Parameterized and Approximation Algorithms for the Load Coloring Problem

Barbero, Florian; Gutin, Gregory; Jones, Mark; Sheng, Bin

doi:10.1007/s00453-016-0259-z

Cited by 8 publications

(3 citation statements)

References 17 publications

(40 reference statements)

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“…Finally, let us also mention the min c-judicious partitioning (which is a maximization problem), called c-Load Coloring, where given a graph G and a positive integer k, the goal is to decide whether V (G) can be partitioned into c parts so that each part has at least k edges. Barbero et al [2] showed that this problem is FPT (also see [21]).…”

Section: Balanced Judicious Bipartition (Bjb)mentioning

confidence: 97%

Balanced Judicious Bipartition is Fixed-Parameter Tractable

Lokshtanov¹,

Saurabh²,

Sharma³

et al. 2019

SIAM J. Discrete Math.

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The family of judicious partitioning problems, introduced by Bollobás and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollobás and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).

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Section: Balanced Judicious Bipartition (Bjb)mentioning

confidence: 97%

Balanced Judicious Bipartition is Fixed-Parameter Tractable

Lokshtanov¹,

Saurabh²,

Sharma³

et al. 2019

SIAM J. Discrete Math.

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show abstract

“…In [6,7], the parameterized and approximation algorithms are proposed to solve the load coloring problem, and theoretically prove their capability in finding the best solution. On the other hand, considering the theoretical intractability of MLCP, several heuristic algorithms are proposed to find the best solutions.…”

Section: Related Workmentioning

confidence: 99%

An Efficient Memetic Algorithm for the Minimum Load Coloring Problem

Zhang

Qiao

2019

Mathematics

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Given a graph G with n vertices and l edges, the load distribution of a coloring q: V → {red, blue} is defined as dq = (rq, bq), in which rq is the number of edges with at least one end-vertex colored red and bq is the number of edges with at least one end-vertex colored blue. The minimum load coloring problem (MLCP) is to find a coloring q such that the maximum load, lq = 1/l × max{rq, bq}, is minimized. This problem has been proved to be NP-complete. This paper proposes a memetic algorithm for MLCP based on an improved K-OPT local search and an evolutionary operation. Furthermore, a data splitting operation is executed to expand the data amount of global search, and a disturbance operation is employed to improve the search ability of the algorithm. Experiments are carried out on the benchmark DIMACS to compare the searching results from memetic algorithm and the proposed algorithms. The experimental results show that a greater number of best results for the graphs can be found by the memetic algorithm, which can improve the best known results of MLCP.

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“…Gutin et al [9] proved that the problem admits a polynomial kernel (with at most 7k vertices) parameterized by k. They also showed that the problem is fixed-parameter tractable when parameterized by the tree-width of the input graph. More recently, Barbero et al [2] obtained a kernel for the problem with at most 4k vertices improving the result of [9].…”

Section: Introductionmentioning

confidence: 98%

On Structural Parameterizations of Load Coloring

Reddy¹

2020

Preprint

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Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring f : V (G) → {r, b} of G such that for every i ∈ {r, b}, there are at least k edges with both end vertices colored i. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (Algorithmica 79:211-229, 2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time f (w)n o(w) , unless ETH fails and it can be solved in time n O(w) , where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to cocluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is NP-complete even on bipartite graphs and split graphs.

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Parameterized and Approximation Algorithms for the Load Coloring Problem

Cited by 8 publications

References 17 publications

Balanced Judicious Bipartition is Fixed-Parameter Tractable

Balanced Judicious Bipartition is Fixed-Parameter Tractable

An Efficient Memetic Algorithm for the Minimum Load Coloring Problem

On Structural Parameterizations of Load Coloring

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