Known Algorithms for Edge Clique Cover are Probably Optimal

Graphs and Algorithms International audience A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

Download Full-text

Edge clique cover of claw-free graphs

Journal of Graph Theory ◽

10.1002/jgt.22403 ◽

2018 ◽

Vol 90 (3) ◽

pp. 311-405 ◽

Cited By ~ 1

Author(s):

Ramin Javadi ◽

Sepehr Hajebi

Keyword(s):

Edge Clique Cover

Download Full-text

On the Complete Width and Edge Clique Cover Problems

Lecture Notes in Computer Science - Computing and Combinatorics ◽

10.1007/978-3-319-21398-9_42 ◽

2015 ◽

pp. 537-547 ◽

Cited By ~ 2

Author(s):

Van Bang Le ◽

Sheng-Lung Peng

Keyword(s):

Edge Clique Cover

Download Full-text

q-Rung Orthopair Fuzzy Competition Graphs with Application in the Soil Ecosystem

Mathematics ◽

10.3390/math7010091 ◽

2019 ◽

Vol 7 (1) ◽

pp. 91 ◽

Cited By ~ 27

Author(s):

Amna Habib ◽

Muhammad Akram ◽

Adeel Farooq

Keyword(s):

Fuzzy Set ◽

Real Life ◽

Fuzzy Model ◽

The Novel ◽

Soil Ecosystem ◽

Practical Applications ◽

Research Article ◽

Fuzzy Graphs ◽

Edge Clique Cover ◽

Competition Number

The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications.

Download Full-text

Known algorithms for Edge Clique Cover are probably optimal

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms ◽

10.1137/1.9781611973105.75 ◽

2013 ◽

Cited By ~ 6

Author(s):

Marek Cygan ◽

Marcin Pilipczuk ◽

Michal Pilipczuk

Keyword(s):

Edge Clique Cover

Download Full-text

Sample Complexity Bounds for RNNs with Application to Combinatorial Graph Problems (Student Abstract)

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i10.7144 ◽

2020 ◽

Vol 34 (10) ◽

pp. 13745-13746

Author(s):

Nil-Jana Akpinar ◽

Bernhard Kratzwald ◽

Stefan Feuerriegel

Keyword(s):

Neural Networks ◽

Recurrent Neural Networks ◽

Theoretical Foundation ◽

Single Layer ◽

Sample Complexity ◽

Complexity Bounds ◽

Hard Edge ◽

Graph Problems ◽

Edge Clique Cover ◽

Primary Contribution

Learning to predict solutions to real-valued combinatorial graph problems promises efficient approximations. As demonstrated based on the NP-hard edge clique cover number, recurrent neural networks (RNNs) are particularly suited for this task and can even outperform state-of-the-art heuristics. However, the theoretical framework for estimating real-valued RNNs is understood only poorly. As our primary contribution, this is the first work that upper bounds the sample complexity for learning real-valued RNNs. While such derivations have been made earlier for feed-forward and convolutional neural networks, our work presents the first such attempt for recurrent neural networks. Given a single-layer RNN with a rectified linear units and input of length b, we show that a population prediction error of ε can be realized with at most Õ(a4b/ε2) samples.1 We further derive comparable results for multi-layer RNNs. Accordingly, a size-adaptive RNN fed with graphs of at most n vertices can be learned in Õ(n6/ε2), i.,e., with only a polynomial number of samples. For combinatorial graph problems, this provides a theoretical foundation that renders RNNs competitive.

Download Full-text