The facility location problem is an NP-hard optimization problem. Therefore, approximation algorithms are often used to solve large instances. Such algorithms often perform much better than worst-case analysis suggests. Therefore, probabilistic analysis is a widely used tool to analyze such algorithms. Most research on probabilistic analysis of NP-hard optimization problems involving metric spaces, such as the facility location problem, has been focused on Euclidean instances, and also instances with independent (random) edge lengths, which are non-metric, have been researched. We would like to extend this knowledge to other, more general, metrics. We investigate the facility location problem using random shortest path metrics. We analyze some probabilistic properties for a simple greedy heuristic which gives a solution to the facility location problem: opening the @math cheapest facilities (with @math only depending on the facility opening costs). If the facility opening costs are such that @math is not too large, then we show that this heuristic is asymptotically optimal. On the other hand, for large values of @math , the analysis becomes more difficult, and we provide a closed-form expression as upper bound for the expected approximation ratio. In the special case where all facility opening costs are equal this closed-form expression reduces to @math or @math or even @math if the opening costs are sufficiently small.
Although a lot of studies have been conducted on random shortest path metrics, or first-passage percolation (e.g. @cite_5 @cite_14 @cite_10 ), systematic research of the behavior of (simple) heuristics and approximation algorithms for optimization problems on random shortest path metrics was initiated only recently @cite_2 . They provide some structural properties of random shortest path metrics, including the existence of a good clustering. These properties are then used for a probabilistic analysis of simple algorithms for several optimization problems, including the minimum-weight perfect matching problem and the @math -median problem. For the facility location problem, several sophisticated polynomial-time approximation algorithms exist, the best one currently having a worst-case approximation ratio of @math @cite_7 . conducted a probabilistic analysis for the facility location problem using Euclidean distances @cite_1 . They expected to show that some polynomial-time approximation algorithms would be asymptotically optimal under these circumstances, but found out that this is not the case. On the other hand, they described a trivial heuristic which is asymptotically optimal in the Euclidean model.
@cite_14 @cite_7 @cite_1 @cite_2 @cite_5 @cite_10
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We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs. We present a 1.488-approximation algorithm for the metric uncapacitated facility location (UFL) problem. Previously, the best algorithm was due to Byrka (2007). Byrka proposed an algorithm parametrized by @c and used it with @c 1.6774. By either running his algorithm or the algorithm proposed by Jain, Mahdian and Saberi ([email protected]?02), Byrka obtained an algorithm that gives expected approximation ratio 1.5. We show that if @c is randomly selected, the approximation ratio can be improved to 1.488. Our algorithm cuts the gap with the 1.463 approximability lower bound by almost 1 3. A flame arrestor is disclosed which includes a conduit extending along inside of a drum arranged to contain a quantity of non-combustible liquid. This conduit is equipped with bubbler nozzles that discharge a combustible gas into the liquid in the form of separate, discrete bubbles. The gas is drawn from the drum through outlets above the liquid level and deflectors are positioned to deflect and distribute a flame front entering the drum through any of the nozzles. Abstract We derive an exact summation formula and a closed-form approximation for the expected length of a shortest path for a complete graph where the arc lengths are independent and exponentially distributed random variables. Experimental data validates both results. The property of completeness allows us to exploit certain symmetries to derive these results, which would otherwise require computing an exponential number of recursive equations. We have also found that this formula is a close approximation for the expected length of a shortest path in complete graphs with uniformly distributed arc lengths. Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n n for two given points, that the maximum if one point is fixed and the other varies is 2 log n n, and that the maximum over all pairs of points is 3 log n n.Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths.