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.

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 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 ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm ◽  
Hard Problems ◽  
Traditional Approaches

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 three 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. High-quality solutions can be obtained with much less time consuming compared to traditional approaches.

COMPLEXITY ASPECTS OF VISIBILITY GRAPHS

1995 ◽  
Vol 05 (03) ◽  
pp. 289-312 ◽  
Author(s):  
YAW-LING LIN ◽  
STEVEN S. SKIENA
Keyword(s):  
Optimization Problems ◽  
Dominating Set ◽  
Vertex Cover ◽  
Independent Set ◽  
Visibility Graph ◽  
Outerplanar Graph ◽  
Maximum Independent Set ◽  
Graph Recognition ◽  
Visibility Graphs ◽  
Necessary And Sufficient

In this paper, we consider two distinct problems related to complexity aspects of the visibility graphs of simple polygons. Recognizing visibility graphs is a long-standing open problem. It is not even known whether visibility graph recognition is in NP. That visibility graph recognition is in NP would be established if we could demonstrate that any n vertex visibility graph is realized by a polygon which can be drawn on an exponentially-sized grid. This motivates a study of the area requirements for realizing visibility graphs. In this paper, we prove: • Θ(n3) area is necessary and sufficient to realize the complete visibility graph Kn. • There exist visibility graphs which require exponential area to realize. • Any maximal outerplanar graph of diameter d can be realized in O(d2 · 2d) area, which can be as small as O(n log2 n) for a balanced mop. Linear maximal outer-planar graphs can be realized in O(n8) area. The second part of this paper considers the complexity of specific optimization problems on visibility graphs. Given a polygon P, we show that finding a maximum independent set, minimum vertex cover, or maximum dominating set in the visibility graph of P are all NP-complete. Further we show that for polygons P1 and P2, the problem of testing if they have isomorphic visibility graphs is isomorphism-complete. These problems remain hard when given the visibility graphs as input.

scholarly journals Smaller Cuts, Higher Lower Bounds

2021 ◽  
Vol 17 (4) ◽  
pp. 1-40
Author(s):  
Amir Abboud ◽  
Keren Censor-Hillel ◽  
Seri Khoury ◽  
Ami Paz
Keyword(s):  
Lower Bounds ◽  
Vertex Cover ◽  
Shortest Paths ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Graph Problems ◽  
All Pairs Shortest Paths ◽  
Dense Graphs ◽  
Hard Problems ◽  
A New Technique

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

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 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 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.

scholarly journals Using synchronized oscillators to compute the maximum independent set

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Antik Mallick ◽  
Mohammad Khairul Bashar ◽  
Daniel S. Truesdell ◽  
Benton H. Calhoun ◽  
Siddharth Joshi ◽  
...  
Keyword(s):  
Integrated Circuit ◽  
Temporal Dynamics ◽  
Optimization Problems ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Analog Computing ◽  
Spatio Temporal ◽  
Computing Platforms ◽  
Maximum Independent Set Problem ◽  
Computational Properties

Abstract Not all computing problems are created equal. The inherent complexity of processing certain classes of problems using digital computers has inspired the exploration of alternate computing paradigms. Coupled oscillators exhibiting rich spatio-temporal dynamics have been proposed for solving hard optimization problems. However, the physical implementation of such systems has been constrained to small prototypes. Consequently, the computational properties of this paradigm remain inadequately explored. Here, we demonstrate an integrated circuit of thirty oscillators with highly reconfigurable coupling to compute optimal/near-optimal solutions to the archetypally hard Maximum Independent Set problem with over 90% accuracy. This platform uniquely enables us to characterize the dynamical and computational properties of this hardware approach. We show that the Maximum Independent Set is more challenging to compute in sparser graphs than in denser ones. Finally, using simulations we evaluate the scalability of the proposed approach. Our work marks an important step towards enabling application-specific analog computing platforms to solve computationally hard problems.

scholarly journals Improving Optimization Bounds Using Machine Learning: Decision Diagrams Meet Deep Reinforcement Learning

2019 ◽  
Vol 33 ◽  
pp. 1443-1451 ◽  
Author(s):  
Quentin Cappart ◽  
Emmanuel Goutierre ◽  
David Bergman ◽  
Louis-Martin Rousseau
Keyword(s):  
Machine Learning ◽  
Reinforcement Learning ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Independent Set ◽  
Upper And Lower Bounds ◽  
Maximum Independent Set ◽  
Critical Element ◽  
Decision Diagrams ◽  
Combinatorial Optimization Problems

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower bounds that can be significantly better than classical bounding mechanisms, such as linear relaxations. It is well known that the quality of the bounds achieved through this flexible bounding method is highly reliant on the ordering of variables chosen for building the diagram, and finding an ordering that optimizes standard metrics is an NP-hard problem. In this paper, we propose an innovative and generic approach based on deep reinforcement learning for obtaining an ordering for tightening the bounds obtained with relaxed and restricted DDs. We apply the approach to both the Maximum Independent Set Problem and the Maximum Cut Problem. Experimental results on synthetic instances show that the deep reinforcement learning approach, by achieving tighter objective function bounds, generally outperforms ordering methods commonly used in the literature when the distribution of instances is known. To the best knowledge of the authors, this is the first paper to apply machine learning to directly improve relaxation bounds obtained by general-purpose bounding mechanisms for combinatorial optimization problems.

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