Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems

Bringmann, Karl; Engels, Christian; Manthey, Bodo; Rao, B. V. Raghavendra

doi:10.1007/s00453-014-9901-9

Cited by 7 publications

(54 citation statements)

References 37 publications

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“…This rather particular value of κ is originates from the following intuitive argument. Based on the results of Bringmann et al [3,Lemma 5.1] (see below) we know that the expected cost of the solution that opens the k cheapest facilities is given by g(k) := F k + H n−1 − H k−1 . This convex function decreases as long as k satisfies f k < 1/(k − 1).…”

Section: A Simple Heuristic and Some Of Its Propertiesmentioning

confidence: 99%

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Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

Klootwijk

Manthey

2019

Computing With Foresight and Industry

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The facility location problem is an N P-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 N P-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 κ cheapest facilities (with κ only depending on the facility opening costs). If the facility opening costs are such that κ is not too large, then we show that this heuristic is asymptotically optimal. On the other hand, for large values of κ, 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 O( 4 ln(n)) or O(1) or even 1 + o(1) if the opening costs are sufficiently small. * An extended abstract of this work will appear in the Proceedings of the 15th Conference on Computability in Europe (CiE).

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Section: A Simple Heuristic and Some Of Its Propertiesmentioning

confidence: 99%

“…If 1 ≤ κ < n, then the distribution of ALG is less trivial. In this case, the total opening costs are given by F κ , whereas, the distribution of the connection costs is known and given by n−1 i=κ Exp(i) [3,Sect. 5].…”

Section: A Simple Heuristic and Some Of Its Propertiesmentioning

confidence: 99%

Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

Klootwijk

Manthey

2019

Computing With Foresight and Industry

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“…For first passage percolation in complete graphs, the expected distance between two fixed vertices is approximately ln(n)/n and the expected distance from a fixed vertex to the vertex that is most distant is approximately 2 ln(n)/n [2,8]. Furthermore, the expected diameter of the metric is approximately 3 ln(n)/n [6,8].…”

Section: Related Workmentioning

confidence: 99%

“…Bringmann et al [2] used this model on the complete graph to analyze heuristics for matching, TSP, and k-median.…”

Section: Related Workmentioning

confidence: 99%

“…The independent, identically distributed edge lengths do not even yield a metric in the first place. In order to overcome these shortcomings, Bringmann et al [2] have proposed and analyzed the following model to generate random metric spaces, which had already been proposed by Karp and Steele in 1985 [10]: given an undirected complete graph, start by drawing random edge weights for each edge independently and then define the distance between any two vertices as the total weight of the shortest path between them, measured with respect to the random weights.…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics

Klootwijk

Manthey

Visser

2018

Lecture Notes in Computer Science

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Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms.The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained by Bringmann et al. (Algorithmica, 2013), who have used random shortest path metrics on complete graphs to analyze heuristics.The goal of this paper is to generalize these findings to non-complete graphs, especially Erdős-Rényi random graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances, we prove that the greedy heuristic for the minimum distance maximum matching problem, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the k-median problem all achieve a constant expected approximation ratio. Additionally, we show a polynomial upper bound for the expected number of iterations of the 2-opt heuristic for the traveling salesman problem.

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Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics

Klootwijk,

Manthey

2023

Algorithmica

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Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, “beyond worst-case analysis” of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős–Rényi random graphs) have been used so far. In this paper we extend these findings to sparse graphs, with a focus on sparse graphs with ‘fast growing cut sizes’, i.e. graphs for which $$|\delta (U)|=\Omega (|U|^\varepsilon )$$ | δ ( U ) | = Ω ( | U | ε ) for some constant $$\varepsilon \in (0,1)$$ ε ∈ ( 0 , 1 ) for all subsets U of the vertices, where $$\delta (U)$$ δ ( U ) is the set of edges connecting U to the remaining vertices. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a sparse graph with fast growing cut sizes, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.

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Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems

Cited by 7 publications

References 37 publications

Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics

Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics

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