MAXIMUM INDEPENDENT SET OF A PERMUTATION GRAPH IN K TRACKS

A maximum weighted independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram (i.e., a set Φ of chords with end-points on two horizontal lines). The problem of finding, among all noncrossing subsets of Φ with density at most k, one with maximum size is considered, where the density of a subset is the maximum number of chords crossing a vertical line and k is a given parameter. A Θ(n log n) time and Θ(n) space algorithm, for solving the problem with n chords, is proposed. As an application, we solve the problem of finding, among all proper subsets with density at most k of an interval graph, one with maximum number of intervals.

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Finding a maximum independent set in a permutation graph

Information Processing Letters ◽

10.1016/0020-0190(90)90180-6 ◽

1990 ◽

Vol 36 (1) ◽

pp. 19-23 ◽

Cited By ~ 16

Author(s):

Haklin Kim

Keyword(s):

Independent Set ◽

Maximum Independent Set ◽

Permutation Graph

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A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems

Journal of the Operational Research Society ◽

10.1038/sj.jors.2600405 ◽

1997 ◽

Vol 48 (6) ◽

pp. 612-622 ◽

Cited By ~ 11

Author(s):

M Hifi

Keyword(s):

Genetic Algorithm ◽

Independent Set ◽

Maximum Independent Set ◽

Equivalent Problems

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ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

Nagoya Mathematical Journal ◽

10.1017/nmj.2021.6 ◽

2021 ◽

pp. 1-40

Author(s):

NICK GILL ◽

BIANCA LODÀ ◽

PABLO SPIGA

Keyword(s):

Permutation Group ◽

Model Theory ◽

Independent Set ◽

Maximum Size ◽

Proper Subset ◽

Permutation Groups ◽

Finite Permutation Group ◽

Pointwise Stabilizer ◽

Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

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Beyond Alice and Bob: Improved Inapproximability for Maximum Independent Set in CONGEST

Proceedings of the 39th Symposium on Principles of Distributed Computing ◽

10.1145/3382734.3405702 ◽

2020 ◽

Author(s):

Yuval Efron ◽

Ofer Grossman ◽

Seri Khoury

Keyword(s):

Independent Set ◽

Maximum Independent Set

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

Computational Social Networks ◽

10.1186/s40649-021-00086-z ◽

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.

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