Additive non-approximability of chromatic number in proper minor-closed classes

Additive non-approximability of chromatic number in proper minor-closed classes

Zdeněk Dvořák Charles University, Prague, Czech Republic. E-mail: rakdver@iuuk.mff.cuni.cz. Supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foundation.    Ken-ichi Kawarabayashi National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan. Supported by JST ERATO Grant Number JPMJER1305, Japan.
Abstract

Robin Thomas asked whether for every proper minor-closed class , there exists a polynomial-time algorithm approximating the chromatic number of graphs from up to a constant additive error independent on the class . We show this is not the case: unless , for every integer , there is no polynomial-time algorithm to color a -minor-free graph using at most colors. More generally, for every and , there is no polynomial-time algorithm to color a -minor-free graph using less than colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.

We also give somewhat weaker non-approximability bound for -minor-free graphs with no cliques of size . On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no -cycles.

The problem of determining the chromatic number, or even of just deciding whether a graph is colorable using a fixed number of colors, is NP-complete [7], and thus it cannot be solved in polynomial time unless . Even the approximation version of the problem is hard: for every , Zuckerman [18] proved that unless , there exists no polynomial-time algorithm approximating the chromatic number of an -vertex graph within multiplicative factor .

There are more restricted settings in which the graph coloring problem becomes more tractable. For example, the well-known Four Color Theorem implies that deciding -colorability of a planar graph is trivial for any ; still, -colorability of planar graphs is NP-complete [7]. From the approximation perspective, this implies that chromatic number of planar graphs can be approximated in polynomial time up to multiplicative factor of (but not better), and additively up to .

More generally, the result of Thomassen [16] on 6-critical graphs implies that the -coloring problem restricted to graphs that can be drawn in any fixed surface of positive genus is polynomial-time solvable for any . The case includes -colorability of planar graphs and consequently is NP-complete, while the complexity of -colorability of embedded graphs is unknown for all surfaces of positive genus. Consequently, chromatic number of graphs embedded in a fixed surface can be approximated up to multiplicative factor of and additively up to .

If a graph can be drawn in a given surface, all its minors can be drawn there as well. Hence, it is natural to also consider the coloring problem in the more general setting of proper minor-closed classes. Further motivation for this setting comes from Hadwiger’s conjecture, stating that all -minor-free graphs are -colorable. This conjecture is open for all , and not even a polynomial-time algorithm to decide -colorability of -minor-free graphs is known (Kawarabayashi and Reed [9] designed an algorithm that for a given input graph finds a -coloring, or a minor of in , or finds a counterexample to Hadwiger’s conjecture). However, any -minor-free graph is -colorable [11]. This implies that for every proper minor-closed class , if is the minimum integer such that , then there exists a constant such that every graph in is -colorable, and thus chromatic number of graphs in can be approximated up to multiplicative factor and additively up to .

On the hardness side, consider for any planar graph and an integer the graph obtained from by adding universal vertices (adjacent to every other vertex of ). Then , and since -colorability of planar graphs is NP-complete, there cannot exist a polynomial-time algorithm to decide whether such a graph is -colorable, unless . Furthermore, does not contain as a minor; indeed, each minor of has an induced planar subgraph containing all but of its vertices, which is not the case for . This outlines the importance of another graph parameter in this context, the apex number: we say a graph is -apex if there exists a set of vertices of of size at most such that is planar, and the apex number of is the minimum such that is -apex. The presented construction shows that if the apex number of is at least , then -colorability is NP-complete even when restricted to the class of -minor-free graphs. On the positive side, Dvořák and Thomas [5] gave, for any -apex graph and integer , a polynomial-time algorithm to decide whether a -connected -minor-free graph is -colorable (the connectivity assumption is necessary, since they also proved that for every integer , there exists a -apex graph such that testing -colorability of -connected -minor-free graphs is NP-complete).

In all the mentioned results for proper minor-closed classes, the number of colors needed and thus also the magnitude of error of the corresponding approximation algorithms depended on the specific class. This contrasts with the case of embedded graphs: the multiplicative () and additive () errors of these approximation algorithms are independent on the fixed surface in that the graphs are drawn. Hence, it is natural to ask the following questions.

Question 1.

Does there exist with the following property: for every proper minor-closed class , there exists a polynomial-time algorithm taking as an input a graph and returning an integer such that ?

Question 2.

Does there exist with the following property: for every proper minor-closed class , there exists a polynomial-time algorithm taking as an input a graph and returning an integer such that ?

That is, is it possible to approximate chromatic number up to a multiplicative or additive error independent on the considered class of graphs , as long as is proper minor-closed? Perhaps a bit surprisingly, the answer to Question 1 is positive. As shown by DeVos et al. [4] and algorithmically by Demaine et al. [3], for every proper minor-closed class , there exists a constant such that the vertex set of any graph can be partitioned in polynomial time into two parts and with both and having tree-width at most . Consequently, can be determined exactly in linear time [1], and we can color and using disjoint sets of colors, obtaining a coloring of using at most colors. That is, has the property described in Question 1.

In the light of this result, Question 2 may seem more tractable. Thomas [15] conjectured that such a constant exists, and Kawarabayashi et al. [10] conjectured that this is the case even for list coloring. As our main result, we disprove these conjectures.

Theorem 3.

Let be a positive integer, let be a -connected graph that is not -apex, and let be a real number. Unless , there is no polynomial-time algorithm taking as an input an -minor-free graph and returning an integer such that .

In particular, in the special case of and being a clique, we obtain the following.

Corollary 4.

Let be a positive integer. Unless , there is no polynomial-time algorithm taking as an input a -minor-free graph and returning an integer such that .

On the positive side, Kawarabayashi et al. [10] showed it is possible to approximate chromatic number of -minor free graphs in polynomial time additively up to . We leave open the question whether a better additive approximation (of course above the bound given by Corollary 4) is possible.

Another positive result was given by Demaine et al. [2], who proved that if is a -apex graph, then the chromatic number of -minor-free graphs can be approximated additively up to . Let us also remark that if is -apex (i.e., planar), then -minor-free graphs have bounded tree-width [13], and thus their chromatic number can be determined exactly in linear time [1]. We generalize these results to excluded minors with larger apex number (the relevance of the apex number in the context is already showcased by Theorem 3).

Theorem 5.

Let be a positive integer and let be a -apex graph. There exists a polynomial-time algorithm taking as an input an -minor-free graph and returning an integer such that .

The construction we use to establish Theorem 3 results in graphs with large clique number (on the order of ). On the other hand, forbidding triangles makes the coloring problem for embedded graphs more tractable—all planar graphs are -colorable [8] and there exists a linear-time algorithm to decide -colorability of a graph embedded in any fixed surface [6]. It is natural to ask whether Question 2 could not have a positive answer for triangle-free graphs, and this question is still open. On the negative side, we show that forbidding cliques of size is not sufficient.

Theorem 6.

Let and be real numbers such that and . Let . There exists a positive integer such that the following holds. Let be a -connected graph with at least vertices that is not -apex. Unless , there is no polynomial-time algorithm taking as an input an -minor-free graph with and returning an integer such that .

In particular, in the special case of and being a complete graph, we get the following.

Corollary 7.

For every positive integer , there exists an integer as follows. Unless , there is no polynomial-time algorithm taking as an input a -minor-free graph with and returning an integer such that .

On the positive side, we offer the following small improvement to the additive error of Theorem 5.

Theorem 8.

Let be a positive integer and let be a -apex graph. There exists a polynomial-time algorithm taking as an input an -minor-free graph with no triangles and returning an integer such that .

What about graphs of larger girth? It turns out that Question 2 has positive answer for graphs of girth at least , with . Somewhat surprisingly, it is not even necessary to forbid triangles to obtain this result, just forbidden 4-cycles are sufficient. Indeed, we can show the following stronger result.

Theorem 9.

Let be positive integers and let be a proper minor-closed class of graphs. There exists a polynomial-time algorithm taking as an input a graph not containing as a subgraph and returning an integer such that .

Let us remark that the multiplicative -approximation algorithm of Demaine et al. [3] can be combined with the algorithms of Theorems 5, 8, and 9 by returning the minimum of their results. E.g., if is a -apex graph, then there is a polynomial-time algorithm coloring an -minor-free graph using at most colors, for any such that ; the combined multiplicative-additive non-approximability bounds of Theorems 3 and 6 are also of interest in this context.

In Section 1, we present a graph construction which we exploit to obtain the non-approximability results in Section 2. The approximation algorithms are presented in Section 3.

1 Tree-like product of graphs

Let and be graphs, and let and . Let denote the rooted tree of depth such that each vertex at depth at most has precisely children (the depth of the tree is the number of vertices of a longest path starting with its root, and the depth of a vertex is the number of vertices of the path from the root to ; i.e., the root has depth ). For each non-leaf vertex , let be a distinct copy of the graph and let be a bijection from to the children of in . If , is a non-leaf vertex of the subtree of rooted in , and , then we say that is a progenitor of . The level of is defined to be the depth of in . Note that a vertex at level has exactly one progenitor at level for all positive . The graph is obtained from the disjoint union of the graphs for non-leaf vertices by, for each edge with , adding all edges from vertices of at level to their progenitors at level . Note that the graph depends on the ordering of the vertices of , which we consider to be fixed arbitrarily.

Lemma 10.

Let and be graphs with and . Let be the maximum integer such that is an independent set in . The graph has vertices and . Furthermore, if is a minor of and is -connected, then there exists a set of size at most such that is a minor of .

Proof.

The tree has non-leaf vertices, and thus .

Consider a clique in , and let be a vertex of of largest level. Let be the vertex of such that . Note that all vertices of are progenitors of , and the vertices of corresponding to their levels are pairwise adjacent. Consequently, and . Therefore, each clique in has size at most . A converse argument shows that cliques in and in give rise to a clique in of size , implying that .

For , let denote the graph obtained from by adding universal vertices. Observe that is obtained from copies of , …, by clique-sums on cliques of size at most (consisting of the progenitors whose level is most ). Hence, each -connected minor of is a minor of one of , …, , and thus a minor of can be obtained from by removing at most vertices. ∎

For an integer , the -blowup of a graph is the graph obtained from by replacing every vertex by an independent set of vertices, and by adding all edges such that and for some . For the purposes of constructing the graph , we order the vertices of so that for each , the vertices of are consecutive in the ordering. The strong -blowup is obtained from the -blowup by making the sets into cliques for each . For integers , an -coloring of is a function that to each vertex of assigns a subset of of size such that for each edge of . The fractional chromatic number is the infimum of . Note that if is the strong -blowup of a graph , then a -coloring of gives a -coloring of . Consequently, we have the following.

Observation 11.

If is the strong -blowup of a graph , then .

We now state a key result concerning the chromatic number of the graph .

Lemma 12.

Let be integers and let be a graph. Let be a graph such that , and let be the -blowup of . Then

and if , then

Proof.

Let , where . Note that . Let be a proper coloring of using colors. Let , …, be pairwise disjoint sets of colors. For each non-leaf vertex of of depth , color properly using the colors in . Observe that this gives a proper coloring of using at most colors, and thus .

Suppose now that and consider a proper coloring of . Let be a path in from its root to one of the leaves and let be a coloring of constructed as follows. Suppose that we already selected , …, for some . Let denote the set of progenitors of level at least of the vertices of . Since and uses at least distinct colors on , there exists such that is different from the colors of all vertices of . We define be the child of in corresponding to , and set .

Note that is a proper coloring of such that for each , assigns vertices in pairwise distinct colors. Consequently, is a proper coloring of the strong -blowup of , and thus (and ) uses at least distinct colors by Observation 11. We conclude that . ∎

For positive integers and , let denote the -blowup of , i.e., the complete -partite graph with parts of size . Let us summarize the results of this section in the special case of the graph with planar.

Corollary 13.

Let be a planar graph with vertices and let be a positive integer. Let . The graph has vertices. If is -colorable, then , and otherwise . Furthermore, every -connected graph appearing as a minor in is -apex.

Proof.

Note that , and the last vertices of form an independent set. The claims follow from Lemma 10 (with , and , using the fact that every minor of a planar graph is planar) and Lemma 12 (with , and ). ∎

2 Non-approximability

The main non-approximability result is a simple consequence of Corollary 13 and NP-hardness of testing -colorability of planar graphs.

Proof of Theorem 3.

Suppose for a contradiction that there exists such a polynomial-time algorithm , taking as an input an -minor-free graph and returning an integer such that .

Let be a planar graph, and let . By Corollary 13, the size of is polynomial in the size of and is -minor-free. Furthermore, if is -colorable, then , and otherwise . Hence, if is -colorable, then the value returned by the algorithm applied to is less than , and if is not -colorable, then the value returned is at least . This gives a polynomial-time algorithm to decide whether is -colorable.

However, it is NP-hard to decide whether a planar graph is -colorable [7], which gives a contradiction unless . ∎

Note that the graphs used in the proof of Theorem 3 have large cliques (of size greater than ). This turns out not to be essential—we can prove somewhat weaker non-approximability result even for graphs with clique number . To do so, we need to apply the construction with both and being triangle-free. A minor issue is that testing -colorability of triangle-free planar graphs is trivial by Grötzsch’ theorem [8]. However, this can be easily worked around.

Lemma 14.

Let denote the class of graphs such that all their -connected minors with at least vertices are planar. The problem of deciding whether a triangle-free graph is -colorable is NP-hard.

Proof.

Let be the Grötzsch graph ( is a triangle-free graph with vertices and chromatic number , and all its proper subgraphs are -colorable). Let be a graph obtained from by removing any edge . Note that is -colorable and the vertices and have the same color in every -coloring.

Let be a planar graph. Let be obtained from by replacing each edge of by a copy of whose vertex is identified with and an edge is added between and (i.e., is obtained from by a sequence of Hajós sums with copies of ). Clearly, is triangle-free, it is -colorable if and only if is -colorable, and . Furthermore, is obtained from a planar graph by clique-sums with on cliques of size two, and thus every -connected minor of is either planar or a minor of (and thus has at most vertices). Hence, belongs to .

Since testing -colorability of planar graphs is NP-hard, it follows that testing -colorability of triangle-free graphs from is NP-hard. ∎

We also need a graph which is triangle-free and its fractional chromatic number is large and equal to its ordinary chromatic number. We are not aware of such a construction being previously studied; in Appendix, we prove the following.

Lemma 15.

For every positive integer , there exists a triangle-free graph with vertices such that .

Theorem 6 now follows in the same way as Theorem 3, using the graphs from Lemma 15 instead of cliques.

Proof of Theorem 6.

Suppose for a contradiction that there exists such a polynomial-time algorithm , taking as an input an -minor-free graph with and returning an integer such that . Recall that . Let be the graph from Lemma 15. Let and let be the -blowup of . Let be the class of graphs such that all their -connected minors with at least vertices are planar.

Consider a triangle-free graph , and let . By Lemma 10, the size of is polynomial in the size of . Consider any -connected minor of . By Lemma 10, there exists a set of size at most such that is a minor of . Since is -connected and , we conclude that either or is planar. Consequently, , and thus does not contain as a minor. Furthermore, .

Recall that . By Lemma 12, if is -colorable, then , and otherwise . Hence, if is -colorable, then the value returned by the algorithm applied to is less than , and if is not -colorable, then the value returned is at least . This gives a polynomial-time algorithm to decide whether is -colorable, in contradiction to Lemma 14 unless . ∎

3 Approximation algorithms

Let us now turn our attention to the additive approximation algorithms. The algorithms we present use ideas similar to the ones of the -approximation algorithm of Demaine et al. [3] and of the additive approximation algorithms of Kawarabayashi et al. [10] and Demaine et al. [2]. We find a partition of the vertex set of the input graph into parts and such that has bounded tree-width (and thus its chromatic number can be determined exactly) and has bounded chromatic number, and color the parts using disjoint sets of colors. The existence of such a decomposition is proved using the minor structure theorem [14], in the variant limiting the way apex vertices attach to the surface part of the decomposition. The proof of this stronger version can be found in [5]. Let us now introduce definitions necessary to state this variant of the structure theorem.

A tree decomposition of a graph is a pair , where is a tree and is a function assigning to each vertex of a subset of , such that for each there exists with , and such that for each , the set induces a non-empty connected subtree of . The width of the tree decomposition is , and the tree-width of a graph is the minimum of the widths of its tree decompositions.

The decomposition is rooted if is rooted. For a rooted tree decomposition and a vertex of distinct from the root, if is the parent of in , we write and . If is the root of , then and . The torso expansion of a graph with respect to its rooted tree decomposition is the graph obtained from by adding edges of cliques on for all .

A path decomposition is a tree decomposition where is a path. A vortex with boundary sequence , …, and depth is a graph with a path decomposition such that and for . An embedding of a graph in a surface is -cell if each face of the embedding is homeomorphic to an open disk.

Theorem 16.

For every non-negative integer and a -apex graph , there exists a constant such that the following holds. For every -minor-free graph , there exists a rooted tree decomposition of with the following properties. Let denote the torso expansion of with respect to . For every , there exists a set of size at most with , a set of size at most , and subgraphs , , …, of for some such that

  • , and for , the graphs and are vertex-disjoint and contains no edges between and ,

  • the graph is -cell embedded in a surface in that cannot be drawn,

  • for , is a vortex of depth intersecting only in its boundary sequence, and this sequence appears in order in the boundary of a face of , and for ,

  • vertices of have no neighbors in , and

  • if is a child of in and , then .

Furthermore, the tree decomposition and the sets and subgraphs as described can be found in polynomial time.

Informally, the graph is a clique-sum of the graphs for , and contains a bounded-size set of apex vertices such that can be embedded in up to a bounded number of vortices of bounded depth. Furthermore, at most of the apex vertices (forming the set ) can have neighbors in the part of drawn in , or in the other bags of the decomposition that intersect . Note that it is also possible that is the null surface, and consequently .

We need the following observation on graphs embedded up to vortices.

Lemma 17.

Let be a surface of Euler genus and let be a non-negative integer. Let be a graph and let , , …, be its subgraphs such that , the subgraphs , …, are pairwise vertex-disjoint and contains no edges between them, is -cell embedded in , and there exist pairwise distinct faces , …, of this embedding such that for , intersects only in a set of vertices contained in the boundary of . If the graphs , …, have tree-width at most , then there exists a subset of vertices of such that is planar and the graph has tree-width at most .

Proof.

If is the sphere, then we can set ; hence, we can assume that . Let be the graph obtained from by, for , adding a new vertex drawn inside and joined by edges to all vertices of . Note that has a -cell embedding in extending the embedding of . Let be a subgraph of such that the embedding of in inherited from the embedding of is -cell and is minimum. Then has only one face, since otherwise it is possible to remove an edge separating distinct faces of , and has minimum degree at least two, since otherwise we can remove a vertex of degree at most from . By generalized Euler’s formula, we have , and thus contains at most vertices of degree greater than two. By considering the graph obtained from by suppressing vertices of degree two, we see that is either a cycle (if ) or a subdivision of a graph with at most edges.

Let be the set of vertices of of degree at least three and their neighbors. We claim that each vertex of is adjacent in to only two vertices of . Indeed, suppose that a vertex has at least three neighbors in belonging to . Let and be the neighbors of in , and let be a vertex distinct from and adjacent to in . The graph has two faces, and by symmetry, we can assume that the edge separates them. Since , the vertex has degree two in , and thus the embedding of is -cell, contradicting the minimality of .

Let be the set of vertices of at distance at most from a vertex of degree greater than two. Note that . For , let denote the set of vertices of that are in adjacent to a vertex of . We claim that . Indeed, suppose for a contradiction that and consider a path in such that belongs to (vertices , , …, , , …, have degree two in , since ). If , then let be a path in of length two between and through a vertex of ; note that , since has at least neighbors in and belongs to by the previous paragraph and . If , then there exists a vertex ; we let be a path of length at most four between and passing only through their neighbors in and possibly through . By symmetry, we can assume that the edge separates the two faces of , and the graph if or if contradicts the minimality of .

Let ; we have . Let . Clearly, is planar. Let be the graph obtained from by adding vertices of as universal ones, adjacent to all other vertices of . The tree-width of is at most . Note that each vertex of has degree two in , and thus is a subgraph of a subdivision of . We conclude that the tree-width of is also at most . ∎

Note that the set can be found in polynomial time. For the clarity of presentation of the proof we selected with minimum; however, it is sufficient to start with an arbitrary inclusionwise-minimal subgraph with exactly one -cell face (obtained by repeatedly removing edges that separate distinct faces and vertices of degree at most ) and repeatedly perform the reductions described in the proof until each vertex of is adjacent in to only two vertices of and until for .

For positive integers and , we say that a rooted tree decomposition of a graph is -restricted if for each vertex of , the subgraph of the torso expansion of induced by is planar, , and each vertex of has at most neighbors in that belong to . Using the decomposition from Theorem 16, we now partition the considered graph into a part of bounded tree-width and a -restricted part.

Theorem 18.

For every positive integer and a -apex graph , there exists a constant with the following property. The vertex set of any -minor-free graph can be partitioned in polynomial time into two parts and such that has tree-width at most and has a -restricted rooted tree decomposition. Additionally, for any such graph and positive integers , there exists a constant such that if is -minor-free and does not contain as a subgraph, then and can be chosen so that has tree-width at most and has a -restricted rooted tree decomposition.

Proof.

Let be the constant from Theorem 16 for , and let be the maximum Euler genus of a surface in that cannot be embedded. Let and .

Since is -minor-free, we can in polynomial time find its rooted tree decomposition , its torso expansion , and for each , find , , , , …, , and as described in Theorem 16. Let be the set of vertices obtained by applying Lemma 17 to , , …, ; i.e., is planar and the graph has tree-width at most . When considering the case that does not contain as a subgraph, let be the set of vertices of that have at least neighbors in belonging to (and thus to ); otherwise, let . Since there are at most ways how to choose a set of neighbors in and no vertices can have the same set of neighbors in , we have . Let .

We define . Note that . Consequently, is obtained from by adding and some of the vertices of , and consequently has tree-width at most when considering the case that does not contain as a subgraph and at most otherwise. The graph is a clique-sum of the graphs for , and thus the tree-width of is also at most or .

Let , and consider the graph . For , let . Then is a rooted tree decomposition of such that for every , the graph is planar, all vertices of adjacent in to a vertex of belong to (and thus there are at most such vertices), and when considering the case that does not contain as a subgraph, each vertex of has at most neighbors in belonging to .

Note that can contain vertices not belonging to , and thus can have size larger than , and the tree decomposition is not necessarily -restricted. However, by the condition (e) from the statement of Theorem 16, the vertices of can only be contained in the bags of descendants of which are disjoint from , and thus we can fix up this issue as follows.

If is a child of and , we say that the edge is skippable; note that in that case . For each vertex of , let be the nearest ancestor of such that the first edge on the path from to in is not skippable. Let be the rooted tree with vertex set where the parent of each vertex is . Observe that is a tree decomposition of . Furthermore, denoting by the child of on the path from to in , note that if a vertex is contained in , then , and since the edge is not skippable, the condition (e) from the statement of Theorem 16 implies that .

Hence, letting for each vertex of , we conclude that is a rooted tree decomposition of which is -restricted, and when considering the case that does not contain as a subgraph, the decomposition is -restricted. ∎

Let us now consider the chromatic number of graphs with a -restricted tree decomposition.

Lemma 19.

Let and be positive integers. Let be a graph with a -restricted rooted tree decomposition . The chromatic number of is at most . Additionally, if is triangle-free, then the chromatic number of is at most .

Proof.

We can color using colors, starting from the root of the tree decomposition, as follows. Suppose that we are considering a vertex such that is already colored. Since , this leaves at least other colors to be used on . Hence, we can extend the coloring to by the Four Color Theorem.

We can also color using colors, starting from the root of the tree decomposition, as follows. For each vertex of , at most colors are used on its neighbors in , leaving with at least available colors not appearing on its neighbors. Since is planar, we can color it from