Local search heuristic for k-median and facility location problems

ABSTRACT

In this paper, we analyze local search heuristics for the k-median and facility location problems. We define the {\em locality gap\/} of a local search procedure as the maximum ratio of a locally optimum solution (obtained using this procedure) to the global optimum. For k-median, we show that local search with swaps has a locality gap of exactly 5. When we permit p facilities to be swapped simultaneously then the locality gap of the local search procedure is exactly 3+2/p. This is the first analysis of local search for k-median that provides a bounded performance guarantee with only k medians. This also improves the previous known 4 approximation for this problem. For Uncapacitated facility location, we show that local search, which permits adding, dropping and swapping a facility, has a locality gap of exactly 3. This improves the 5 bound of Korupolu et al. We also consider a capacitated facility location problem where each facilitym has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new operation which opens one or more copies of a facility and drops zero or more facilities. We prove that local search which permits this new operation has a locality gap between 3 and 4. instances where it is not necessary to satisfy every demand. Our algorithms provide the optimum total profit, while stretching the definition of locality by a constant and violating the required demands by a constant. We prove that without this stretch, the problem becomes NP-Hard to approximate. facility location, we show that local search, which permits adding, dropping and swapping a facility, has a locality gap of exactly 3. This improves the 5 bound of Korupolu et al. We also consider a capacitated facility location problem where each facilitym has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new operation which opens one or more copies of a facility and drops zero or more facilities. We prove that local search which permits this new operation has a locality gap between 3 and 4.

REFERENCES

• 1: Moses Charikar , Sudipto Guha, Improved Combinatorial Algorithms for the Facility Location and k-Median Problems, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, p.378, October 17-18, 1999
• 2: Fabián A. Chudak, Improved Approximation Algorithms for Uncapitated Facility Location, Proceedings of the 6th International IPCO Conference on Integer Programming and Combinatorial Optimization, p.180-194, June 22-24, 1998
• 3: Fabián A. Chudak , David B. Shmoys, Improved approximation algorithms for a capacitated facility location problem, Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, p.875-876, January 17-19, 1999, Baltimore, Maryland, United States
• 4: Fabián A. Chudak , David P. Williamson, Improved Approximation Algorithms for Capacitated Facility Location Problems, Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization, p.99-113, June 09-11, 1999
• 5: Sudipto Guha , Samir Khuller, Greedy strikes back: improved facility location algorithms, Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms, p.649-657, January 25-27, 1998, San Francisco, California, United States
• 6: Kamal Jain , Vijay V. Vazirani, Primal-Dual Approximation Algorithms for Metric Facility Location and k-Median Problems, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, p.2, October 17-18, 1999
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• 8: Madhukar R. Korupolu , C. Greg Plaxton , Rajmohan Rajaraman, Analysis of a local search heuristic for facility location problems, Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms, p.1-10, January 25-27, 1998, San Francisco, California, United States
• 9: Madhukar R. Korupolu , C. G Plaxton , Rajmohan Rajaraman, Analysis of a Local Search Heuristic for Facility Location Problems, Certer for Discrete Mathematics & Theoretical Computer Science, 1998
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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
      I.2.8 Problem Solving, Control Methods, and Search
          Subjects: Heuristic methods

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Sorting and searching

  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
      I.2.8 Problem Solving, Control Methods, and Search
          Subjects: Graph and tree search strategies

General Terms:
Algorithms, Measurement, Performance, Theory, Verification