Improved Approximation for Maximum Edge Colouring Problem

10/25/2019 ∙ by L. Sunil Chandran, et al. ∙ 0

The anti-Ramsey number, ar(G, H) is the minimum integer k such that in any edge colouring of G with k colours there is a rainbow subgraph isomorphic to H, i.e., a copy of H with each of its edges assigned a different colour. The notion was introduced by Erdös and Simonovits in 1973. Since then the parameter has been studied extensively in combinatorics, also the particular case when H is a star graph. Recently this case received the attention of researchers from the algorithm community because of its applications in interface modelling of wireless networks. To the algorithm community, the problem is known as maximum edge q-colouring problem. In this paper, we study the maximum edge 2-colouring problem from the approximation algorithm point of view. The case q=2 is particularly interesting due to its application in real-life problems. Algorithmically, this problem is known to be NP-hard for q> 2. For the case of q=2, it is also known that no polynomial-time algorithm can approximate to a factor less than 3/2 assuming the unique games conjecture. Feng et al. showed a 2-approximation algorithm for this problem. Later Adamaszek and Popa presented a 5/3-approximation algorithm with the additional assumption that the input graph has a perfect matching. Note that the obvious but the only known algorithm issues different colours to the edges of a maximum matching (say M) and different colours to the connected components of G ∖ M. In this article, we give a new analysis of the aforementioned algorithm leading to an improved approximation bound for triangle-free graphs with perfect matching. We also show a new lower bound when the input graph is triangle-free. The contribution of the paper is a completely new, deeper and closer analysis of how the optimum achieves a higher number of colours than the matching based algorithm, mentioned above.

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1 Introduction

A -edge colouring of a graph is a function . Note that doesn’t need to be a proper colouring of the edges, i.e., edges incident to the same vertex may receive the same colour. A subgraph of is called a rainbow subgraph (heterochromatic subgraph) with respect to a -edge colouring if all the edges of are coloured distinctly. For a pair of graphs and the anti-Ramsey number, , denotes the minimum number of colours such that in any -edge colouring of there exists at least one subgraph isomorphic to which is a rainbow subgraph. Equivalently if is the maximum possible number of colours in an edge colouring of such that there exists no rainbow subgraph isomorphic to with respect to then . We call the first parameter of , as the input graph and the second parameter as the pattern graph.

The notion, anti-Ramsey number, was introduced by Erdös and Simonovits in 1973 Erdös et al. (1975). Most of the initial research on this topic focused on complete graphs () as the input graph and pattern graphs that possesses certain nice structure, for example, path, cycle, complete graph etc. The exact expression of , when the pattern graph is a path of length (), was reported in the article written by Simonovits and Sós Simonovits and Sós (1984). Whereas the simple case of the pattern graph is a cycle of length () took years to get solved completely. It was proved by Erdös, Simonovits and Sós that Erdös et al. (1975). In the same paper it was conjectured that for . The conjecture was verified affirmatively for the case by Alon Alon (1983). Later it was studied by Jiang and West Jiang and West (2003). Almost thirty years after it was conjectured, Montellano-Ballesteros and Neumann-Lara reported proof of the statement in 2005 Montellano-Ballesteros and Neumann-Lara (2005). A lower bound considering the pattern graph as the clique of size () was reported in Manoussakis et al. (1996). Schiermeyer and Montellano-Ballesteros together with Neumann-Lara independently reported the exact value of Schiermeyer (2004); Montellano-Ballesteros and Neumann-Lara (2002). In the same article Schiermeyer also studied the case when pattern graph is a matching. Haas and Young later studied the case when pattern graph is a perfect matching Haas and Young (2012). A tighter bound on matching was reported in the article by Fujita et al. Fujita et al. (2009). The article by Jiang and West reported bounds on when is a tree Jiang and West (2004). Jiang also derived an upper bound on when is a complete subdivided graph relating the parameter with Turán number, that is the maximum cardinality of edges of an -vertex graph that does not contain a subgraph Jiang (2002a).

The study of anti-Ramsey number was not entirely restricted to the case when the input graph is a complete graph. Axenovich et al. studied the case when the input graph is a complete bipartite graph Axenovich et al. (2004). A -round variant of anti-Ramsey number was introduced and studied in Blokhuis et al. (2001).

In this paper, we consider the pattern graph as the claw graph, i.e. the star graph with exactly 3 leaves, denoted by . Study of anti-Ramsey number where the pattern graph is the claw or more generally the star graph was initiated in the work of Manoussakis et al. Manoussakis et al. (1996). The bound was later improved in Jiang (2002b). In the same article exact value of the bipartite variant of the problem was also reported. Gorgol and Lazuka computed the exact value of when is with an edge added to it Gorgol and Lazuka (2010). Montellano-Ballesteros relaxed the condition on input graph and considered any graph as input in their study Montellano-Ballesteros (2006). The study of anti-Ramsey number with claw graph as pattern graph was revisited recently due to its application in modelling channel assignment in a network equipped with a multi-channel wireless interface Raniwala and Chiueh (2005). They introduced the problem as maximum edge -colouring problem, thus initiating the exploration of the algorithmic aspects of this parameter, .

For a graph , an edge -colouring of is an assignment of colours to edges of such that no more than distinct colours are incident at any vertex. An optimal edge -colouring is one which uses the maximum number of colours. It is easily seen that the number of colours in maximum edge -colouring of is .

In Adamaszek and Popa (2016), it was reported that the problem is -hard for every . Moreover, they showed that it is hard to approximate within a factor of for every , assuming the unique games conjecture. A simple -factor algorithm for maximum -colouring problem was reported in Feng et al. (2009). A description of the algorithm is provided in Algorithm 1. Henceforth we refer this algorithm as the matching based algorithm. In a recent article Adamaszek and Popa (2016), authors reported a approximation factor for the same algorithm assuming that the input graphs have a perfect matching. Approximation bounds for the matching based algorithm when the input graph has certain degree constraints were reported in Chandran et al. (2018). A fixed-parameter tractable algorithm was reported for the case in Goyal et al. (2013).

In the present article, our focus is on the case when , and when the graph has a perfect matching. It is worth mentioning here, although Montellano-Ballesteros reported bounds on , their expression is not enough to draw any inference in this particular scenario. Their technique is useful for deriving bounds when the input graph has certain regular structures such as complete graph, complete -partite graph, hypercube etc.

0:  A graph which has perfect matching.
1:  Compute the maximum matching of .
2:  Assign distinct colour to each edge of .
3:  Assign a new colour to each component of .
4:  return  The colour assigned graph.
Algorithm 1 Matching based algorithm for edge -colouring

2 Key Notation and Main Result

Throughout this article (except possibly the last section), we consider to be a graph which has a perfect matching . We use to denote the components of , where is the number of such components.

Let be an optimal edge -colouring of using colours . Let and denote the colours used in the matching , and those not used in the matching respectively. Clearly . We call colours in as matching colours and colours in as non-matching colours. For an edge , we denote the colour assigned to in colouring by . For a vertex , the colour assigned to the matching edge at is denoted by .

For a colour , let denote the subgraph spanned by the edges coloured . We note that, in an optimal colouring , is connected for all colours , since otherwise we can increase the number of colours. For a non-matching colour , is a subgraph of a unique component . For convenience, we often refer to a connected subgraph spanned by edges of a colour as colour component. Subgraphs corresponding to matching colours are called matching colour components, while those corresponding to non matching colours are called non matching colour components. With the notation as discussed, the following are the main contributions of this paper:

Theorem 1.

Let be a graph having a perfect matching . Let denote the number of colors in an optimal max -edge coloring of . Then:

  1. Adamaszek and Popa (2016).

  2. when is additionally triangle-free.

Corollary 1.

Algorithm 1 guarantees an approximation factor for graphs with perfect matching and an approximation factor of for triangle free graphs with perfect matching.

3 Overview

We start with an overview of some structural observations about an optimal colouring which help us establish the approximation factor. Let be one of the components of and let and be connected subgraphs of spanned by some non-matching colours and . Note that . Our next lemma shows that any - path contains two distinct vertices and such that the matching edges incident at and have the same colour. Let us call as the colour repetition pair.

Lemma 1.

Let be a path in such that and . Then there exist such that .

Proof.

For the lemma is obvious. Let . Let be maximal so that the path is monochromatic. If then and satisfy the assertion of the lemma. Otherwise, clearly and . Hence, the lemma follows by applying induction on the path . ∎

We observe that any - path satisfies the conditions in Lemma 1 at its end-points. This is because if is an edge coming out of the non-matching colour component , with and , has to be the same as . Now suppose has such non-matching colours. Then we can find at least paths connecting all of these colour components. If these paths are all disjoint (as in Figure 1), we would have colour repetition pairs by applying the Lemma 1 on each of these paths. Intuitively, a lot of such pairs should imply “repetition” of colours among matching edges, and hence help us bound the number of distinct matching colours. In later sections, we try to quantify the repetition implied by these colour repetition pairs. Our focus here and in the next section is to exhibit a large number of such pairs.

Figure 1: Circles denote subgraphs formed by non-matching colours. The figure shows the case when all the subgraphs can be connected by disjoint paths between them

Evidently, the non-matching components may not be nicely connected as in Figure 1. In the Section 4, we show that one can still find distinct colour repetition pairs from a component containing non-matching colours. In the Section 5

, we estimate quantitatively the repetition in matching colours. The result is given in Lemma

8. We use it to prove our main theorem in the same section.

4 Colour Repetition Pairs

We begin by generalizing the Lemma 1 to rooted trees, where the endpoint conditions in the Lemma 1 are satisfied at the root and the leaves of the tree. For a tree , let and denote the root and the set of leaves of respectively. Further, we assume that has a depth first ordering, where denotes the index of vertex in the ordering. We assume that when is a descendant of in , we have . Thus the root has minimum index. We use the depth first ordering to define an ordering on vertices of , where if . We will use to denote with . Note that this ordering contains the usual hierarchical ordering with root as the maximum element. This is same as the post order traversal of trees found in the literature. We are now in a position to state our next lemma.

Lemma 2.

Let , , be a rooted tree in with and . Further for , the colour of the edges in incident at is the same as the matching colour at . Then we have a set of pairs of vertices of such that:

  1. For , we have .

  2. for all .

  3. for all .

  4. The path is monochromatic, with edges coloured for all .

  5. For an internal vertex in the path , we have for .

  6. For and , the paths and do not share an internal vertex.

We call the tuples in as the repetition pairs. Statements (b) and (c) can be seen as generalization of the notion of colour repetition pairs for paths in a tree. As in the proof of Lemma 1, for a repetition pair , was the vertex with the same incident matching colour as but with an index to the right of closer to root. Similarly in a tree, we have ’s “closer” to the root in depth first ordering than ’s. The statements (d)-(f) establish further structural properties of these repetition pairs that are needed subsequently.

Proof of Lemma 2.

We first prove the Lemma for the case when is monochromatic. Let be the colour of all the edges of . From the colouring constraints that satisfies at the root and the leaves, it follows that for . For , define to be the closest vertex to on the path such that . Observe that exists for each as the root satisfies . Let . Then is trivially seen to satisfy statements (a)-(e). To prove (f), we assume the contrapositive, i.e, there exist such that , and the paths - and - in have a common internal vertex. Let be the common internal vertex. Then and are descendants of , while are ancestors of . But and are the first vertices on and respectively with matching colour as . Since the paths and share the sub-path , it must be that . But then their matching partners must also be the same, i.e., , which contradicts the assumption that . This contradiction proves (f).

Next consider the case when is not monochromatic. We proceed by induction. Let be a vertex of minimum height, such that witnesses edges of two colours in . Let and be the colours incident at . Without loss of generality, let be the colour of the matching edge at , i.e., . Let be the subtree of rooted at . Let and be the subtrees of consisting of edges of colours and respectively. We consider three cases, viz:

is non-empty and is also non-empty: Let denote the leaves in . Now is monochromatic and satisfies the end point constraints required by the lemma. Thus we have the set with of pairs of vertices of satisfying the statements (a)-(f) because of the previous case. Let be the tree obtained from by deleting all descendants of in . Let be the set of leaves in . Notice that . Now, by induction hypothesis, we have a set of pairs of vertices of satisfying (a)-(f). Thus both and satisfy the statements (a)-(f) with respect to the trees and . We claim that satisfies (a)-(f) for the tree . It is trivially seen that (a)-(e) hold. To see that (f) also holds we observe that only vertex that is possibly shared between and is , and is not an internal vertex of any path in where .

is non-empty and is empty: Let and let be the subpath of such that has degree more than in and height of is minimum among all the vertices with this property. Let the subtree rooted at which contains be . Let be the tree obtained from by removing all descendants of in (possibly ). Since is not an internal vertex of we see that satisfies the end point constraints required by the lemma. Let leaves of be . Observe that . Now, by induction hypothesis, we have a set of pairs of vertices of satisfying (a)-(f) and a set of pairs of vertices of satisfying (a)-(f). We claim that satisfies (a)-(f) for the tree . It is trivially seen that (a)-(e) hold. To see that (f) also holds we observe that only vertex that is possibly shared between and is , and is not an internal vertex of any path in where .

is empty: Clearly, must be non-empty in this case. Let denote the leaves in . Note that the colour of incident edges at in is not the same as the matching colour at , and hence we cannot define the pairs as in the monochromatic case. Let be the leaf such that is maximum in . For each define as the closest vertex to on the path such that . Again, observe that exists for all because . Define . Statements (a)-(e) follow for (w.r.t the tree ) almost by definition of function . Statement (f) can be proven in the following way. There is a unique path between any two pairs of vertices. Then at some point, the two paths have a vertex in common. Thus, the pairs satisfy (a)-(f) for the tree , except that the number of pairs is one short of the number of leaves in . We recover the deficit in the remaining tree. As before, let be the tree obtained from by removing the descendants of in . Let denote the leaves in . Since was empty and witnesses two colours, we have and it is incident with an -coloured edge in . By induction hypothesis, we have pairs of vertices of with satisfying statements (a)-(f). Again, we claim that satisfy the requirements of the lemma for . Statements (a)-(e) are easily verified. For (f), we note that is a leaf of and hence is not an internal vertex of any path in for . Thus, (f) also holds for the pairs . Finally, we note that , as we compensate the loss of leaf in with an extra leaf in since and . ∎

Lemma 2 can be easily extended to forest consisting of rooted trees. Let be a forest containing rooted trees. Let denote the set of roots of trees in the forest, and denote the set of leaves in the forest . We define a partial order on the forest which restricts to the (total) order (as defined earlier) on each component tree . If two vertices belong to different component trees, they are incomparable under . As before denotes , but . We now state the extension of Lemma 2 to forests.

Lemma 3.

Let be a forest in with and . Suppose that for each , the colour of edges in incident at is the same as the matching colour at . Then, there exists set of pairs of vertices of satisfying:

  1. For , we have .

  2. for all . In particular, the path exists for all .

  3. for all .

  4. The path is monochromatic, with edges coloured for all .

  5. For an internal vertex in the path , we have for .

  6. For and , the paths and do not share an internal vertex.

Proof.

The proof follows by taking union of pairs satisfying Lemma 2 for each component tree in the forest. ∎

Let us return to the question of finding color repetition pairs in a component containing non-matching color components. We could use Lemma 1, when the non-matching components can be connected using pairwise disjoint paths as in Figure 1. Lemma 3 allows us to exhibit repetition pairs, as long as we have a non-matching color component from which we can reach all other non-matching color components without passing through other colour components. Figure 2 illustrates such an arrangement, and the corresponding forest to which we can apply Lemma 3. Note that root of both trees in the forest is in . The path doesn’t intersect with any other colour component other than and . For each , we can construct a - path avoiding vertices of other colour component where .

Figure 2: Non-matching colour components which can be connected in a tree-like fashion from a component. The dashed lines denote paths in , where we have not explicitly marked all vertices outside the non-matching components. The requisite number of repetition pairs are obtained by applying Lemma 3 to the forest consisting of dashed trees.

Our final technical effort in this section is to show that we can find repetition pairs in a component with non-matching colours, even if we do not have a non-matching colour component, from which all other non-matching components are accessible without passing through some vertex of an intermediate component. The situation is illustrated in Figure 3.

Figure 3: There is no component which can directly reach all other components. Choosing as root, the first level forest consists of two trees indicated by thin dashed lines. Choosing leaf components in as root components, we reach other components using forest denoted by thick lines.

4.1 Cascading sequence of forests

We call an ordered pair

of rooted trees to be a cascading pair if and are vertex disjoint or and . We call a cascading sequence of forests if (i) for and (ii) For, and , the pair is a cascading pair for .

Let be a cascading sequence of forests. We use the notation to denote the vertices in the collection . We call to be an internal vertex of if is an internal vertex of some (at most one) forest . The notation will denote the set of internal vertices in the collection . We record the following easy observation as a lemma.

Lemma 4.

Let be a cascading sequence of forests. Then for any two distinct vertices , there exists at most one forest such that contains the - path.

Definition 1 (Order on cascading sequence of forests).

Let be an cascading sequence of forests. We define the partial order on as: if and only if for some (at-most one) . Otherwise and are incomparable under . Transitive closure of this relation is the order we consider. (By abuse of notation, we will use for transitive closure of this relation.)

For a cascading sequence of forests and vertices and , the notation denotes the path if there exists a forest (at-most one) such that contains - path. We now state a version of Lemma 3 for a cascading sequence of forests.

Lemma 5.

Let be a cascading sequence of forests in . Then, there exist pairs of vertices in where , satisfying:

  1. For , we have .

  2. for all . In particular for all , the path exists.

  3. for all .

  4. The path is monochromatic, with edges coloured for all .

  5. For an internal vertex in the path , we have for .

  6. For and , the paths and do not share an internal vertex.

Proof.

The proof follows by taking union of pairs satisfying Lemma 3 for each forest in the collection . The properties (a)-(e) are easily verified. The property (f) holds for the union of pairs because the limited intersection of trees between different forests implies that a vertex can be an internal vertex in at most one forest. ∎

We will exhibit a cascading sequence of forests in which essentially link up all the non matching color components.

Lemma 6.

There exists a collection of cascading sequence of forests in such that:

  1. .

  2. for all non-matching colour components .

  3. for all and for all , the colour of edges in incident at is the same as the matching colour at .

Proof.

For each , we will exhibit a cascading sequence of forests in satisfying and for all non-matching color components contained in . Then it is easily seen that satisfies the requirements of the lemma. Let be an arbitrary but fixed component of . Let (where ) be the non-matching color components contained in . Let . Let be a maximal forest with i) , ii) for iii) is maximal under this condition. Suppose forests have been constructed such that they form a cascading sequence. Define , and let . Intuitively, is the indices of non-matching color components already visited by the collection , while is the indices which have not been visited so far. Let be the maximal forest in such that , and . Note that by connectedness of , is non-empty for . Thus unless . Let be minimal such that . Then is a cascading sequence of forests in with . To complete the proof, we need to show that . This follows from the fact that for each , there exists minimum such that . Then . In other words, for each , there exists a forest which has a leaf in the component . The claim follows. ∎

The above proof can be intuitively understood using Figure 3, where the forest consists of two dashed trees, and the forest consists of the two solid trees. The collection is the cascading sequence for the arrangement in Figure 3.

5 Repetition Content

In the previous section we exhibited pairs of vertices having the same matching colour incident at them. Intuitively, a lot of such pairs should imply certain amount of repetition of colours among matching edges. In this section we attempt to quantitatively estimate the repetition.

Call a set -monochromatic if all the edges in incident with vertices in have the same colour. Let denote the number of edges in which are incident with a vertex in . For an -monochromatic set , let repetition content of , denoted by be defined as . For , we observe that . The lower bound is attained when every vertex in has its matching partner also in . The upper bound is obtained for sets where no edge in has both vertices in . Following is an easy consequence of the definition of repetition content.

Lemma 7.

For (the set of matching colours in ), let be a collection of -monochromatic sets such that matching edges incident with are coloured . Then .

5.1 Repetition by matching colours

We recall the notation from Section 2. Throughout the remainder of this section let denote a cascading sequence of forests satisfying Lemma 6. Let denote the vertex set . Then by Lemma 5, there exists set of repetition pairs of vertices with , where is the number of non-matching colour components in component of .

We consider a partition of the pairs according to the incident matching colour. Let for . Let be the repetition pairs consisting of matching pairs, i.e. . Let and denote the projections of the set on first and second coordinate. Let . We use as the repetition associated with colour . We call a colour as high if . Otherwise, we call the colour as low. Let and denote the high and low colours respectively.

Lemma 8.

Let the sets and be as defined in this section. Then we have,

  1. for all .

  2. for all .

  3. If then .

  4. For , all vertices in have their matching partner in also in . Furthermore, has a unique maximum element with respect to ordering .

  5. for .

Proof.

From part (a) of Lemma 5, we have . Let be a maximal element in under . Clearly, by part (b) of Lemma 5, . Thus . This proves claim (a). Since for all , it follows using claim (a) that for all . The claim (b) is thus proved. Claim (c) also follows from claim (b) and definition of . For claim (d), observe that for , we have by claim (c), , and hence from the definition of low. Also by claim (b), we have . Since is an integer, we conclude , and hence . Since no element can be maximum, together with claim (a), we conclude and thus . Clearly then is the unique maximal element in . Also since satisfies , each element in must have its matching partner in in . This proves claim (d). From claim (d) we see that or for . To prove claim (e), we rule out the possibility . Note that implies where . But then , which contradicts the assumption by claim (c). This proves claim (e). ∎

Lemma 9.

If is triangle free and we have either (a) , or (b) and there exist vertices in with such that the path has an interior vertex .

Proof.

From parts (d) and (e) of Lemma 8 we see that if , we must have . Let be the maximum element of according to the order . Clearly . Let be the matching partner of . Clearly as . Let be a vertex such that and let be the matching partner of . Recalling is the unique maximum element of , . Thus . We also must have . Since and , by part (e) of Lemma 5, and . Thus . Since is triangle free, one of the paths must have an internal vertex, thus satisfying condition (b) of the lemma. ∎

Lemma 10.

Let and for some . Then the paths and do not share an internal vertex.

Proof.

If possible, let be an internal vertex of both and . By part (e) of Lemma 5, and . This is a contradiction, as and are the only two colors incident at . Thus, the lemma is proved. ∎

Finally, we prove our main result:

Proof of Theorem 1.

We have . From Lemma 7, we have,

(1)

Moreover from Lemma 8(b) we have,

(2)

Observing that , from (1) and (5.1), we have,

(3)

Let . Note that the path contains at least one internal vertex for . From Lemma 10 and part (f) of Lemma 5 it follows that all these internal vertices are distinct. Thus, the collection has at least internal vertices. Since was chosen according to Lemma 6 and each non-matching color component contains at least two vertices, we have:

(4)

Substituting, in (3), we have:

(5)

Since each determines at least matching edges of the same color we have . Combining with (4) we have:

(6)

Equating the upperbounds on