# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Near-optimal solutions to large-scale facility location problems. (English) Zbl 1140.90442
Summary: We investigate the solution of large-scale instances of the capacitated and uncapacitated facility location problems. Let $n$ be the number of customers and $m$ the number of potential facility sites. For the uncapacitated case we solved instances of size $m\times n=3000\times 3000$; for the capacitated case the largest instances were $1000\times 1000$. We use heuristics that produce a feasible integer solution and use a Lagrangian relaxation to obtain a lower bound on the optimal value. In particular, we present new heuristics whose gap from optimality was generally below $1\%$. The heuristics combine the volume algorithm and randomized rounding. For the uncapacitated facility location problem, our computational experiments show that our heuristic compares favorably against DUALOC.

##### MSC:
 90B85 Continuous location 90C59 Approximation methods and heuristics
##### Keywords:
Volume algorithm; Randomized rounding; Facility location
Full Text:
##### References:
 [1] Aardal, K.: Capacitated facility locationseparation algorithms and computational experience. Math. programming 81, 149-175 (1998) · Zbl 0919.90096 [2] Ahn, S.; Cooper, C.; Cornuéjols, G.; Frieze, A. M.: Probabilistic analysis of a relaxation for the k-median problem. Math. oper. Res. 13, 1-31 (1988) · Zbl 0653.90049 [3] Balinski, M. L.: Integer programmingmethods, uses, computation. Management sci. 12, No. 3, 253-313 (1965) · Zbl 0129.12004 [4] Barahona, F.; Anbil, R.: The volume algorithmproducing primal solutions with a subgradient method. Math. programming 87, 385-399 (2000) · Zbl 0961.90058 [5] F.A. Chudak, Improved approximation algorithms for the uncapacitated facility location problem, Ph.D. Thesis, Cornell University, 1998. · Zbl 0910.90201 [6] Chudak, F. A.: Improved approximation algorithms for uncapacitated facility location. Proceedings of the 6th IPCO conference, 180-194 (1998) · Zbl 0910.90201 [7] F.A. Chudak, D.B. Shmoys, Improved approximation algorithms for the uncapacitated facility location problem, SIAM J. Comput. 33 (2003) 1, 1 -- 25. · Zbl 1044.90056 [8] Cornuéjols, G.; Nemhauser, G. L.; Wolsey, L. A.: The uncapacitated facility location problem. Discrete location theory, 119-171 (1990) [9] Cornuéjols, G.; Sridharan, R.; Thizy, J. M.: A comparison of heuristics and relaxations for the capacitated plant location problem. European J. Oper. res. 50, 280-297 (1991) · Zbl 0734.90046 [10] Erlenkotter, D.: A dual-based procedure for uncapacitated facility location. Oper. res. 26, 992-1009 (1978) · Zbl 0422.90053 [11] D. Erlenkotter, Program DUALOC --- Version II, Distributed on request, 1991. [12] A.V. Goldberg, An efficient implementation of a scaling minimum-cost flow algorithm, Technical Report STAN-CS-92-1439, Stanford University, 1992. [13] Held, M.; Wolfe, P.; Crowder, H. P.: Validation of subgradient optimization. Math. programming 49, 62-88 (1991) · Zbl 0284.90057 [14] C. Lemaréchal, Nondifferential optimization, in: G.L. Nemhauser, A.H.G. Rinnoy Kan, M.J. Todd (Eds.), Optimization, Handbooks in Operations Research, North Holland, Amsterdam, 1989, pp. 529 -- 572. [15] P. Mirchandani, R. Francis (Eds.), Discrete Location Theory, Wiley, New York, 1990. · Zbl 0718.00021 [16] Raghavan, P.; Thompson, C. D.: Randomized rounding. Combinatorica 7, 365-374 (1987) · Zbl 0651.90052 [17] D.B. Shmoys, É. Tardos, K. Aardal, Approximation algorithms for facility location problems, in: Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 265 -- 274. · Zbl 0962.68008 [18] Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable functions. Math. programming study 3, 145-173 (1975) · Zbl 0369.90093