Vertex Coloring Problem Research Articles

Iterated Clique Reductions in Vertex Weighted Coloring for Large Sparse Graphs.

The Minimum Vertex Weighted Coloring (MinVWC) problem is an important generalization of the classic Minimum Vertex Coloring (MinVC) problem which is NP-hard. Given a simple undirected graph G=(V,E), the MinVC problem is to find a coloring s.t. any pair of adjacent vertices are assigned different colors and the number of colors used is minimized. The MinVWC problem associates each vertex with a positive weight and defines the weight of a color to be the weight of its heaviest vertices, then the goal is the find a coloring that minimizes the sum of weights over all colors. Among various approaches, reduction is an effective one. It tries to obtain a subgraph whose optimal solutions can conveniently be extended into optimal ones for the whole graph, without costly branching. In this paper, we propose a reduction algorithm based on maximal clique enumeration. More specifically our algorithm utilizes a certain proportion of maximal cliques and obtains lower bounds in order to perform reductions. It alternates between clique sampling and graph reductions and consists of three successive procedures: promising clique reductions, better bound reductions and post reductions. Experimental results show that our algorithm returns considerably smaller subgraphs for numerous large benchmark graphs, compared to the most recent method named RedLS. Also, we evaluate individual impacts and some practical properties of our algorithm. Furthermore, we have a theorem which indicates that the reduction effects of our algorithm are equivalent to that of a counterpart which enumerates all maximal cliques in the whole graph if the run time is sufficiently long.

Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallel

AbstractThe Lovász local lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection of “bad” events which are mostly independent and have low probability. A seminal algorithm of Moser and Tardos (J. ACM, 2010, 57, 11) (which we call the MT algorithm) gives nearly‐automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (Combinatorica, 1985, 5, 241–245); this is more powerful than alternate LLL criteria and also leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility. Third, we provide a derandomized version of the MT‐distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non‐repetitive vertex coloring, independent transversals, strong coloring, and other problems.

Large‐Scale Cardiac Muscle Cell‐Based Coupled Oscillator Network for Vertex Coloring Problem

Modern computers require an exponential increase in resources when solving computationally hard problems, motivating the need for an alternative computing platform to solve such problems in an energy‐efficient manner. Vertex coloring, a nondeterministic polynomial time (NP‐hard) combinatorial optimization problem, is one such problem. Herein, an experimental demonstration of using cardiac cell‐based bio‐oscillator network coupling dynamics to solve a vertex coloring problem in various scales of graphs using a simple cell patterning method to construct scalable and controlled cardiac cell networks is presented. Although there are limitations to using these cardiac cells as oscillators, such as their low frequency compared to complementary metal–oxide–semiconductor (CMOS) oscillators, that result in longer processing times, the accuracy in large graph instances, the significantly less amount of energy consumption, and the ease of fabrication and potential to extend this system to massively parallel 3D structures make the bio‐oscillators a promising new platform for collective computing applications.

Experimental implementation of Hardy-like quantum pigeonhole paradoxes

We present the general Hardy-like quantum pigeonhole paradoxes for \textit{n}-particle states, and find that each of such paradoxes can be simply associated to an un-colorable solution of a specific vertex-coloring problem induced from the projected-coloring graph (a kind of unconventional graph). Besides, as a special kind of Hardy's paradox, several kinds of Hardy-like quantum pigeonhole paradoxes can even give rise to higher success probability in demonstrating the conflict between quantum mechanics and local or noncontextual realism than the previous Hardy's paradoxes. Moreover, not only multi-qubit states, but high-dimensional states can exhibit the paradoxes. In contrast to only one type of contradiction presented in the original quantum pigeonhole paradox, two kinds of three-qubit projected-coloring graph states as the minimal illustration are discussed in our work, and an optical experiment to verify such stronger paradox is performed. This quantum paradox provides innovative thoughts and methods in exploring new types of stronger multi-party quantum nonlocality and may have potential applications in multi-party untrusted communications and device-independent random number generation.

Vertex coloring (4K1, hole-twin, 5-wheel)-free graphs

There has been recently keen interest in finding polynomial-time algorithms for the VERTEX COLORING problem for graphs G that are F-free for a given set F of graphs. If L is a set of four-vertex graphs, then the complexity of VERTEX COLORING for L-free graphs is known with three exceptions: L1 = {claw, 4K1}, L2 = {claw, 4K1, co-diamond}, and L3 = {4K1, C4}. In this paper, we study a problem arising from the class L3. A hole is an induced cycle with at least four vertices. A hole-twin is the graph obtained from a hole by adding a vertex that forms true twins with some vertex of the hole. A 5-wheel is the graph obtained from a C5 by adding a vertex that is adjacent to all vertices of the C5. We prove that a (4K1, hole-twin, 5-wheel)-free graph is perfect or has bounded clique width. As consequence we obtain a polynomial time algorithm for VERTEX COLORING for (4K1, hole-twin, 5-wheel)-free graphs.

Resorting to Context-Aware Background Knowledge for Unveiling Semantically Related Social Media Posts

Social media networks have become a prime source for sharing news, opinions, and research accomplishments in various domains, and hundreds of millions of posts are announced daily. Given this wealth of information in social media, finding related announcements has become a relevant task, particularly in trending news (e.g., COVID-19 or lung cancer). To facilitate the search of connected posts, social networks enable users to annotate their posts, e.g., with hashtags in tweets. Albeit effective, an annotation-based search is limited because results will only include the posts that share the same annotations. This paper focuses on retrieving context-related posts based on a specific topic, and presents PINYON, a knowledge-driven framework, that retrieves associated posts effectively. PINYON implements a two-fold pipeline. First, it encodes, in a graph, a CORPUS of posts and an input post; posts are annotated with entities for existing knowledge graphs and connected based on the similarity of their entities. In a decoding phase, the encoded graph is used to discover communities of related posts. We cast this problem into the Vertex Coloring Problem, where communities of similar posts include the posts annotated with entities colored with the same colors. Built on results reported in the graph theory, PINYON implements the decoding phase guided by a heuristic-based method that determines relatedness among posts based on contextual knowledge, and efficiently groups the most similar posts in the same communities. PINYON is empirically evaluated on various datasets and compared with state-of-the-art implementations of the decoding phase. The quality of the generated communities is also analyzed based on multiple metrics. The observed outcomes indicate that PINYON accurately identifies semantically related posts in different contexts. Moreover, the reported results put in perspective the impact of known properties about the optimality of existing heuristics for vertex graph coloring and their implications on PINYON scalability.

Adjacent Upper Triangular Matrix Algorithm for Vertex Colouring Problem

Adjacent Upper Triangular Matrix Algorithm for Vertex Colouring Problem

Algorithms for the rainbow vertex coloring problem on graph classes

Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed p≥3 both variants of the problem become NP-complete when restricted to split (S3,…,Sp)-free graphs, where Sq denotes the q-sun graph.

A stand-alone branch-and-price algorithm for identical parallel machine scheduling with conflicts

We consider a scheduling problem where a set of n jobs has to be processed in a non-preemptive way on a set of m identical parallel machines. Each job j is associated with a positive integer processing time pj. The problem is also characterized by a conflict graph where adjacent nodes in the graph represent conflicting jobs that cannot be processed on the same machine. A schedule is an assignment of a time interval of pj time units on m machines for each job j. The schedule is feasible if intervals on the same machine do not overlap, and jobs to be processed on the same machine are pairwise not conflicting. The aim is to find a feasible schedule that minimizes the maximum completion time of the jobs. The problem is NP-hard as it generalizes both the problem of scheduling jobs on identical parallel machines to the aim of minimizing the makespan (P||Cmax) and the vertex coloring problem (VCP), two well-known NP-hard problems. We present the first stand-alone branch-and-price (BP) algorithm that works directly on the problem. In comprehensive computational experiments with benchmark instances, we prove that the new BP algorithm, even without using ad hoc primal heuristics and feasibility check algorithms, is competitive with the best exact solution algorithm proposed so far in the literature. These results are mainly achieved thanks to the branching scheme we propose.

Efficient Solvability of the Weighted Vertex Coloring Problem for Some Two Hereditary Graph Classes

Efficient Solvability of the Weighted Vertex Coloring Problem for Some Two Hereditary Graph Classes

Integer and constraint programming approaches for providing optimality to the bandwidth multicoloring problem

In this paper, constraint and integer programming techniques are applied to solving bandwidth coloring problems, in the sense of proving optimality or finding better feasible solutions for benchmark instances from the literature. The Bandwidth Coloring Problem (BCP) is a generalization of the classic vertex coloring problem (VCP), where, given a graph, its vertices must be colored such that not only adjacent ones do not share the same color, but also their colors must be separated by a minimum given value. BCP is further generalized to the Bandwidth Multicoloring Problem (BMCP), where each vertex can receive more than one different color, also subject to separation constraints. BMCP is used to model the Minimum Span Channel Assignment Problem (MS-CAP), which arises in the planning of telecommunication networks. Research on algorithmic strategies to solve these problems focus mainly on heuristic approaches and the performance of such methods is tested on artificial and real scenarios benchmarks, such as GEOM, Philadelphia and Helsinki sets. We achieve optimal solutions or provide better upper bounds for these well-known instances, We also compare the effects of multicoloring demands on the performance of each exact solution approach, based on empirical analysis.

Building a Maximal Independent Set for the Vertex-coloring Problem on Planar Graphs

We analyze the vertex-coloring problem restricted to planar graphs and propose to consider classic wheels and polyhedral wheels as basic patterns for the planar graphs. We analyze the colorability of the composition among wheels and introduce a novel algorithm based on three rules for the vertex-coloring problem. These rules are: 1) Selecting vertices in the frontier. 2) Processing subsumed wheels. 3) Processing centers of the remaining wheels. Our method forms a maximal independent set S1 ⊂ V (G) consisting of wheel's centers, and a maximum number of vertices in the frontier of the planar graph. Thus, we show that if the resulting graph G′ = (G − S1) is 3-colorable, then this implies the existence of a valid 4-coloring for G.

P.039 Cell type-specific DNA methylation is associated with childhood attention-deficit hyperactivity disorder symptoms

We analyze the vertex-coloring problem restricted to planar graphs and propose to consider classic wheels and polyhedral wheels as basic patterns for the planar graphs. We analyze the colorability of the composition among wheels and introduce a novel algorithm based on three rules for the vertex-coloring problem. These rules are: 1) Selecting vertices in the frontier. 2) Processing subsumed wheels. 3) Processing centers of the remaining wheels. Our method forms a maximal independent set S1 ⊂ V (G) consisting of wheel's centers, and a maximum number of vertices in the frontier of the planar graph. Thus, we show that if the resulting graph G′ = (G − S1) is 3-colorable, then this implies the existence of a valid 4-coloring for G.

Efficient Solvability of the Weighted Vertex Coloring Problem for Some Hereditary Class of Graphs with $$\boldsymbol {5}$$ -Vertex Prohibitions

We consider the problem of minimizing the number of colors in the colorings of the vertices of a given graph so that, to each vertex there is assigned some set of colors whose number is equal to the given weight of the vertex; and adjacent vertices receive disjoint sets. For all hereditary classes defined by a pair of forbidden induced connected subgraphs on $$5 $$ vertices but four cases, the computational complexity of the weighted vertex coloring problem with unit weights is known. We prove the polynomial solvability on the sum of the vertex weights for this problem and the intersection of two of the four open cases. We hope that our result will be helpful in resolving the computational complexity of the weighted vertex coloring problem in the above-mentioned forbidden subgraphs.

A distributed algorithm for vertex coloring problems in wireless networks

Abstract A lot of problems in computer networks are modeled by the vertex coloring problem (VCP), the multicoloring problem (MCP), the bandwidth coloring problem (BCP), and the bandwidth multicoloring problem (BMCP). To solve the VCP with Δ + 1 colors, a myriad distributed algorithms have been proposed to reduce time complexity (the number of rounds), where Δ is the maximum vertex degree in the graph. Time and communication complexities of these algorithms are functions of n and Δ , where n is the number of vertices in the graph. The MCP can be converted into VCP after transformation of the graph; the transformation increases the time and communication complexities. However, no distributed algorithms for BCP and BMCP have been proposed in the literature. In this paper, we propose a General Distributed Vertex Coloring Algorithm (GDVCA) to solve four problems in wireless networks. In GDVCA, some (and not necessarily all) vertices assign colors to themselves and their neighbors, which we refer to them as assigning vertices. The communication cost of GDVCA is very low; the number of messages transmitted by each assigning vertex is one and by the other vertices is zero. Assigning vertices assign the first available colors to uncolored vertices; moreover assigning vertices can use heuristic methods to choose the next proper vertex for coloring. Therefore, the number of colors used by the distributed algorithm GDVCA is significantly low. The number of time slots required to color a graph is O ( Δ 2 ) and at most n, where each round comprises several time slots.

Fuzzy-Clustering-Based Circular Topological Via Minimization in PCB Designs

It is necessary for reliability and yield to minimize the number of used vias on nets in printed circuit board (PCB) designs. To our knowledge, the proposed fuzzy-clustering-based algorithm is the first work for a general routing region with multiple-pin nets in k -layer circular topological via minimization ( k -CTVM). In this article, given the topological connections in a set of routing nets and a set of k available layers in a routing plane, first, all the multipin nets can be transformed into a set of two-pin nets by introducing a set of preassigned vias onto the branch points on multipin nets and a conflict graph can be constructed for a final set of two-pin nets. Furthermore, the probabilistic similarity between two connected vertices using the same color can be computed for the constrained vertex-coloring problem with k colors in a conflict graph. Next, based on the definition of the clustering distance between two connected vertices in a conflict graph, fuzzy graph clustering can be developed to obtain a fuzzy matrix on k clusters. Finally, all the given nets can be assigned onto the k available layers by introducing a set of necessary vias on two-pin nets and eliminating the unnecessary preassigned vias on multipin nets. Compared with the combination of Cong's algorithm and an iterative net postassignment, NetInsertion1 , in the k -CTVM problem, the experimental results show that our proposed fuzzy-clustering- based algorithm can use less CPU time to reduce 57.3% of the number of the total used vias for eight tested PCB designs. Compared with the combination of Yan's algorithm and an iterative net postassignment, NetInsertion2 , in the k -CTVM problem, the experimental results show that our proposed fuzzy-clustering-based algorithm can reduce 34.0% of the number of the total used vias for eight tested PCB designs.

An enhanced formulation for solving graph coloring problems with the Douglas–Rachford algorithm

We study the behavior of the Douglas-Rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In spite of the combinatorial nature of this problem, the Douglas-Rachford algorithm was recently shown to be a successful heuristic for solving a wide variety of graph coloring instances, when the problem was cast as a feasibility problem on binary indicator variables. In this work we consider a different formulation, based on semidefinite programming. The much improved performance of the Douglas-Rachford algorithm, with this new approach, is demonstrated through various numerical experiments.

The Intersection of Two Vertex Coloring Problems

A hole is a chordless cycle with at least four vertices. A hole is even if its number of vertices is even. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. Currently, the following two problems are unresolved: the complexity of coloring even hole-free graphs, and the complexity of coloring $$(4K_1, C_4)$$-free graphs. The intersection of these two problems is the problem of coloring $$(4K_1, C_4, C_6)$$-free graphs. In this paper we present partial results on this problem.

DIAMOND: a distributed algorithm for vertex coloring problems and resource allocation

The vertex colouring problem (VCP) and its generalisations have myriad applications in computer networks. To solve the VCP with Δ + 1 colours, numerous distributed algorithms based on LOCAL model have been proposed to reduce time complexity (the number of rounds), where Δ is the maximum vertex degree in the graph. In this paper, the authors present a distributed algorithm based on modified LOCAL model (DIAMOND) that reduces the number of rounds to one. It greedily solves the VCP with at most Δ + 1 colours. Computational results on Geometry (GEOM) graphs show that the number of used colours to colour each instance using DIAMOND is about Δ + 1 / 2 . DIAMOND is easily extended to solve greedily generalised VCPs in only one round. Moreover, they present two efficient resource allocation algorithms using DIAMOND. They allocate more resource to the graph compared with ( Δ + 1 ) -colouring and even to ( d ¯ + 1 ) -colouring algorithms, where d ¯ is the average vertex degree of the graph. They run in two and Δ rounds.

A fuzzy irregular cellular automata-based method for the vertex colouring problem

ABSTRACTVertex colouring is among the most important problems in graph theory which has been widely applied across different real-world problems. In vertex colouring problem (VCP), the goal is to assign a distinct colour to each vertex of the graph in such a way that no two adjacent vertices have the same colour. This paper presents a fuzzy irregular cellular automaton (FICA) for finding a near-optimal solution for the VCP. FICA is an extension fuzzy cellular automaton (FCA) in which the cells of the automaton can be arranged in an irregular structure. The aim of the proposed method is to reap the benefits of both FCA and irregular cellular automata while minimising their drawbacks. To evaluate the proposed method, various computer simulations have been conducted on a variety of graphs. The results suggest that the proposed method is able to achieve better results in terms of the minimum number of required colours and the execution time of the algorithm, compared to other peer algorithms.