Approximation bounds on maximum edge 2-coloring of dense graphs

10/01/2018 ∙ by L. Sunil Chandran, et al. ∙ ibm indian institute of science 0

For a graph G and integer q≥ 2, an edge q-coloring of G is an assignment of colors to edges of G, such that edges incident on a vertex span at most q distinct colors. The maximum edge q-coloring problem seeks to maximize the number of colors in an edge q-coloring of a graph G. The problem has been studied in combinatorics in the context of anti-Ramsey numbers. Algorithmically, the problem is NP-Hard for q≥ 2 and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than 1+1/q. The case q=2, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A 2-factor algorithm is known for general graphs, and recently a 5/3-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: "Find a maximum matching M of G. Give distinct colors to the edges of M. Let C_1,C_2,..., C_t be the connected components that results when M is removed from G. To all edges of C_i give the (|M|+i)th color." In this paper, we first show that the approximation guarantee of the matching based algorithm is (1 + 2/δ) for graphs with perfect matching and minimum degree δ. For δ> 4, this is better than the 5/3 approximation guarantee proved in AAAP. For triangle free graphs with perfect matching, we prove that the approximation factor is (1 + 1/δ - 1), which is better than 5/3 for δ> 3.

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1 Introduction

For a graph and an integer , the edge -coloring problem seeks to maximise the number of colors used to color the edges of , subject to the constraint that a vertex , is incident with edges of at most different colors. We note that this problem differs from the classical edge coloring, where all edges incident at a vertex have distinct color. The problem has also been found useful in modelling channel assignment in networks equipped with multi-channel wireless interfaces [8].

In a combinatorial setting, the number of colors used in maximum edge -coloring relates to anti-Ramsey number. For graphs and , the anti-Ramsey number denotes the maximum number of colors , such that in an edge coloring of with colors, all subgraphs of isomorphic to have at least two edges of the same color. Then, it is seen that the number of colors in maximum edge -coloring of is where is a star with vertices. Further details on anti-Ramsey numbers can be found in Erdös et. al [3]. Combinatorial bounds for edge -coloring for several classes of graphs are obtained in [7].

Recently, the maximum edge -coloring problem has also been studied from an algorithmic perspective. Anna and Popa [1], proved that the problem is NP-Hard for every . Moreover, they proved that, assuming the unique games conjecture, it cannot be approximated within a factor less than for every and assuming just , it cannot be approximated within a factor less than for every [2]. A -factor approximation algorithm (cf. Algorithm 1) for the maximum edge -coloring for general graphs was given by Feng et. al. in [4, 5, 6]. The algorithm of Feng et. al. is simple and intuitive and is given below (Algorithm 1). Hereafter we refer to this algorithm as the matching based algorithm. The same authors also showed that the problem is polynomial time solvable for trees and complete graphs. Adamaszek and Popa [1] showed that the matching based algorithm of Feng et. al. improves to a -factor algorithm for graphs with perfect matching.

  Let be the input graph.
  Let be a maximum matching in .
  Assign distinct color to each edge of .
  Assign a new color to each component of .
Algorithm 1 Matching Based Algorithm for edge 2-coloring [1]

1.1 Our Results:

Our contribution in this paper is to show that the approximation guarantee of the matching based algorithm (Algorithm 1) improves significantly for graphs with large minimum degree. Specifically, we obtain the following results:

Result 1: For graphs with perfect matching, the matching based algorithm has an approximation guarantee of where is the minimum degree. Recall that Adamaszek and Popa [1] proved an approximation guarantee of for this case. The approximation gurantee proved by us is better than that of Adamaszek and Popa for , and equals to theirs for .

Comment: For graphs with , Algorithm 1 is almost optimal. In fact it is an additive 1 approximation algorithm. If we further assume that is even, for the matching based algorithm is optimal. (This follows from result 1, noting that such graphs always have a perfect matching.)

Result 2: For triangle free graphs with perfect matching, we show a better approximation guarantee, namely . This is better than that of [1] for .

Result 3: For the general case we show that the matching based algorithm is a -factor approximation algorithm. Here where is the number of vertices in and is a maximum matching in . Considering the attempts of Adamaszek and Popa to get a better approximation guarantee for graphs with perfect matching, it is natural to consider this ratio. This factor is better than the previous known 2 factor for the general case, when .

From the technical looking approximation factor , we can easily get the following corollaries for interesting special cases:

  • Corollary 1 of Result 3: It is easy to see that where is the maximum degree of . (Consider the set of vertices spanned by the edges of . Since due to maximality of , all edges of should be adjacent to at least one vertex in , we get .) Therefore for -regular graphs, the approximation guarantee is . This is better than the 2 factor known for the general case [4, 5, 6], when . (Note that all -regular graphs need not have a perfect matching and thus 5/3 factor of [1] or the factor from our result 1, is not applicable.)

  • Coroallary 2 of Result 3: For graphs of minimum degree , we have an approximation factor of . To see this note that . Now substituting and , we see that the approximation factor is at most . For , this is better than the 2 factor known for the general case [4, 5, 6].

2 Notation and Preliminaries

Throughout this paper, we consider connected graphs with minimum degree . An edge -coloring of a graph with colors is a map , such that a vertex is incident with edges of at most two colors. Let denote the number of colors in the coloring returned by Algorithm 1 for the graph , and let be the number of colors in the maximum edge -coloring of .

2.1 Characteristic subgraph

Let be an edge -coloring of the graph . A subgraph of containing exactly one edge of each color in is called a characteristic subgraph of with respect to coloring . Note that the maximum degree of a characteristic graph is at most two.

Lemma 2.1.

For a graph and an edge -coloring of , there exists a characteristic graph which is disjoint union of paths.

Proof.

Let be a characteristic graph with minimum number of cycles. We claim that has no cycles. For sake of contradiction, suppose has a cycle. Let be one of the vertices in the cycle, and be its two neighbors in the cycle. Since , there is neighbor of in , which is not incident with an edge in . Now the edge must have the same color as or , say . Then is a characteristic subgraph with smaller number of cycles. Hence, did not have any cycles. ∎

In the remainder of the paper, we will tacitly assume that characteristic subgraphs do not contain cycles. The components of the characteristic subgraphs (which are paths) will be called characteristic paths.

3 The Case of Graphs with perfect matching

In this section, we derive some useful bounds for graphs with perfect matching. The proofs here also help to illustrate key ideas in simpler setting, which will be extended further in the proof of Theorem 4.1.

Theorem 3.1.

Let be an -vertex graph with perfect matching, where . Then, we have:

  1. . Furthermore, if is triangle-free then .

  2. for .

The above theorem yeilds the approximation factor of given by Anna and Popa in [1] for graphs with perfect matching (under added assumption of ). When graphs are also traingle-free, we get an improved approximation factor of .

Proof of Theorem 3.1.

Let be an optimal edge -coloring of , and let be a characteristic subgraph of with respect to coloring . Let , denote the vertices of with degree in . Let . Clearly . The number of colors in which is same as the number of edges in is given by . For , let denote the neighbors of through edges not in . Clearly for all .

Claim: For , . Let and be the colors incident at vertices and respectively. If then the color of belongs to . By the definition of characteristic graph, if is not an edge in , then . Thus edge cannot get a color if . We infer that .

From the above claim, it follows that for . Consider the bipartite graph with bipartition and edge set of , given by . We show that for and for . Assume . Now, there are at most two colors incident at in , say colors and . If is a neighbor of in , we must have or . Since there are at most two vertices such that , if follows that has at most neighbors in . For , let be the unique neighbor of in . Let be the color of edge . Notice that since it is an edge of . Also for . Thus is not incident with -colored edges in . Thus edges incident on in must have the same color, and by the previous arguments, there can be at most two of them. Thus for . Counting the edges across the bipartition in two ways we have:

(1)

The result now follows from some algebra, as shown below.

(2)
From , using we see that . Therefore we have .
Substituting, we get
(3)

This proves the claimed approximation factor in part (i) of the theorem for graphs with perfect matching. If the graph is further assumed to be triangle-free, we prove that a vertex can have atmost one edge of a given color incident on it in . Let and let be one of the colors incident at in . Then, the only possible -colored edges incident at in are edges where with . Let be such that . Then is an edge in . As is triangle-free we conclude at most one of is a neighbor of . Thus is incident with at most one edge of color in . Now it follows that, in this case, we have for and for . The Equation (1) can now be written as:

(4)

It follows that . Making similar substitutions in Equation (3), we get

(5)

This proves the approximation factor claimed for triangle-free graphs in part (i) of the theorem.

To prove part (ii), observe that by Dirac’s theorem, a graph with has a hamilton cycle, and hence a maximum matching of size . Thus . Further from part (i), we have . Thus since is an integer we get . Note that if we assume that is even, and , this proves that the algorithm is optimal.

4 Result for general graphs

In this section, we prove our main result, which is the following:

Theorem 4.1.

Let be an -vertex connected graph with . Let be a maximum matching of . Then where , with being the ratio of number of vertices to the size of maximum matching of the graph.

Proof.

Let be an optimal coloring of using colors. Let be a charactersistic subgraph of with respect to coloring , with the maximum number of characteristic paths. We will say a vertex is an internal vertex of , if it is an internal vertex of one of the characteristic paths. Similarly, a vertex will be called a terminal vertex of if it is a terminal vertex of one of the characteristic paths. In the proof of Theorem 3.1, we bound the number of edges in the characteristic subgraph (which is same as the number of colors in the coloring) by . In the case of graphs with perfect matching, is a lower bound on the number of colors returned by the Algorithm 1, and thus intuitively, the term was the “excess”. We tried to upper bound this excess in the previous proof. However in the general case,

could be a gross over-estimate of the maximum matching, so the previous strategy does not work. Instead, we consider the following excess: We pick a matching

from within by selecting alternate edges in each characteristic path, starting with the first edge in each path. Let be the number of unselected edges. Then .

The remainder of the proof attempts to upper bound the excess term . In fact we show that from which the theorem immediately follows. Let be the set of vertices consisting of left endpoint of each unselected edge (see Figure 1). Thus we have . First we note that is an independent set in . This is because vertices in are mutually non-adjacent internal vertices of , and hence have mutually disjoint incident colors. For each , choose a set of edges incident at , which are not present in (this is possible, as each vertex has at most two neighbors in ). Let us call these edges as special edges. Let and be sets of vertices of which are incident with , and of these special edges. Let . Since, the vertices in are incident with mutually disjoint sets of colors, a vertex in is incident with at most two special edges emanating from . Thus , or . Counting the special edges across the bipartition we have:

(6)

Moreover, as , we can rewrite the above equality as , and thus,

(7)

Figure 1: Characterisitic paths are shown on the left. The unselected edges are marked with , and corresponding vertices in are indicated by a box. The special edges are indicated with dashed lines. The set is indicated by the ellipse.

To obtain a bound on , we consider the neighbors of vertices in which are not in . Note that each vertex has at least neighbors outside . Let be the set of terminal vertices of the characteristic subgraph and be the set of internal vertices of , which are not in . Let denote the set of vertices which are not incident with an edge in . Note that . This is because a vertex incident with an edge in can only receive special edges of at most one more color. But then it has at most one neighbor in , and hence it is not a vertex in . We show the following:

Claim (a): Let be a color present in a characteristic path of length at least two in . Then there is no -colored edge between two vertices of . We prove by contradiction. Let be an edge in a characteristic path of length at least two, and let be color of . Assume that is not a terminal vertex of . If possible let and be two vertices in with edge having color . Then is a characteristic subgraph with more characteristic paths than , a contradiction. The claim follows.

Claim (b): Let be a color present in a characteristic path of length at least two in . Then there is no -colored triangle formed by the -colored edge in and a vertex from . To prove, again assume that is an -colored edge in where is not a terminal vertex. If possible let be an -colored triangle with . Again, is a characteristic subgraph with more characteristic paths than , contradicting the choice of . The claim follows.

Claim (c): The neighbors of vertices in , which are not in , lie in . To prove, we observe that the colors incident on vertices in appear in a characteristic path of length at least two (as vertices in are on such paths). Then as , from Claim (a), we have that a vertex is not incident with a vertex in . Further, is not incident with a vertex in , as the only internal vertices in that are incident with colors at are its neighbors in . Hence, all the neighbors of , not in are in .

Consider the bipartite graph with bipartition with edge set consisting of - edges in . Clearly for . We now prove that for . Let , and let be the edge of incident at , where is the color of . Let be a neighbor of in . We show that is not of color . For sake of contradiction suppose that color of is . Then as , it has a neighbor in which is incident with edge of color . But then as is the only internal vertex in incident with edge of color . Now form an -colored triangle, which contradicts Claim (b). Thus, all the edges of incident at must have the color different from , and hence must have the same color (say ). Clearly all the vertices in incident with edge of color , must be incident with a -colored special edge from a vertex in . As vertices in are incident with mutually disjoint colors, we infer that all such vertices are incident with -colored special edges from a single vertex (see Figure 2). Since there are exactly special edges emanating from a vertex in , we conclude there are at most vertices in incident with color . Hence for . Now, we have . Finally, observe that and hence . Substituting in Equation (7), we have:

(8)

where . The claimed approximation now follows from the inequality . ∎

Figure 2: The vertex in is incident with -colored edge in . Its neighbors in are through the “other” color at , namely . Moreover, these neighbors are also incident with -colored special edges from a vertex in .

5 Tightness of Result 3

In this section we give a construction to show that the approximation factor in Theorem 4.1 is tight.

Theorem 5.1.

Given and , where , and a sufficiently large with respect to say , there exists a graph with number of vertices , and minimum degree such that it is possible to get a -edge coloring for this graph that uses at least colors.

Proof.

It is enough to construct a graph with the cardinality of maximum matching equals to minimum degree and demonstrate a 2-edge coloring for this graph that uses at least colors. Construct a bipartite graph with set of vertices on one side and on the other side (see Figure 3 for schematic representation). Let be the disjoint union of two sets of vertices and . Also let be the disjoint union of two sets of vertices and . Let , and .

In , we will first add a matching between and . In our coloring of , the edges of this matching will get distinct colors, say from to . All the vertices in are made adjacent to all the vertices in , so that all the vertices in has degree at least . These edges between and other than the matching edges will be given the color .

Let be chosen such that , is a positive integer. (We can assume that , so that ). Let , and let , where . We make each , adjacent to , so that the degree of is , for . All the edges from to , and is given the same color, say .

For , make adjacent to . Thus each of these vertices gets an exclusive neigbourhood of vertices each in . All the edges from to the vertices of ,will be given the color .

The total number of colors used is clearly . Note that by this construction the degree of each is . Thus the minimum degree of is . Moreover the number of vertices in is . Since G is a bipartite graph with vertices on one side, the cardinality of the maximum matching is . It is easy to verify that , by substituting in the expression for .

The number of colors used is , as required.

:

:

:

:
Figure 3: The figure illustrates the non-matching edges of the tight example in Theorem 5.1. An ellipse represents a group of vertices, the number at the center being the cardinality of the group. A represents a single vertex. An edge between two groups indicates the complete set of edges between them.

6 Tight example for Result 1

Let be any -regular graph on vertices such that it can be properly edge colored using colors. (Clearly such graphs exist, for example all -regular bipartite graphs are -edge colorable). Properly edge color using colors to . We then construct a new graph from as follows: Replace each vertex of by a clique of vertices . Thus has vertices. Note that is an even number.

Add an edge in if and is colored with color in the edge coloring of . Clearly, is -regular and has a perfect matching: The set of all edges of which are not part of any clique clearly form a perfect matching of . More precisely, . But it is easy to see that removing all the edges of the perfect matching of leaves connected components, namely . Coloring the edges in the matching with distinct colors and coloring the edges of each of the components with a new color yields a -coloring using colors.

On the other hand, it is easy to see that is not the only perfect matching available in . Suppose is even. Then another simple way to get a perfect matching of is as follows: From each clique pick a matching of size . The union of all these matchings clearly is a perfect matching of . Let us name this perfect matching as . Note that the matching based algorithm picks up an arbitrary perfect matching, colors its edges with distinct colors and then gives new colors to the connected componets that results when that perfect matching is removed from the graph: one new color per component. Suppose the matching based algorithm picks up instead of to start with. It is obvious that if is removed from , the resulting graph has only one connected component. Therefore the matching based algorithm (Algorithm 1) yields a -coloring of size , where is the number of vertices of . (To deal with the case when is odd, we can assume that the graph was a -regular bipartite graph such that there exists a perfect matching in it such that is still connected. When properly edge coloring we can make sure that forms the set of edges colored . Now to get we pick a sized matching of the first vertices from each clique , and the set of edges . Clearly remains connected.)

Thus for , which is very close to for large .

References

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  • [4] Wangsen Feng, Ping Chen and Bei Zhang. Approximate maximum edge coloring within factor 2: a further analysis. ISORA, 182–189, 2008.
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