Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3

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.

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Spanning k-tree with specified vertices

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500435 ◽

2019 ◽

Vol 11 (04) ◽

pp. 1950043

Author(s):

Feifei Song ◽

Jianjie Zhou

Keyword(s):

Maximum Degree ◽

Spanning Trees ◽

Independent Set ◽

Disjoint Paths ◽

Maximum Independent Set ◽

Local Connectivity

A [Formula: see text]-tree is a tree with maximum degree at most [Formula: see text]. For a graph [Formula: see text] and [Formula: see text] with [Formula: see text], let [Formula: see text] be the cardinality of a maximum independent set containing [Formula: see text] and [Formula: see text]. For a graph [Formula: see text] and [Formula: see text], the local connectivity [Formula: see text] is defined to be the maximum number of internally disjoint paths connecting [Formula: see text] and [Formula: see text] in [Formula: see text]. In this paper, we prove the following theorem and show the condition is sharp. Let [Formula: see text], [Formula: see text] and [Formula: see text] be integers with [Formula: see text], [Formula: see text] and [Formula: see text]. For any two nonadjacent vertices [Formula: see text] and [Formula: see text] of [Formula: see text], we have [Formula: see text] and [Formula: see text]. Then for any [Formula: see text] distinct vertices of [Formula: see text], [Formula: see text] has a spanning [Formula: see text]-tree such that each of [Formula: see text] specified vertices has degree at most [Formula: see text]. This theorem implies H. Matsuda and H. Matsumura’s result in [on a [Formula: see text]-tree containing specified cleares in a graph, Graphs Combin. 22 (2006) 371–381] and V. Neumann-Lara and E. Rivera-Campo’s result in [Spanning trees with bounded degrees, Combinatorica 11 (1991) 55–61].

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

10.21203/rs.3.rs-22822/v2 ◽

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.

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Smaller Cuts, Higher Lower Bounds

ACM Transactions on Algorithms ◽

10.1145/3469834 ◽

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.

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COMPLEXITY ASPECTS OF VISIBILITY GRAPHS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195995000179 ◽

1995 ◽

Vol 05 (03) ◽

pp. 289-312 ◽

Cited By ~ 6

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.

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On the Maximum Independent Set Problem in Graphs of Bounded Maximum Degree

Acta Mathematica Vietnamica ◽

10.1007/s40306-020-00368-0 ◽

2020 ◽

Vol 45 (2) ◽

pp. 463-475

Author(s):

Ngoc C. Lê ◽

Trung Tran

Keyword(s):

Maximum Degree ◽

Independent Set ◽

Maximum Independent Set ◽

Independent Set Problem ◽

Maximum Independent Set Problem

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

10.21203/rs.3.rs-22822/v1 ◽

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.

Download Full-text