Approximation Algorithms for Minimum-Load k-Facility Location
We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F' of k facilities, and assign each client j to a facility f(j) in F' so as to minimize maximum, over all facilities in F', of the sum of distances of clients j assigned to F' to F'. We call
this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.
approximation algorithms
min-max star cover
facility location
line metrics
17-33
Regular Paper
Sara
Ahmadian
Sara Ahmadian
Babak
Behsaz
Babak Behsaz
Zachary
Friggstad
Zachary Friggstad
Amin
Jorati
Amin Jorati
Mohammad R.
Salavatipour
Mohammad R. Salavatipour
Chaitanya
Swamy
Chaitanya Swamy
10.4230/LIPIcs.APPROX-RANDOM.2014.17
H.-C. An, A. Bhaskara, C. Chekuri, S. Gupta, V. Madan, and O. Svensson. Centrality of trees for capacitated k-center. In Proceedings of APPROX, 2014.
E.M. Arkin, R. Hassin, and A. Levin. Approximations for minimum and min-max vehicle routing problems. Journal of Algorithms, 59(1):1-18, 2006.
V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544-562, 2004.
M. Charikar, S. Guha, É. Tardos, and D. B. Shmoys. A constant-factor approximation algorithm for the k-median problem. Journal of Computer and System Sciences, 65(1):129-149, 2002.
M. Cygan, M. T. Hajiaghayi, and S. Khuller. Lp rounding for k-centers with non-uniform hard capacities. Arxiv preprint arXiv:1208.3054, 2012.
G. Even, N. Garg, J. Könemann, R. Ravi, and A. Sinha. Covering graphs using trees and stars. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 24-35, 2003.
D. Hochbaum and D. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM Journal on Computing, 17:539-551, 1988.
K. Jain, M. Mahdian, E. Markakis, A. Saberi, and V. V. Vazirani. Greedy facility location algorithms analyzed using dual-fitting with factor-revealing lp. Journal of the ACM, 50(6):795-824, 2003.
K. Jain and V. V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM, 48(2):274-296, 2001.
A. Jorati. Approximation algorithms for min-max vehicle routing problems. Master’s thesis, University of Alberta, Department of Computing Science, 2013.
M. R. Khani and M. R. Salavatipour. Improved approximation algorithms for the min-max tree cover and bounded tree cover problems. In APPROX, 2011.
S. Li and O. Svensson. Approximating k-median via pseudo-approximation. In Symposium on Theory of Computing (STOC), 2013.
P. Mirchandani and R. Francis, editors. Discrete location theory. Jown Wiley and Sons, 1990.
H. Nagamochi and K. Okada. Approximating the minmax rooted-tree cover in a tree. Information Processing Letters, 104(5):173-178, 2007.
R. Ravi. Workshop on Flexible Network Design, 2012. URL: http://fnd2012.mimuw.edu.pl/qa/index.php?qa=4&qa_1=approximating-star-cover-problems.
http://fnd2012.mimuw.edu.pl/qa/index.php?qa=4&qa_1=approximating-star-cover-problems
D. B. Shmoys. The design and analysis of approximation algorithms: facility location as a case study. In S. Hosten, J. Lee, and R. Thomas, editors, Trends in Optimization, AMS Proceedings of Symposia in Applied Mathematics 61, pages 85-97. 2004.
D. B. Shmoys and E. Tardos. An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62(3):461-474, 1993.
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