A Warning Propagation-Based Linear-Time-and-Space Algorithm for the Minimum Vertex Cover Problem on Giant Graphs
- 1.6k Downloads
A vertex cover (VC) of a graph \(G\) is a subset of vertices in \(G\) such that at least one endpoint vertex of each edge in \(G\) is in this subset. The minimum VC (MVC) problem is to identify a VC of minimum size (cardinality) and is known to be NP-hard. Although many local search algorithms have been developed to solve the MVC problem close-to-optimally, their applicability on giant graphs (with no less than 100,000 vertices) is limited. For such graphs, there are two reasons why it would be beneficial to have linear-time-and-space algorithms that produce small VCs. Such algorithms can: (a) serve as preprocessing steps to produce good starting states for local search algorithms and (b) also be useful for many applications that require finding small VCs quickly. In this paper, we develop a new linear-time-and-space algorithm, called MVC-WP, for solving the MVC problem on giant graphs based on the idea of warning propagation, which has so far only been used as a theoretical tool for studying properties of MVCs on infinite random graphs. We empirically show that it outperforms other known linear-time-and-space algorithms in terms of sizes of produced VCs.
The research at the University of Southern California was supported by the National Science Foundation (NSF) under grant numbers 1724392, 1409987, and 1319966. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsoring organizations, agencies or the U.S. government.
- 1.Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: theory and experiments. In: The Workshop on Algorithm Engineering and Experiments (2004)Google Scholar
- 3.Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D. (eds.): Graph Partitioning and Graph Clustering. Discrete Mathematics and Theoretical Computer Science. American Mathematical Society and Center, Providence (2013)Google Scholar
- 5.Cai, S.: Balance between complexity and quality: local search for minimum vertex cover in massive graphs. In: The International Joint Conference on Artificial Intelligence, pp. 747–753 (2015)Google Scholar
- 11.Filiol, E., Franc, E., Gubbioli, A., Moquet, B., Roblot, G.: Combinatorial optimisation of worm propagation on an unknown network. Int. J. Comput. Electr. Autom. Control Inf. Eng. 1(10), 2931–2937 (2007)Google Scholar
- 12.Finch, S.R.: Mathematical Constants, Encyclopedia of Mathematics and Its Applications, vol. 94. Cambridge University Press, Cambridge (2003)Google Scholar
- 14.Goyal, A., Lu, W., Lakshmanan, L.V.S.: SIMPATH: an efficient algorithm for influence maximization under the linear threshold model. In: The IEEE International Conference on Data Mining, pp. 211–220 (2011). https://doi.org/10.1109/ICDM.2011.132
- 15.Haynsworth, E.V., Goldberg, K.: Bernoulli and Euler polynomials-Riemann zeta function. In: Abramowitz, M., Stegun, I.A. (eds.) Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, pp. 803–819. Dover Publications, Inc., Mineola (1965)Google Scholar
- 21.Niskanen, S., Östergård, P.R.J.: Cliquer user’s guide, version 1.0. Technical report T48, Communications Laboratory, Helsinki University of Technology, Espoo, Finland (2003)Google Scholar
- 23.Rossi, R.A., Ahmed, N.K.: The network data repository with interactive graph analytics and visualization. In: the AAAI Conference on Artificial Intelligence, pp. 4292–4293 (2015). http://networkrepository.com
- 28.Xu, H., Kumar, T.K.S., Koenig, S.: A linear-time and linear-space algorithm for the minimum vertex cover problem on giant graphs. In: The International Symposium on Combinatorial Search, pp. 173–174 (2017)Google Scholar
- 29.Yamaguchi, K., Masuda, S.: A new exact algorithm for the maximum weight clique problem. In: The International Technical Conference on Circuits/Systems, Computers and Communications. pp. 317–320 (2008)Google Scholar