## Solving the Uncapacitated Facility Location Problem Using Message Passing Algorithms

Citations: | 2 - 0 self |

### BibTeX

@MISC{Lazic_solvingthe,

author = {Nevena Lazic and Brendan J. Frey and Parham Aarabi},

title = {Solving the Uncapacitated Facility Location Problem Using Message Passing Algorithms},

year = {}

}

### OpenURL

### Abstract

The Uncapacitated Facility Location Problem (UFLP) is one of the most widely studied discrete location problems, whose applications arise in a variety of settings. We tackle the UFLP using probabilistic inference in a graphical model- an approach that has received little attention in the past. We show that the fixed points of max-product linear programming (MPLP), a convexified version of the max-product algorithm, can be used to construct a solution with a 3-approximation guarantee for metric UFLP instances. In addition, we characterize some scenarios under which the MPLP solution is guaranteed to be globally optimal. We evaluate the performance of both max-sum and MPLP empirically on metric and non-metric problems, demonstrating the advantages of the 3-approximation construction and algorithm applicability to non-metric instances.

### Citations

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(Show Context)
Citation Context ...nds to the optimal assignment of customers to facilities. We perform inference by running two message passing algorithms on the factor graph problem representation: the standard maxproduct algorithm (=-=Kschischang et al., 2001-=-) and its convexified variant max-product linear programming (MPLP) (Globerson & Jaakkola, 2007). The max-product algorithm is guaranteed to converge to the optimal MAP assignment on trees, and has em... |

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Citation Context ...s worse than optimal, for some constant ρ (called the approximation ratio). For UFLP, the O(ln |C|)approximation of (Hochbaum, 1982) cannot be improved in general unless NP ⊆ DT IME(n O(log log n) ) (=-=Arora et al., 1998-=-). However, (Guha & Khuller, 1999) have shown that for the metric UFLP, constant ρapproximation algorithms do exist and that ρ > 1.463 unless NP ⊆ DT IME(n O(log log n) ). Most ρ-approximation algorit... |

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Citation Context ... to derive an effective algorithm for exemplar-based clustering problem, un-Solving the Uncapacitated Facility Location Problem Using Message Passing Algorithms der the name of Affinity Propagation (=-=Frey & Dueck, 2007-=-). However, the theoretical guarantees on maxproduct convergence and solution optimality for general graphs are still an open area of research. MPLP (Globerson & Jaakkola, 2007) is one of several rece... |

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Citation Context ...to choose cluster centers and in graph pre-processing; (Byrka, 2007) obtains best-sofar approximation ratio of 1.5. Primal-dual approximation algorithms, such as the 3approximation UFLP algorithm of (=-=Jain & Vazirani, 2001-=-), start with a feasible solution v to the dual LP, and modify it until it is possible to construct an integral primal feasible solution x that satisfies certain relaxed complementary slackness condit... |

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Citation Context ...eneral graphs are still an open area of research. MPLP (Globerson & Jaakkola, 2007) is one of several recently developed linear programming (LP) based message passing algorithms for graphical models (=-=Kolmogorov, 2006-=-), (Werner, 2007), (Komodakis & Paragios, 2008). The iterative updates of these algorithms are quite similar to those of max-product and correspond to coordinate ascent in the LP dual; the relationshi... |

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Citation Context ... √ N/100, or √ N/1000. We also run the algorithms on two non-metric data sets previously used to evaluate Affinity Propagation in (Dueck et al., 2008): images derived from the Olivetti Face database (=-=Samaria & Harter, 1994-=-), and Document Summarization. 1 The results are shown in Fig. 5, Fig. 6 and Fig. 7, where the error measures the percentage by which the solution cost exceeds the optimal cost for ORLIB, and the LP l... |

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Citation Context ...constant ρ (called the approximation ratio). For UFLP, the O(ln |C|)approximation of (Hochbaum, 1982) cannot be improved in general unless NP ⊆ DT IME(n O(log log n) ) (Arora et al., 1998). However, (=-=Guha & Khuller, 1999-=-) have shown that for the metric UFLP, constant ρapproximation algorithms do exist and that ρ > 1.463 unless NP ⊆ DT IME(n O(log log n) ). Most ρ-approximation algorithms for UFLP are based on the sta... |

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Citation Context ... integral solution is obtained by greedily clustering the customer nodes, and assigning all cluster members to the cluster center’s closest facility. LP-rounding algorithms of (Shmoys et al., 1997), (=-=Chudak & Shmoys, 2003-=-), (Sviridenko, 2002), (Byrka, 2007) differ in the greedy criterion used to choose cluster centers and in graph pre-processing; (Byrka, 2007) obtains best-sofar approximation ratio of 1.5. Primal-dual... |

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Citation Context ...variables for which the complementary slackness conditions are tight. (Jain et al., 2002) use primal-dual analysis to prove that their greedy JMS heuristic guarantees an approximation ratio of 1.61. (=-=Mahdian et al., 2007-=-) further combine the JMS algorithm with the greedy augmentation procedure introduced by (Guha & Khuller, 1999) to obtain the 1.52-approximation MYZ algorithm. In comparison to existing approximation ... |

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Citation Context ...algorithms are polynomial-time algorithms whose solution is provably at most ρ times worse than optimal, for some constant ρ (called the approximation ratio). For UFLP, the O(ln |C|)approximation of (=-=Hochbaum, 1982-=-) cannot be improved in general unless NP ⊆ DT IME(n O(log log n) ) (Arora et al., 1998). However, (Guha & Khuller, 1999) have shown that for the metric UFLP, constant ρapproximation algorithms do exi... |

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Citation Context ... message passing algorithms on the factor graph problem representation: the standard maxproduct algorithm (Kschischang et al., 2001) and its convexified variant max-product linear programming (MPLP) (=-=Globerson & Jaakkola, 2007-=-). The max-product algorithm is guaranteed to converge to the optimal MAP assignment on trees, and has empirically shown excellent performance on graphs with cycles in numerous applications, most nota... |

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Citation Context ...to converge to the optimal MAP assignment on trees, and has empirically shown excellent performance on graphs with cycles in numerous applications, most notably in the area of error-correcting codes (=-=Bendetto et al., 1996-=-). More recently, it has been used to derive an effective algorithm for exemplar-based clustering problem, un-Solving the Uncapacitated Facility Location Problem Using Message Passing Algorithms der ... |

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(Show Context)
Citation Context ...ustering the customer nodes, and assigning all cluster members to the cluster center’s closest facility. LP-rounding algorithms of (Shmoys et al., 1997), (Chudak & Shmoys, 2003), (Sviridenko, 2002), (=-=Byrka, 2007-=-) differ in the greedy criterion used to choose cluster centers and in graph pre-processing; (Byrka, 2007) obtains best-sofar approximation ratio of 1.5. Primal-dual approximation algorithms, such as ... |

35 |
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(Show Context)
Citation Context ... of research. MPLP (Globerson & Jaakkola, 2007) is one of several recently developed linear programming (LP) based message passing algorithms for graphical models (Kolmogorov, 2006), (Werner, 2007), (=-=Komodakis & Paragios, 2008-=-). The iterative updates of these algorithms are quite similar to those of max-product and correspond to coordinate ascent in the LP dual; the relationship between the different dual problems is descr... |

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(Show Context)
Citation Context ...The Uncapacitated Facility Location Problem (UFLP) is one of the most widely studied discrete location problems. Its applications arise in a variety of settings, including distribution system design (=-=Klose & Drexl, 2003-=-), self-configuration in wireless sensor networks (Frank & Romer, 2007), computational biology (Dueck et al., 2008) and computer vision (Li, 2007; Lazic et al., 2009). The UFLP can be stated as follow... |

27 | An improved approximation algorithm for the metric uncapacitated facility location problem
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(Show Context)
Citation Context ...ained by greedily clustering the customer nodes, and assigning all cluster members to the cluster center’s closest facility. LP-rounding algorithms of (Shmoys et al., 1997), (Chudak & Shmoys, 2003), (=-=Sviridenko, 2002-=-), (Byrka, 2007) differ in the greedy criterion used to choose cluster centers and in graph pre-processing; (Byrka, 2007) obtains best-sofar approximation ratio of 1.5. Primal-dual approximation algor... |

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Citation Context ...erative updates of these algorithms are quite similar to those of max-product and correspond to coordinate ascent in the LP dual; the relationship between the different dual problems is described in (=-=Sontag & Jaakkola, 2009-=-). MPLP has several desirable properties: it is guaranteed to converge, the objective function is monotonically non-increasing, and it gives an upper bound on the optimal MAP solution. We use these pr... |

13 |
Two-view motion segmentation from linear programming relaxation
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Citation Context ...ings, including distribution system design (Klose & Drexl, 2003), self-configuration in wireless sensor networks (Frank & Romer, 2007), computational biology (Dueck et al., 2008) and computer vision (=-=Li, 2007-=-; Lazic et al., 2009). The UFLP can be stated as follows: given a set of customers C, a set of facilities F , cost fj of opening each facility j ∈ F , and cost cij of connecting customer i to facility... |

11 | P.: FLoSS: Facility Location for Subspace Segmentation
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Citation Context ...uding distribution system design (Klose & Drexl, 2003), self-configuration in wireless sensor networks (Frank & Romer, 2007), computational biology (Dueck et al., 2008) and computer vision (Li, 2007; =-=Lazic et al., 2009-=-). The UFLP can be stated as follows: given a set of customers C, a set of facilities F , cost fj of opening each facility j ∈ F , and cost cij of connecting customer i to facility j, open a subset of... |

8 | Constructing Treatment Portfolios Using Affinity Propagation
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Citation Context ...s applications arise in a variety of settings, including distribution system design (Klose & Drexl, 2003), self-configuration in wireless sensor networks (Frank & Romer, 2007), computational biology (=-=Dueck et al., 2008-=-) and computer vision (Li, 2007; Lazic et al., 2009). The UFLP can be stated as follows: given a set of customers C, a set of facilities F , cost fj of opening each facility j ∈ F , and cost cij of co... |

8 | K.: Distributed Facility Location Algorithms for Flexible Configuration of Wireless Sensor Networks
- Frank, Römer
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(Show Context)
Citation Context ...widely studied discrete location problems. Its applications arise in a variety of settings, including distribution system design (Klose & Drexl, 2003), self-configuration in wireless sensor networks (=-=Frank & Romer, 2007-=-), computational biology (Dueck et al., 2008) and computer vision (Li, 2007; Lazic et al., 2009). The UFLP can be stated as follows: given a set of customers C, a set of facilities F , cost fj of open... |