ABSTRACT
This paper studies the problem of efficiently computing a maximum independent set from a large graph, a fundamental problem in graph analysis. Due to the hardness results of computing an exact maximum independent set or an approximate maximum independent set with accuracy guarantee, the existing algorithms resort to heuristic techniques for approximately computing a maximum independent set with good performance in practice but no accuracy guarantee theoretically. Observing that the existing techniques have various limits, in this paper, we aim to develop efficient algorithms (with linear or near-linear time complexity) that can generate a high-quality (large-size) independent set from a graph in practice. In particular, firstly we develop a Reducing-Peeling framework which iteratively reduces the graph size by applying reduction rules on vertices with very low degrees (Reducing) and temporarily removing the vertex with the highest degree (Peeling) if the reduction rules cannot be applied. Secondly, based on our framework we design two baseline algorithms, BDOne and BDTwo, by utilizing the existing reduction rules for handling degree-one and degree-two vertices, respectively. Both algorithms can generate higher-quality (larger-size) independent sets than the existing algorithms. Thirdly, we propose a linear-time algorithm, LinearTime, and a near-linear time algorithm, NearLinear, by designing new reduction rules and developing techniques for efficiently and incrementally applying reduction rules. In practice, LinearTime takes similar time and space to BDOne but computes a higher quality independent set, similar in size to that of an independent set generated by BDTwo. Moreover, in practice NearLinear has a good chance to generate a maximum independent set and it often generates near-maximum independent sets. Fourthly, we extend our techniques to accelerate the existing iterated local search algorithms. Extensive empirical studies show that all our algorithms output much larger independent sets than the existing linear-time algorithms while having a similar running time, as well as achieve significant speedup against the existing iterated local search algorithms.
- T. Akiba and Y. Iwata. Branch-and-reduce exponential/fpt algorithms in practice: A case study of vertex cover. Theor. Comput. Sci., 609, 2016. Google ScholarDigital Library
- D. V. Andrade, M. G. Resende, and R. F. Werneck. Fast local search for the maximum independent set problem. J. Heuristics, 18(4), 2012. Google ScholarDigital Library
- F. Araujo, J. Farinha, P. Domingues, G. C. Silaghi, and D. Kondo. A maximum independent set approach for collusion detection in voting pools. J. Parallel and Distributed Computing, 71(10), 2011.Google ScholarCross Ref
- A.-L. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286, 1999.Google Scholar
- S. Basagni. Finding a maximal weighted independent set in wireless networks. Telecommunication Systems, 18(1--3), 2001. Google ScholarDigital Library
- U. Benlic and J.-K. Hao. Breakout local search for maximum clique problems. Computers & Operations Research, 40(1), 2013. Google ScholarDigital Library
- P. Berman and T. Fujito. On approximation properties of the independent set problem for low degree graphs. Theor. Comput. Sys., 32(2), 1999.Google Scholar
- F. Bi, L. Chang, X. Lin, L. Qin, and W. Zhang. Efficient subgraph matching by postponing cartesian products. In Proc. of SIGMOD'16, 2016. Google ScholarDigital Library
- S. Butenko. Maximum independent set and related problems with applications. PhD thesis, University of Florida, 2003. Google ScholarDigital Library
- J. M. Byskov. Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett., 32(6), 2004. Google ScholarDigital Library
- S. Cai. Balance between complexity and quality: local search for minimum vertex cover in massive graphs. In Proc. of IJCAI'15, 2015. Google ScholarDigital Library
- L. Chang, J. X. Yu, and L. Qin. Fast maximal cliques enumeration in sparse graphs. Algorithmica, 2012.Google Scholar
- L. Chang, J. X. Yu, L. Qin, X. Lin, C. Liu, and W. Liang. Efficiently computing k-edge connected components via graph decomposition. In Proc. of SIGMOD'13, 2013. Google ScholarDigital Library
- J. Cheng, Y. Ke, S. Chu, and C. Cheng. Efficient processing of distance queries in large graphs: a vertex cover approach. In Proc. of SIGMOD'12, 2012. Google ScholarDigital Library
- J. Cheng, Y. Ke, A. W.-C. Fu, J. X. Yu, and L. Zhu. Finding maximal cliques in massive networks by h*-graph. In Proc. of SIGMOD'10, 2010. Google ScholarDigital Library
- N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput., 14(1), 1985. Google ScholarDigital Library
- R. H. Chitnis, G. Cormode, M. T. Hajiaghayi, and M. Monemizadeh. Parameterized streaming: Maximal matching and vertex cover. In Proc. of SODA'15, 2015. Google ScholarDigital Library
- T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. McGraw-Hill Higher Education, 2001. Google ScholarDigital Library
- J. Dahlum, S. Lamm, P. Sanders, C. Schulz, D. Strash, and R. F. Werneck. Accelerating local search for the maximum independent set problem. In Proc. of SEA'16, 2016. Google ScholarDigital Library
- U. Feige. Approximating maximum clique by removing subgraphs. SIAM J. Discrete Math., 18(2), 2004. Google ScholarDigital Library
- F. V. Fomin, F. Grandoni, and D. Kratsch. A measure & conquer approach for the analysis of exact algorithms. J. ACM, 56(5), 2009. Google ScholarDigital Library
- A. W.-C. Fu, H. Wu, J. Cheng, and R. C.-W. Wong. Is-label: an independent-set based labeling scheme for point-to-point distance querying. PVLDB, 6(6), 2013. Google ScholarDigital Library
- M. R. Garey and D. S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1990. Google ScholarDigital Library
- A. Gibbons. Algorithmic Graph Theory. Cambridge University Press, 1985.Google Scholar
- M. M. Halldórsson and J. Radhakrishnan. Greed is good: Approximating independent sets in sparse and bounded-degree graphs. Algorithmica, 18(1), 1997.Google Scholar
- J. Håstad. Clique is hard to approximate within n 1ε. In Proc. of FOCS'96, 1996. Google ScholarDigital Library
- S. Lamm, P. Sanders, and C. Schulz. Graph partitioning for independent sets. In Proc. of SEA'15, 2015. Google ScholarDigital Library
- S. Lamm, P. Sanders, C. Schulz, D. Strash, and R. F. Werneck. Finding near-optimal independent sets at scale. In Proc. of ALENEX'16, 2016.Google ScholarCross Ref
- Y. Lim, U. Kang, and C. Faloutsos. Slashburn: Graph compression and mining beyond caveman communities. IEEE Trans. Knowl. Data Eng., 26(12), 2014.Google Scholar
- Y. Liu, J. Lu, H. Yang, X. Xiao, and Z. Wei. Towards maximum independent sets on massive graphs. PVLDB, 8(13), 2015. Google ScholarDigital Library
- P. M. Pardalos and J. Xue. The maximum clique problem. J. global Optimization, 4(3), 1994.Google Scholar
- D. Puthal, S. Nepal, C. Paris, R. Ranjan, and J. Chen. Efficient algorithms for social network coverage and reach. In 2015 IEEE International Congress on Big Data, 2015. Google ScholarDigital Library
- P. S. Segundo, A. Lopez, and P. M. Pardalos. A new exact maximum clique algorithm for large and massive sparse graphs. Computers & Operations Research, 66, 2016. Google ScholarDigital Library
- S. S. Skiena. The Algorithm Design Manual. Springer Publishing Company, Incorporated, 2nd edition, 2008. Google ScholarDigital Library
- E. Tomita, Y. Sutani, T. Higashi, S. Takahashi, and M. Wakatsuki. A simple and faster branch-and-bound algorithm for finding a maximum clique. In Proc. of WALCOM'10, 2010. Google ScholarDigital Library
- P.-J. Wan, B. Xu, L. Wang, S. Ji, and O. Frieder. A new paradigm for multiflow in wireless networks: Theory and applications. In Proc. of INFOCOM'15, 2015.Google ScholarCross Ref
- J. Xiang, C. Guo, and A. Aboulnaga. Scalable maximum clique computation using mapreduce. In Proc. of ICDE'13, 2013. Google ScholarDigital Library
- M. Xiao and H. Nagamochi. Confining sets and avoiding bottleneck cases: A simple maximum independent set algorithm in degree-3 graphs. Theor. Comput. Sci., 469, 2013. Google ScholarDigital Library
- Z. Zhang, L. Qin, and J. X. Yu. Contract & expand: I/O efficient sccs computing. In Proc. of ICDE'14, 2014.Google ScholarCross Ref
- Y. Zhu, L. Qin, J. X. Yu, Y. Ke, and X. Lin. High efficiency and quality: large graphs matching. VLDB J., 22(3), 2013. Google ScholarDigital Library
Index Terms
- Computing A Near-Maximum Independent Set in Linear Time by Reducing-Peeling
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