An Efficient Local Search for the Minimum Independent Dominating Set Problem
In the present paper, we propose an efficient local search for the minimum independent dominating set problem. We consider a local search that uses k-swap as the neighborhood operation. Given a feasible solution S, it is the operation of obtaining another feasible solution by dropping exactly k vertices from S and then by adding any number of vertices to it. We show that, when k=2, (resp., k=3 and a given solution is minimal with respect to 2-swap), we can find an improved solution in the neighborhood or conclude that no such solution exists in O(n Delta) (resp., O(n Delta^3)) time, where n denotes the number of vertices and Delta denotes the maximum degree. We develop a metaheuristic algorithm that repeats the proposed local search and the plateau search iteratively, where the plateau search examines solutions of the same size as the current solution that are obtainable by exchanging a solution vertex and a non-solution vertex. The algorithm is so effective that, among 80 DIMACS graphs, it updates the best-known solution size for five graphs and performs as well as existing methods for the remaining graphs.
Minimum independent dominating set problem
local search
plateau search
metaheuristics
Mathematics of computing~Graph algorithms
13:1-13:13
Regular Paper
The source code of the proposed algorithm is written in C++ and available at http://puzzle.haraguchi-s.otaru-uc.ac.jp/minids/.
https://arxiv.org/abs/1802.06478
Kazuya
Haraguchi
Kazuya Haraguchi
Otaru University of Commerce, Midori 3-5-21/Otaru, Hokkaido, Japan
10.4230/LIPIcs.SEA.2018.13
D.V. Andrade, M.G.C. Resende, and R.F. Werneck. Fast local search for the maximum independent set problem. Journal of Heuristics, 18:525-547, 2012.
C. Berge. Theory of Graphs and its Applications. Methuen, London, 1962.
BHOSLIB: Benchmarks with hidden optimum solutions for graph problems. http://sites.nlsde.buaa.edu.cn/~kexu/benchmarks/graph-benchmarks.htm. accessed on February 1, 2018.
http://sites.nlsde.buaa.edu.cn/~kexu/benchmarks/graph-benchmarks.htm
N. Bourgeois, F.D. Croce, B. Escoffier, and V.Th. Paschos. Fast algorithms for MIN independent dominating set. Discrete Applied Mathematics, 161(4):558-572, 2013.
P.P. Davidson, C. Blum, and J. Lozano. The weighted independent domination problem: ILP model and algorithmic approaches. In Proc. EvoCOP 2017, pages 201-214, 2017. URL: http://dx.doi.org/10.1007/978-3-319-55453-2_14.
http://dx.doi.org/10.1007/978-3-319-55453-2_14
M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman &Company, 1979.
I.P. Gent and T. Walsh. The SAT phase transition. In Proc. ECAI-94, pages 105-109, 1994.
W. Goddard and M.A. Henning. Independent domination in graphs: A survey and recent results. Discrete Mathematics, 313:839-854, 2013.
C.P. Gomes and B. Selman. Problem structure in the presence of perturbations. In Proc. AAAI-97, pages 221-227, 1997.
C.P. Gomes and D.B. Shmoys. Completing quasigroups or latin squares: a structured graph coloring problem. In Proc. Computational Symposium on Graph Coloring and Generalizations, 2002.
M.M. HalldÃ³rsson. Approximating the minimum maximal independence number. Information Processing Letters, 46(4):169-172, 1993.
IBM ILOG CPLEX. https://www.ibm.com/analytics/data-science/prescriptive-analytics/cplex-optimizer. accessed on February 1, 2018.
https://www.ibm.com/analytics/data-science/prescriptive-analytics/cplex-optimizer
F. Kuhn, T. Nieberg, T. Moscibroda, and R. Wattenhofer. Local approximation schemes for ad hoc and sensor networks. In Proc. the 2005 Joint Workshop on Foundations of Mobile Computing, pages 97-103, 2005.
C. Laforest and R. Phan. Solving the minimum independent domination set problem in graphs by exact algorithm and greedy heuristic. RAIRO-Operations Research, 47(3):199-221, 2013.
C. Liu and Y. Song. Exact algorithms for finding the minimum independent dominating set in graphs. In Proc. ISAAC 2006, LNCS 4288, pages 439-448, 2006.
LocalSolver. http://www.localsolver.com/. accessed on February 1, 2018.
http://www.localsolver.com/
F. Mascia. dimacs benchmark set. http://iridia.ulb.ac.be/~fmascia/maximum_clique/DIMACS-benchmark. accessed on February 1, 2018.
http://iridia.ulb.ac.be/~fmascia/maximum_clique/DIMACS-benchmark
W. Pullan. Optimisation of unweighted/weighted maximum independent sets and minimum vertex covers. Discrete Optimization, 6(2):214-219, 2009.
W. Pullan and H.H. Hoos. Dynamic local search for the maximum clique problem. Journal of Artificial Intelligence Research, 25:159-185, 2006.
Y. Wang, J. Chen, H. Sun, and M. Yin. A memetic algorithm for minimum independent dominating set problem. Neural Computing and Applications, in press. URL: http://dx.doi.org/10.1007/s00521-016-2813-7.
http://dx.doi.org/10.1007/s00521-016-2813-7
Y. Wang, R. Li, Y. Zhou, and M. Yin. A path cost-based grasp for minimum independent dominating set problem. Neural Computing and Applications, 28(1):143-151, 2017. URL: http://dx.doi.org/10.1007/s00521-016-2324-6.
http://dx.doi.org/10.1007/s00521-016-2324-6
M. Zehavi. Maximum minimal vertex cover parameterized by vertex cover. SIAM Journal on Discrete Mathematics, 31(4):2440-2456, 2017.
Kazuya Haraguchi
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