On the Nearest Neighbor Rule for the Metric Traveling Salesman Problem

We present a very simple family of traveling salesman instances with $n$ cities where the nearest neighbor rule may produce a tour that is $\Theta(\log n)$ times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time a lower bound in the rectilinear case

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

TSP--Infrastructure for the Traveling Salesperson Problem

The traveling salesperson (or, salesman) problem (TSP) is a well known and important combinatorial optimization problem. The goal is to find the shortest tour that visits each city in a given list exactly once and then returns to the starting city. Despite this simple problem statement, solving the TSP is difficult since it belongs to the class of NP-complete problems. The importance of the TSP arises besides from its theoretical appeal from the variety of its applications. Typical applications in operations research include vehicle routing, computer wiring, cutting wallpaper and job sequencing. The main application in statistics is combinatorial data analysis, e.g., reordering rows and columns of data matrices or identifying clusters. In this paper, we introduce the R package TSP which provides a basic infrastructure for handling and solving the traveling salesperson problem. The package features S3 classes for specifying a TSP and its (possibly optimal) solution as well as several heuristics to find good solutions. In addition, it provides an interface to Concorde, one of the best exact TSP solvers currently available.

Probabilistic Analyses of Combinatorial Optimization Problems on Random Shortest Path Metrics

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 by Bringmann et al. (Algorithmica, 2015), who have used random shortest path metrics generated from complete graphs to analyse heuristics.In this thesis we look at several variations of the random shortest path metrics, and perform probabilistic analyses for some simple heuristics for several combinatorial optimization problems on these random metric spaces. A random shortest path metric is constructed by drawing independent random edge weights for each edge in a 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.We provide some basic properties of the distances between vertices in random shortest path metrics. Using these properties, we perform several probabilistic analyses. For random shortest path metrics generated from (dense) Erdős-Rényi random graphs we show that the greedy heuristic for the minimum-distance perfect 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 in this model.For random shortest path metrics generated from sparse graphs we show that the greedy heuristic for the minimum-distance perfect matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio in this model. For random shortest path metrics generated from complete graphs we analyse a simple greedy heuristic for 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. For large values of κ we provide a closed-form expression as upper bound for the expected approximation ratio and we evaluate this expression for the special case where all facility opening costs are equal.Moreover, we show in this model that a simple 2-approximation algorithm for the Steiner tree problem is asymptotically optimal as long as the number of terminals is not too large. We also present some numerical results that imply that the 2-opt heuristic for the traveling salesman problem seems to perform rather poorly in this model