Local Search Heuristics for k -Median and Facility Location Problems

We analyze local search heuristics for the metric k-median and facility location problems. We define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution (obtained using this procedure) to the global optimum. For k-median, we show...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 33; no. 3; pp. 544 - 562
Main Authors: Arya, Vijay, Garg, Naveen, Khandekar, Rohit, Meyerson, Adam, Munagala, Kamesh, Pandit, Vinayaka
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2004
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Approximation
Computer science
Computer science; control theory; systems
Costs
Exact sciences and technology
Heuristic
Mathematical programming
Operational research and scientific management
Operational research. Management science
Optimization. Search problems
Permits
Theoretical computing
Traveling salesman problem
Facility location
Local search
Operations research
Minimization
Global optimum
Approximation algorithm
Location problem
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:We analyze local search heuristics for the metric k-median and facility location problems. We define the locality gap of a local search procedure for a minimization problem 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 5. Furthermore, if we permit up to p facilities to be swapped simultaneously, then the locality gap is 3+2/p. This is the first analysis of a 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 3. This improves the bound of 5 given by M. Korupolu, C. Plaxton, and R. Rajaraman [Analysis of a Local Search Heuristic for Facility Location Problems, Technical Report 98-30, DIMACS, 1998]. We also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities. We prove that this local search has a locality gap between 3 and 4.
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ISSN:0097-5397
1095-7111
DOI:10.1137/S0097539702416402