[Paper Review] Local-search based approximation algorithms for mobile facility location problems

Sara Ahmadian, Zachary Friggstad+1 more|arXiv (Cornell University)|Jan 6, 2013

Facility Location and Emergency Management29 references31 citations

TL;DR

This paper presents the first local-search-based approximation algorithm for the mobile facility location (MFL) problem, achieving a (3 + ε)-approximation for any ε > 0. By introducing a novel tree-like structure to guide local-search moves and allowing optimal rematching of facilities to destinations at each step, the method matches the best-known approximation guarantee for the k-median problem and proves tightness up to o(1) factors.

ABSTRACT

We consider the mobile facility location (MFL) problem. We are given a set of facilities and clients located in a common metric space G = (V, c). The goal is to move each facility from its initial location to a destination (in V) and assign each client to the destination of some facility so as to minimize the sum of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs.We give the first local-search based approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is (3 + e)-approximation for this problem for any constant e > 0 using local search. The previous best guarantee for MFL was an 8-approximation algorithm due to Friggstad and Salavatipour [12] based on LP-rounding. Our guarantee matches the best-known approximation guarantee for the k-median problem. Since there is an approximation-preserving reduction from the k-median problem to MFL, any improvement of our result would imply an analogous improvement for the k-median problem. Furthermore, our analysis is tight (up to o(1) factors) since the tight example for the local-search based 3-approximation algorithm for k-median can be easily adapted to show that our local-search algorithm has a tight approximation ratio of 3. Our results extend to the weighted generalization wherein each facility i has a non-negative weight wi and the movement cost for i is wi times the distance traveled by i.In contrast to the k-median problem, the local search procedure that moves, at each step, a constant number of facilities (to chosen destinations) and assigns each client to the nearest destination, is known to have an unbounded locality gap. Our local-search algorithm is a natural and simple variant, where we only select the destinations of the facilities in each step and optimally rematch the facilities to these destinations (which might entail moving all facilities). One of the chief novelties in the analysis is that in order to generate a suitable collection of local-search moves whose resulting inequalities yield the desired bound on the cost of a local-optimum, we define a tree-like structure that (loosely speaking) functions as a recursion tree, using which we spawn off local-search moves by exploring this tree to a constant depth.

Motivation & Objective

• To develop a local-search-based approximation algorithm for the mobile facility location (MFL) problem, which allows facilities to be relocated to reduce client assignment and movement costs.
• To close the gap between the best-known approximation guarantees for MFL and the k-median problem, which is known to be reducible to MFL.
• To overcome the unbounded locality gap of standard local search (moving only a constant number of facilities) by introducing a refined local-search procedure with optimal rematching.
• To extend the algorithm to the weighted MFL variant, where facility movement cost scales with facility weights.
• To provide a tight analysis of the approximation ratio by constructing a suitable collection of local-search moves using a tree-like recursion structure.

Proposed method

• The algorithm uses a local-search framework where, at each step, a set of facility destinations is selected and all facilities are optimally rematched to these destinations, rather than moving only a few.
• A novel tree-like structure is constructed to guide the generation of local-search moves, simulating a recursion tree that explores potential improvements up to a constant depth.
• The tree structure enables the creation of a collection of local-search moves whose cost inequalities are used to bound the cost of a local optimum.
• The analysis leverages the fact that any improvement in the k-median problem's local-search algorithm can be adapted to MFL, ensuring tightness of the (3 + ε)-approximation ratio.
• The method handles the weighted MFL variant by scaling movement costs by facility weights, maintaining the same approximation guarantee.
• The algorithm avoids the unbounded locality gap of standard local search by not restricting the number of facilities moved per iteration and instead reassigning all facilities optimally to chosen destinations.

Experimental results

Research questions

• RQ1 Can a local-search-based algorithm achieve the best-known approximation ratio for the mobile facility location problem?
• RQ2 Why does standard local search (moving only a constant number of facilities) fail for MFL, and how can this be remedied?
• RQ3 Can the approximation guarantee for the k-median problem be matched or improved for MFL using local search?
• RQ4 What structural properties of the solution space allow for tight analysis of local-search algorithms in MFL?
• RQ5 How can a tree-like structure be used to generate effective local-search moves that yield bounded approximation ratios?

Key findings

• The proposed local-search algorithm achieves a (3 + ε)-approximation for the mobile facility location problem for any ε > 0, matching the best-known approximation guarantee for the k-median problem.
• The algorithm's approximation ratio is tight up to o(1) factors, as the tight example for the k-median local-search 3-approximation can be adapted to MFL.
• The method overcomes the unbounded locality gap of standard local search by allowing full rematching of facilities to selected destinations at each step.
• The tree-like structure used in the analysis enables the systematic generation of local-search moves that yield cost-bounding inequalities.
• The algorithm extends naturally to the weighted MFL variant, preserving the (3 + ε)-approximation guarantee when movement costs are scaled by facility weights.
• Any improvement over the (3 + ε)-approximation would imply a corresponding improvement for the k-median problem, due to an approximation-preserving reduction.