scholarly journals Latest Algorithms on Particular Graph Classes

2020 ◽  
pp. 21-35
Author(s):  
Phan Thuan DO ◽  
Ba Thai PHAM ◽  
Viet Cuong THAN
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Maximum Clique ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Np Hard ◽  
Induced Matching ◽  
Graph Classes ◽  
Maximum Induced Matching ◽  
Circular Arc Graphs

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

scholarly journals Shifting Coresets: Obtaining Linear-Time Approximations for Unit Disk Graphs and Other Geometric Intersection Graphs

2017 ◽  
Vol 27 (04) ◽  
pp. 255-276 ◽  
Author(s):  
Guilherme D. da Fonseca ◽  
Vinícius Gusmão Pereira de Sá ◽  
Celina Miraglia Herrera de Figueiredo
Keyword(s):  
Approximation Algorithms ◽  
Unit Disk ◽  
Optimization Problems ◽  
Linear Time ◽  
Dominating Set ◽  
Independent Set ◽  
Constant Factor ◽  
Intersection Graphs ◽  
Unit Disk Graphs ◽  
Graph Classes

Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy that allows us to obtain linear-time constant-factor approximation algorithms for such problems. To illustrate the applicability of the proposed variation, we obtain results for three well-known optimization problems. Among such results, the proposed method yields linear-time [Formula: see text]-approximations for the maximum-weight independent set and the minimum dominating set of unit disk graphs, thus bringing significant performance improvements when compared to previous algorithms that achieve the same approximation ratios. Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.

scholarly journals Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

The 0–1 inverse maximum independent set problem on forests and unicyclic graphs

2016 ◽  
Vol 08 (02) ◽  
pp. 1650019
Author(s):  
Shuaifu Liu ◽  
Zhao Zhang
Keyword(s):  
Linear Time ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Independent Set Problem ◽  
Unicyclic Graphs ◽  
Maximum Independent Set Problem ◽  
Bounded Size ◽  
Linear Time Algorithms ◽  
The Given ◽  
Few Vertices

Given a graph [Formula: see text] and an independent set [Formula: see text] of [Formula: see text], the 0–1 inverse maximum independent set problem (IMIS[Formula: see text]) is to delete as few vertices as possible such that [Formula: see text] becomes a maximum independent set of [Formula: see text]. It is known that IMIS[Formula: see text] is NP-hard even when the given independent set has a bounded size. In this paper, we present linear-time algorithms for IMIS[Formula: see text] on forests and unicyclic graphs.

scholarly journals Gumbel-softmax-based Optimization: A Simple General Framework for Optimization Problems on Graphs

2020 ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

Abstract In computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington-Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time consuming compared to traditional approaches.

scholarly journals Graph Neural Networks for Maximum Constraint Satisfaction

2021 ◽  
Vol 3 ◽  
Author(s):  
Jan Tönshoff ◽  
Martin Ritzert ◽  
Hinrikus Wolf ◽  
Martin Grohe
Keyword(s):  
Constraint Satisfaction ◽  
Network Architecture ◽  
Optimization Problems ◽  
Independent Set ◽  
Constraint Satisfaction Problems ◽  
Maximum Independent Set ◽  
Combinatorial Optimization Problems ◽  
Maximum Cut ◽  
Binary Constraint ◽  
Graph Neural Networks

Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.

scholarly journals Efficient Algorithms for Maximum Induced Matching Problem in Permutation and Trapezoid Graphs

2021 ◽  
Vol 182 (3) ◽  
pp. 257-283
Author(s):  
Viet Dung Nguyen ◽  
Ba Thai Pham ◽  
Phan Thuan Do
Keyword(s):  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Matching Problem ◽  
Induced Matching ◽  
Graph Classes ◽  
Permutation Models ◽  
Maximum Induced Matching ◽  
Sweep Line ◽  
Trapezoid Graphs ◽  
Better Than

We first design an 𝒪(n2) solution for finding a maximum induced matching in permutation graphs given their permutation models, based on a dynamic programming algorithm with the aid of the sweep line technique. With the support of the disjoint-set data structure, we improve the complexity to 𝒪(m+n). Consequently, we extend this result to give an 𝒪(m+n) algorithm for the same problem in trapezoid graphs. By combining our algorithms with the current best graph identification algorithms, we can solve the MIM problem in permutation and trapezoid graphs in linear and 𝒪(n2) time, respectively. Our results are far better than the best known 𝒪(mn) algorithm for the maximum induced matching problem in both graph classes, which was proposed by Habib et al.

scholarly journals Solving Combinatorial Optimization Problems on Quantum Computers

2020 ◽  
pp. 5-13
Author(s):  
Vyacheslav Korolyov ◽  
Oleksandr Khodzinskyi
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Independent Set ◽  
Cloud Services ◽  
Quantum Computers ◽  
Maximal Independent Set ◽  
Maximum Independent Set ◽  
Combinatorial Optimization Problems ◽  
D Wave

Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.

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