A note on maximum differential coloring of planar graphs

Bekos, Michael A.; Kaufmann, Michael; Kobourov, Stephen G.; Veeramoni, Sankar

doi:10.1016/j.jda.2014.06.004

Cited by 6 publications

(9 citation statements)

References 26 publications

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“…-Since it is NP-complete to determine the 1-differential chromatic number of a planar graph [20], a natural question to ask is whether it is possible to compute in polynomial time the 1-differential chromatic number of an outerplanar graph.…”

Section: Discussionmentioning

confidence: 99%

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The maximum k-differential coloring problem

Bekos

Kaufmann

Kobourov

et al. 2017

Journal of Discrete Algorithms

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Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NPcomplete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the ( 2n/3 , 2n)-differential coloring problem, even in the case where the input graph is planar.

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Section: Discussionmentioning

confidence: 99%

“…Note that closely related is also the radio frequency assignment problem, where n transmitters have to be assigned n frequencies, so that interfering transmitters have frequencies as far apart as possible, see e.g., [18,19]. For a more detailed bibliographic overview related to the maximum differential coloring problem refer to [20].…”

Section: Related Workmentioning

confidence: 99%

The maximum k-differential coloring problem

Bekos

Kaufmann

Kobourov

et al. 2017

Journal of Discrete Algorithms

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“…-As all planar graphs are ( n 3 + 1, 2n)-differential colorable, is it possible to characterize which planar graphs are ( n 3 + 2, 2n)-differential colorable? -Since it is NP-complete to determine the 1-differential chromatic number of a planar graph [20], a natural question to ask is whether it is possible to compute in polynomial time the 1-differential chromatic number of an outerplanar graph.…”

Section: Discussionmentioning

confidence: 99%

The Maximum k-Differential Coloring Problem

Bekos

Kaufmann

Kobourov

et al. 2015

Lecture Notes in Computer Science

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Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NPcomplete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (2n/3 , 2n)-differential coloring problem, even in the case where the input graph is planar.

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“…In Miller and Pritikin (1989) and Yixun and JinJiang (2003) various theoretical bounds for general graphs based on graph parameters like size, order, degree, stability number and chromatic number are presented. For certain classes of graphs like Hamming graphs (Dobrev et al 2013), hypercubes (Raspaud et al 2009;Wang et al 2009), complete k-ary trees (Calamoneri et al 2009), caterpillars and spiders (Bekos et al 2013(Bekos et al , 2014 there exist tighter bounds and/or exact algorithms. For general graphs, a variety of (meta-)heuristic approaches exist: Bansal and Srivastava (2011) proposed a memetic algorithm, Duarte et al (2011) develops a generalized randomized adaptive search procedure with path relinking, Lozano et al (2012) presented a variable neighborhood search and Scott and Hu (2014) designed a hill-climbing algorithm.…”

Section: Ab(g)mentioning

confidence: 99%

“…Problems in this class include the bandwidth problem (Cuthill and McKee 1969;Caprara and Salazar-González 2005) and variants of it like cyclic bandwidth (Rodriguez-Tello et al 2015), the linear arrangement problem (Caprara et al 2011;Rodriguez-Tello et al 2008) and the cutwidth problem (Martí et al 2013), see also the surveys (Díaz et al 2002;Gallian 2009). In this work, we consider the antibandwidth problem (ABP), also known as dual bandwidth problem (Yixun and JinJiang 2003), separation problem (Miller and Pritikin 1989) and maximum differential coloring problem (Bekos et al 2014). The ABP is NP-hard in general graphs and has applications in scheduling (Leung et al 1984), radio frequency assignment (Hale 1980), obnoxious facility location (Cappanera 1999) and map-coloring (Gansner et al 2010).…”

Section: Introductionmentioning

confidence: 99%

A note on computational approaches for the antibandwidth problem

Sinnl

2020

Cent Eur J Oper Res

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In this note, we consider the antibandwidth problem, also known as dual bandwidth problem, separation problem and maximum differential coloring problem. Given a labeled graph (i.e., a numbering of the vertices of a graph), the antibandwidth of a node is defined as the minimum absolute difference of its labeling to the labeling of all its adjacent vertices. The goal in the antibandwidth problem is to find a labeling maximizing the antibandwidth. The problem is NP-hard in general graphs and has applications in diverse areas like scheduling, radio frequency assignment, obnoxious facility location and map-coloring. There has been much work on deriving theoretical bounds for the problem and also in the design of metaheuristics in recent years. However, the optimality gaps between the best known solution values and reported upper bounds for the HarwellBoeing Matrix-instances, which are the commonly used benchmark instances for this problem, are often very large (e.g., up to 577%). Moreover, only for three of these 24 instances, the optimal solution is known, leading the authors of a state-of-the-art heuristic to conclude "HarwellBoeing instances are actually a challenge for modern heuristic methods". The upper bounds reported in literature are based on the theoretical bounds involving simple graph characteristics, i.e., size, order and degree, and a mixed-integer programming (MIP) model. We present new MIP models for the problem, together with valid inequalities, and design a branch-and-cut algorithm and an iterative solution algorithm based on them. These algorithms also include two starting heuristics and a primal heuristic. We also present a constraint programming approach, and calculate upper bounds based on the stability number and chromatic number. Our computational study shows that the developed approaches allow to find the proven optimal solution for eight instances from literature, where the optimal solution was unknown and also provide reduced gaps for eleven additional instances, including improved solution values for seven instances, the largest optimality gap is now 46%.

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A note on maximum differential coloring of planar graphs

Cited by 6 publications

References 26 publications

The maximum k-differential coloring problem

The maximum k-differential coloring problem

The Maximum k-Differential Coloring Problem

A note on computational approaches for the antibandwidth problem

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