Algorithm for Relatively Small Planted Clique with
 Small Edge Probability

doi:10.16183/j.cnki.jsjtu.2017.10.008

Abstract

Abstract: Planted clique problem is a central problem in averagecase complexity theory. The random graph with planted clique model is extended, and small edge probability conditions are invented to allow the edge probability to decrease. It is shown that under such conditions there exists a polynomialtime randomized algorithm to find a relatively small planted clique by using probability tools for analysis of randomized algorithms. The result improves analysis of the algorithm in the literature.

Key words: , random graph, planted clique, small edge probability, randomized algorithm, average case complexity

CLC Number:

TP 301.6

HUANG Xuangui. Algorithm for Relatively Small Planted Clique with
Small Edge Probability[J]. Journal of Shanghai Jiaotong University, 2017, 51(10): 1202-1206.

References

［1］HASTAD J. Clique is hard to approximate within n1－ε［C］∥Proceedings of The 37th Annual Symposium on Foundations of Computer Science. Washington, DC: IEEE Computer Society, 1996: 627636.
［2］JANSON S, LUCZAK T, RUCINSKI A. Random graphs［M］. New York: John Wiley & Sons, 2000.
［3］ARORA S, BARAK B. Computational complexity: A modern approach［M］. New York: Cambridge University Press, 2009: 361372.
［4］KARP R M. The probabilistic analysis of some combinatorial search algorithms［C］∥Algorithms and Complexity: New Directions and Recent Results. New York: Academic Press, 1976: 119.
［5］JERRUM M. Large cliques elude the Metropolis process［J］. Random Structures and Algorithms, 1992, 3(4): 347359.
［6］KUERA L. Expected complexity of graph partitioning problems［J］. Discrete Applied Mathematics, 1995, 57(2): 193212.
［7］ALON N, KRIVELEVICH M, SUDAKOV B. Finding a large hidden clique in a random graph［J］. Random Structures and Algorithms, 1998, 13(3/4): 457466.
［8］FEIGE U, RON D. Finding hidden cliques in linear time［C］∥Proceedings of The 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms. Nancy, France: Discrete Mathematics and Theoretical Computer Science (DMTCS), 2010: 189204.
［9］DEKEL Y, GURELGUREVICH O, PERES Y. Finding hidden cliques in linear time with high probability［C］∥Proceedings of the Meeting on Analytic Algorithmics and Combinatorics. San Francisco: Society for Industrial and Applied Mathematics, 2011: 6775.
［10］AMES B P W, VAVASIS S A. Nuclear norm minimization for the planted clique and biclique problems［J］. Mathematical Programming, 2011, 129(1): 6989.
［11］DESHPANDE Y, MONTANARI A. Finding hidden cliques of size N/e in nearly linear time［J］. Foundations of Computational Mathematics, 2015, 15(4): 10691128.
［12］FELDMAN V, GRIGORESCU E, REYZIN L, et al. Statistical algorithms and a lower bound for detecting planted cliques［C］∥Proceedings of The 45th Annual ACM Symposium on Theory of Computing. Palo Alto, USA: Association for Computing Machinery, 2013: 655664.
［13］MEKA R, POTECHIN A, WIGDERSON A. Sumofsquares lower bounds for planted clique［C］∥Proceedings of the 47th Annual ACM Symposium on Theory of Computing. Portland, USA: Association for Computing Machinery, 2015: 8796.
［14］DESHPANDE Y, MONTANARI A. Improved sumofsquares lower bounds for hidden clique and hidden submatrix problems［C］∥Proceedings of The 28th Conference on Learning Theory. Paris, France: JMLR, Inc, 2015: 523562.
［15］HAZAN E, KRAUTHGAMER R. How hard is it to approximate the best Nash equilibrium?［J］. SIAM Journal on Computing, 2011, 40(1): 7991.
［16］SANTHANAM R. The complexity of explicit constructions［J］. Theory of Computing Systems, 2012, 51(3): 297312.
［17］HAJEK B, WU Y, XU J. Computational lower bounds for community detection on random graphs［C］∥Proceedings of The 28th Conference on Learning Theory. Paris, France: JMLR, Inc, 2015: 899928.

[1]	LIUYongfei. Simulation Analysis of a Subsea Hydraulic Control System Based on SimulationX [J]. Ocean Engineering Equipment and Technology, 2022, 9(1): 1-7.
[2]	WANGJuan, ZHENG Maoyao, ZHANGZiliang, SONGGuangxing, ZHANGLei. Precision Control Technology for Large Semi-Submersible Platforms [J]. Ocean Engineering Equipment and Technology, 2022, 9(1): 21-26.
[3]	WANGJuan, YANG Mingwang, ZHENG Maoyao, LIULingyun, ZHAOLijun. Application of High Strength Steelin Construction of Large Semi-Submersible Platforms [J]. Ocean Engineering Equipment and Technology, 2022, 9(1): 27-31.
[4]	LIU Hao , ZHANGNing , WANG Huoping , ZHU Liyun , ZHANG Yu . A Method for Improving the Motion Performance of Steel Catenary Riser by Adding Inertia Bodies to the Sagbend Section #br# [J]. Ocean Engineering Equipment and Technology, 2022, 9(1): 37-45.
[5]	YIN Yankun, YI Difei. Engineering Criticality Assessment of Key Joint for Hull Structure of Semi-Submersible Floating Production Unit [J]. Ocean Engineering Equipment and Technology, 2022, 9(1): 52-57.
[6]	LUO Ruiqiao. Application of Down hole Throttle Technology in High Temperature Gas Fields in Eastern South China Sea [J]. Ocean Engineering Equipment and Technology, 2022, 9(1): 58-66.
[7]	MA Qunsheng (马群圣), CEN Xingxing (岑星星), YUAN Junyi (袁骏毅), HOU Xumin (侯旭敏). Word Embedding Bootstrapped Deep Active Learning Method to Information Extraction on Chinese Electronic Medical Record [J]. J Shanghai Jiaotong Univ Sci, 2021, 26(4): 494-502.
[8]	ZHANG Shengfa (张胜发), TANG Na (唐纳), SHEN Guofeng (沈国峰), WANG Han (王悍), QIAO Shan (乔杉). Universal Software Architecture of Magnetic Resonance-Guided Focused Ultrasound Surgery System and Experimental Study [J]. J Shanghai Jiaotong Univ Sci, 2021, 26(4): 471-481.
[9]	KONG Xiangqiang (孔祥强), MENG Xiangxi (孟祥熙), LI Jianbo (李见波), SHANG Yanping (尚燕平), CUI Fulin (崔福林) . Comparative Study on Two-Stage Absorption Refrigeration Systems with Different Working Pairs [J]. J Shanghai Jiaotong Univ Sci, 2021, 26(2): 155-162.
[10]	ZHUANG Weimin (庄蔚敏), WANG Pengyue (王鹏跃), AO Wenhong (熬文宏), CHEN Gang (陈刚) . Experiment and Simulation of Impact Response of Woven CFRP Laminates with Different Stacking Angles [J]. J Shanghai Jiaotong Univ Sci, 2021, 26(2): 218-230.
[11]	ZHOU Xuhui (周旭辉), ZHANG Wenguang (张文光), XIE Jie (谢颉). Effects of Micro-Milling and Laser Engraving on Processing Quality and Implantation Mechanics of PEG-Dexamethasone Coated Neural Probe [J]. J Shanghai Jiaotong Univ Sci, 2021, 26(1): 1-9.
[12]	HUANG Ningning (黄宁宁), MA Yixin (马艺馨), ZHANG Mingzhu (张明珠), GE Hao (葛浩), WU Huawei (吴华伟). Finite Element Modeling of Human Thorax Based on MRI Images for EIT Image Reconstruction [J]. J Shanghai Jiaotong Univ Sci, 2021, 26(1): 33-39.
[13]	WANG Xianjin, GAO Xu, YU Kuigang . Fixture Locating Modelling and Optimization Research of Aluminum Alloy Sidewall in a High-Speed Train Body [J]. J Shanghai Jiaotong Univ Sci, 2020, 25(6): 706-713.
[14]	QIAO Xing, MA Dan, YAO Xuliang, FENG Baolin. Stability and Numerical Analysis of a Standby System [J]. J Shanghai Jiaotong Univ Sci, 2020, 25(6): 769-778.
[15]	WU Jin, MIN Yu, YANG Xiaodie, MA Simin . Micro-Expression Recognition Algorithm Based on Information Entropy Feature [J]. Journal of Shanghai Jiao Tong University(Science), 2020, 25(5): 589-599.