Solving Graph Coloring Problems with the Douglas-Rachford Algorithm

Artacho, Francisco J. Aragón; Campoy, Rubén

doi:10.1007/s11228-017-0461-4

Cited by 15 publications

(6 citation statements)

References 27 publications

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“…Thus, we first consider a comparison that focuses on the forms of graph coloring problem that are solved, the ability to determine the chromatic number, and metrics used in order to determine the algorithm’s performance. Additionally, we provide a comparison that focuses on the solution count between RQGA and the algorithms presented in Silva et al (2020) ; we also compare our approach with the algorithms presented in Silva et al (2020) , Tabi et al (2020) , and Aragón Artacho & Campoy (2018) from the perspective of the number of iterations required to find a solution that colors a 5-node graph.…”

Section: Discussionmentioning

confidence: 99%

“…Therefore, considering the large number of applications, solving graph coloring has received a great deal of research interest Dokeroglu & Sevinc (2021) . For instance, in Aragón Artacho & Campoy (2018) , the authors present a method for solving the graph coloring problem using the Douglas-Rachford Algorithm. Indeed, they prove that the algorithm is an effective heuristic for solving this NP-hard problem.…”

Section: Introductionmentioning

confidence: 99%

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Graph coloring using the reduced quantum genetic algorithm

Ardelean¹,

Udrescu²

2022

PeerJ Computer Science

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Genetic algorithms (GA) are computational methods for solving optimization problems inspired by natural selection. Because we can simulate the quantum circuits that implement GA in different highly configurable noise models and even run GA on actual quantum computers, we can analyze this class of heuristic methods in the quantum context for NP-hard problems. This paper proposes an instantiation of the Reduced Quantum Genetic Algorithm (RQGA) that solves the NP-hard graph coloring problem in O(N1/2). The proposed implementation solves both vertex and edge coloring and can also determine the chromatic number (i.e., the minimum number of colors required to color the graph). We examine the results, analyze the algorithm convergence, and measure the algorithm's performance using the Qiskit simulation environment. Our Reduced Quantum Genetic Algorithm (RQGA) circuit implementation and the graph coloring results show that quantum heuristics can tackle complex computational problems more efficiently than their conventional counterparts.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Graph coloring using the reduced quantum genetic algorithm

Ardelean¹,

Udrescu²

2022

PeerJ Computer Science

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“…Several other methodologies have also find in literature to solve the problem of coloring of graphs that involves: improved cuckoo optimization technique [28], using coupled oscillatory networks [29], using local anti-magic labeling in the graph [30], cellular learning automata based solution [31], solving graph-b coloring problem using hybrid genetic algorithms [32], Douglas-Rachford Algorithm based solution [33] for graph coloring problem and branch-and-cut algorithm for the equitable coloring problem [34].…”

Section: Algorithmmentioning

confidence: 99%

A Linked List-Based Exact Algorithm for Graph Coloring Problem

Shukla¹,

Bharti²,

Garag³

2019

RIA

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Graph coloring is an NP-hard problem. There is ample room to further reduce the number of colors being used under the strict constraints of the problem. This paper proposes an algorithm that stores the information of the vertices in the graph, together with the color and the next address field of a node in a singly linked list. The coloring of a particular vertex is decided on the content of color field of node corresponding to the searched vertex for coloring. The adjacency matrix of the graph was adopted to select the feasible color to be allocated to a vertex, without violating the constraints for coloring of the vertices in the graph. The proposed algorithm was tested on DIMACS benchmark graphs, using Python 3.6. The results show that our algorithm produced the result with the optimal number of colors required to solve the problem. This research provides a new strategy to find the optimal coloring solution for each type of graphs.

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“…(3) For boundary value ODEs, we show how to reformulate the N-hypersurface feasibility problem as a three-set feasibility problem. (4) We analyse the behaviour for both AP and DR experimentally on hypersurface problems for varying N, which for boundary value ODEs corresponds to partition fineness. We catalogue the characteristics of oscillation so frequently observed for DR in particular.…”

Section: Introductionmentioning

confidence: 99%

“…While this article is an important extension of the analysis of projection methods (and DR in particular) for nonconvex hypersurface feasibility problems, its approach is comparable to other experimental works which analyse proximal point algorithms in the absence of convexity by cataloguing the performance of the method for a selection of examples. These include the recent work of Aragón Artacho et al [3] applying DR to solve matrix completion problems and Sudoku puzzles [2,10], the work of Aragón Artacho and Campoy [4] with graph colouring problems and the seminal work of Elser et al [16].…”

Section: Introductionmentioning

confidence: 99%

Application of Projection Algorithms to Differential Equations: Boundary Value Problems

Lamichhane¹,

Lindstrom²,

Sims³

2019

ANZIAM J.

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The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well-suited to parallelization. We explore the stability of the method by applying it to several examples of BVPs, including cases where the traditional Newton's method fails.

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Solving Graph Coloring Problems with the Douglas-Rachford Algorithm

Cited by 15 publications

References 27 publications

Graph coloring using the reduced quantum genetic algorithm

Graph coloring using the reduced quantum genetic algorithm

A Linked List-Based Exact Algorithm for Graph Coloring Problem

Application of Projection Algorithms to Differential Equations: Boundary Value Problems

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