Abstract
This paper studies a new version of the location problem called the mixed center location problem. Let P be a set of n points in the plane. We first consider the mixed 2-center problem, where one of the centers must be in P, and we solve it in \(O(n^2\log n)\) time. Second, we consider the mixed k-center problem, where m of the centers are in P, and we solve it in \(O(n^{m+O(\sqrt{k-m})})\) time. Motivated by two practical constraints, we propose two variations of the problem. Third, we present a 2-approximation algorithm and three heuristics solving the mixed k-center problem (\(k>2\)).
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References
Agarwal PK, Sharir M, Welzl E (2000) The discrete 2-center problem. Discrete Comput Geom 20:287–305
Aurenhammer F, Klein R, Lee DT (2013) Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Company, Singapore
Chan TM (1999) More planar two-center algorithms. Comput Geom Theory Appl 13:189–198
Chen D, Chen R (2009) New relaxation-based algorithms for the optimal solution of the continuous and discrete p-center problems. Comput Oper Res 36:1646–1655
Daskin MS (2000) A new approach to solving the vertex p-center problem to optimality: algorithm and computational results. Commun Oper Res Soc Jpn 45(9):428–36
Drezener Z (1984) The p-center problem-heuristics and optimal algorithms. J Oper Res Soc 35:741–748
Elloumi S, Labb M, Pochet Y (2004) A new formulation and resolution method for the p-center problem. IINFORMS J Comput 16:84–94
Eppstein D (1997) Faster construction of planar two-centers. In: Proceedings of the 8th ACM-SIAM symposium on discrete algorithms, pp 131–138
Feder T, Greene D (1988) Optimal algorithms for approximate clustering. In: Proceedings of the 20th ACM symposium on theory of computing, pp 434–444
Franti P et al (2015) Clustering datasets. http://cs.uef.fi/sipu/datasets/
Gonzalez TF (1985) Clustering to minimize the maximum intercluster distance. Theor Comput Sci 38:293–306
Hershberger J (1993) A faster algorithm for the two-center decision problem. Inf Process Lett 47:23–29
Hershberger J, Suri S (1991) Finding tailored partitions. J Algorithms 12:431–463
Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the k-center problem. Math Oper Res 10(2):180–184
Hwang RZ, Lee RCT, Chang RC (1993) The slab dividing approach to solve the Euclidean P-center problem. Algorithmica 9:1–22
Ilhan T, Ozsoy FA, Pinar MC (2002) An efficient exact algorithm for the vertex p-center problem and computational experiments for different set covering subproblems. http://www.optimization-online.org/DB_HTML/2002/12/588.html
Jaromczyk J, Kowaluk M (1994) An efficient algorithm for the Euclidean two-center problem. In: Proceedings 10th ACM symposium on computational geometry, pp 303–311
Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems, Part I. The p-centers. SIAM J Appl Math 37:513–538
Megiddo N (1983) Linear-time algorithms for the linear programming in \(R^3\) and related problems. SIAM J Comput 12:759–776
Megiddo N, Supowit K (1984) On the complexity of some common geometric location problems. SIAM J Comput 13:1182–1196
Nagarajan V, Schieber B, Shachnai H (2013) The Eucildean k-supplier problem. In: Goemans M, Correa J (eds) IPCO 2013. LNCS, vol 7801. Springer, Heidelberg, pp 290–301
Nenad M, Martine L, Pierre H (2003) Solving the p-center problem with tabu search and variable neighborhood search. Networks 42:48–64
Shamos M, Michael I, Hoey D (1975) Closest-point problems. In: 16th annual symposium on IEEE foundations of computer science, pp 151–162
Sharir M (1997) A near-linear algorithm for the planar 2-center problem. Discrete Comput Geom 18:125–134
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Xu, Y., Peng, J. & Xu, Y. The mixed center location problem. J Comb Optim 36, 1128–1144 (2018). https://doi.org/10.1007/s10878-017-0183-4
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DOI: https://doi.org/10.1007/s10878-017-0183-4