Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics

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 Erd\H{o}s-R\'enyi 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.Comment: An extended abstract appeared in the proceedings of WALCOM 201

The Euclidian Traveling Salesman Selection Problem

The traveling salesman problem (TSP) is one of the famous problems of combinatorial optimization. In this paper a generalization of the TSP, the traveling salesman selection problem (TSSP) is introduced. The problem is restricted to the Euclidian case where the TSP can be formulated as follows: Given n cities in the plane and their Euclidian distances, the problem is to find the shortest TSP-tour, i.e. a closed path visiting each of the n cities exactly once. In addition to that the TSSP specifies a number k<n of cities and the shortest TSP-tour through any subset of k cities shall be found. Existing heuristics are based on approximations for the k-Minimal Spanning Tree to find the node cluster containing the shortest k-tour. Unlike many related problems, the TSSP does not include a depot that has to be visited by the k-tour. The main purpose of this paper is the presentation of an exact algorithm for the TSSP. It is based on a geometrical procedure searching for all clusters of about k nodes which can contain the shortest k-tour. These clusters are the start set for a branch-and-bound procedure that determines the shortest k-tour in each cluster and therefore the exact solution of the TSSP. A dynamic programming approach for calculating shortest t-chains is used to obtain a lower bound for the branch-and-bound algorithm

Incremental Matrix Reordering for Similarity-Based Dynamic Data Sets

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