Bounded Max-Colorings of Graphs (original) (raw)

Approximating the max-edge-coloring problem

Theoretical Computer Science, 2010

The max edge-coloring problem is a natural weighted generalization of the classical edge-coloring problem arising in the domain of communication systems. In this problem each color class is assigned the weight of the heaviest edge in this class and the objective is to find a proper edge-coloring of the input graph minimizing the sum of all color classes' weights. We present new approximation results, that improve substantially the known ones, for several variants of the problem with respect to the class of the underlying graph. In particular, we deal with variants which either are known to be NP-hard (general and bipartite graphs) or are proven to be NP-hard in this paper (complete graphs with bi-valued edge weights) or their complexity question still remains open (trees).

Weighted Coloring on Planar, Bipartite and Split Graphs: Complexity and Improved Approximation

Lecture Notes in Computer Science, 2004

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-complete in planar graphs, even if they are trianglefree and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-complete in P8-free bipartite graphs, but polynomial for P5-free bipartite graphs. We next focus ourselves on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-complete, even in the case where the input-graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6 − ε, for any ε > 0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

2009

Weighted Coloring: Further Complexity and Approximability Results

Abstract. Given a vertex-weighted graph G = (V, E; w), w(v) � 0 for any v 2 V , we consider a weighted version of the coloring problem which consists in finding a partition S = (S1, . . . , Sk) of the vertex set V of G into stable sets and minimizing P, i=1 w(Si) where the weight of S is defined as max{w(v) : v 2 S}. In this paper, we keep on with the investigation of the complexity and the approximability of this problem by mainly answering one of the questions raised by D. J. Guan and X. Zhu (”A Coloring Problem for Weighted Graphs”, Inf. Process. Lett. 61(2):77-81 1997). Keywords: Approximation algorithm; NP-complete problems; weighted

Approximation and Hardness Results for the Maximum Edge q-coloring Problem

Lecture Notes in Computer Science, 2010

In this paper we study the following problem, named Maximum Edges in Transitive Closure, which has applications in computational biology. Given a simple, undirected graph G = (V, E) and a coloring of the vertices, remove a collection of edges from the graph such that each connected component is colorful (i.e., it does not contain two identically colored vertices) and the number of edges in the transitive closure of the graph is maximized. The problem is known to be APX-hard, and no approximation algorithms are known for it. We improve the hardness result by showing that the problem is NP-hard to approximate within a factor of |V | 1/3−ε , for any constant ε > 0. Additionally, we show that the problem is APX-hard already for the case when the number of vertex colors equals 3. We complement these results by showing the first approximation algorithm for the problem, with approximation factor √ 2 · OPT.

Approximations for -Colorings of Graphs

The Computer Journal, 2004

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-coloring with small or optimal λ arises in the context of radio frequency assignment. We show that the problem of finding the minimum λ for planar graphs, bipartite graphs, chordal graphs and split graphs are NP-Complete. We then give approximation algorithms for λ-coloring and compute upper bounds of the best possible λ for outerplanar graphs, graphs of treewidth k, permutation and split graphs. With the exception of the split graphs, all the above bounds for λ are linear in ∆, the maximum degree of the graph. For split graphs, we give a bound of λ ≤ ∆ 1.5 + 2∆ + 2 and show that there are split graphs with λ = Ω(∆ 1.5 ). We also give a bound of λ = Ω(∆ 2 ) for bipartite graphs. Similar results are also given for variations of the λ-coloring problem.

Maximization coloring problems on graphs with few

Discrete Applied Mathematics, 2014

q, q − 4)-graphs Primeval decomposition Fixed parameter tractable algorithms a b s t r a c t Given a graph G, a greedy coloring of G is a proper coloring such that, for each two colors i < j, every vertex of G colored j has a neighbor colored i. The Grundy number is the maximum number of colors in a greedy coloring of G. proved that determining the Grundy number is NP-hard even for complements of bipartite graphs. A b-coloring of G is a proper coloring such that every color class contains a vertex which is adjacent to at least one vertex in every other color class. The b-chromatic number is the maximum number of colors in a b-coloring of G. proved that determining the b-chromatic number is NP-hard. In this paper, we obtain polynomial time algorithms to determine the Grundy number and the b-chromatic number of (q, q − 4)-graphs, for every fixed q, which are the graphs such that every set of at most q vertices induces at most q − 4 distinct P 4 . These algorithms are fixed parameter tractable on the parameter q(G), where q(G) is the minimum q such that G is a (q, q − 4)-graph.

Approximating the maximum 3-edge-colorable subgraph problem

Discrete Mathematics, 2009

We offer the following structural result: every triangle-free graph G of maximum degree 3 has 3 matchings which collectively cover at least 1 − 2 3 γo(G) of its edges, where γ o (G) denotes the odd girth of G. In particular, every triangle-free graph G of maximum degree 3 has 3 matchings which cover at least 13/15 of its edges. The Petersen graph, where we can 3-edge-color at most 13 of its 15 edges, shows this to be tight. We can also cover at least 6/7 of the edges of any simple graph of maximum degree 3 by means of 3 matchings; again a tight bound.

Improved approximation for maximum edge colouring problem

Discrete Applied Mathematics, 2021

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, namely, 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. The case when H is a star graph was considered by several graph theorists from the combinatorial point of view. 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: Find a coloring of the edges of G, maximizing the number of colors satisfying the constraint that each vertex spans at most q colors on its incident edges. It is easy to see that the maximum value of the above optimization problem equals ar(G, K 1,q+1) − 1. In this paper we study the maximum edge 2-coloring 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 5/3approximation 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 higher number of colors than the matching based algorithm, mentioned above.

Bounded Max-Colorings of Graphs (original) (raw)

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