Then is not simplicial.
First note that the product of polytopes and is simplicial if and only if either or one is simplicial and the other one is a point. The latter case is not relevant for us, since a graph with edges has a cut polytope of dimension at least .
Assume that is disconnected with connected components . Then is simplicial if and only if and and are -dimensional. This holds if and only if and are copies of . Hence, if is disconnected, is not simplicial for .
If is connected, we consider the following two cases:
(a) contains no triangle: Let be an edge of . As, by assumption, is not contained in a triangle, defines a facet of . For each , the indicator vector of the cut is a vertex of this facet. Hence, this facet contains at least vertices. On the other hand, since contains no triangle, Turáns Theorem (see [Die18, Theorem 7.1.1.]) yields . Since , we have which implies that is not simplicial.
(b) contains a triangle: Let with and be a triangle in . By creftypecap 2.3, the inequality is facet-defining for . For each and , the indicator vector of the cut is a vertex of this facet. This gives vertices on this facet. As , we have . Hence, is not simplicial. ∎
Using the previous proposition, we are able to give a characterization of all graphs whose cut polytopes are simplicial (see also Table 1).
Let be a graph with no isolated nodes. Then the following are equivalent:
is one of the following graphs:
Note that is -dimensional, hence simplicial. By Equation 2.1 and Equation 2.2, this yields that and are simplicial. It is straight-forward to verify that is a 3-simplex, is affine isomorphic to the -dimensional cyclic polytope on vertices, and is affinely isomorphic to a cross-polytope, all of which are simplicial polytopes.
We have determined the linear description of cut polytopes of -minor-free graphs and classified all graphs with a simple or simplicial cut polytope.
Throughout this paper one can see that besides graph minors, the decomposition of graphs into clique-sums of specific graphs is a useful tool to understand cut polytopes. This motivates several questions discussed in the following.
In [Kam12], it was shown that for each single-crossing graph , MaxCut can be solved in polynomial time on the class of -minor-free graphs. For and the linear description of cut polytopes of -minor-free graphs is now known. This naturally leads to the following question:
Can one give the linear description of cut polytopes of -minor-free graphs, for single-crossing graphs ?
By creftypecap 2.1 for the linear description of a strict -sum of two graphs is given by taking all facet-defining inequalities of both graphs and identifying common variables. This can be traced back to the fact that in these cases is a simplex. Although this does not hold for , the cut polytope of is a cyclic polytope and as such well understood. Therefore, the following question arises:
Can one give a linear description of the -sum of two graphs in terms of their linear descriptions?
While we give a linear description of cut polytopes of -minor-free graphs in Section 3, there are further graphs that fall under the same facet regime (interestingly, even itself). We thus ask:
Can one characterize all graphs whose cut polytopes are described by the inequalities from Section 3?
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