Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs

We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n-object set, maintaining the closest pair, in O(n log^2 n) time per update and O(n) space. With quadratic space, we can instead use a quadtree-like structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Groebner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp. 619-628. For source code and experimental results, see http://www.ics.uci.edu/~eppstein/projects/pairs

The 2013 Multi-objective Physical Travelling Salesman Problem Competition

This paper presents the game, framework, rules and results of the Multi-objective Physical Travelling Salesman Problem (MO-PTSP) Competition, that was held at the 2013 IEEE Conference on Computational Intelligence in Games (CIG). The MO-PTSP is a real-time game that can be seen as a modification of the Travelling Salesman Problem, where the player controls a ship that must visit a series of waypoints in a maze while minimizing three opposing goals: Time spent, fuel consumed and damage taken. The rankings of the competition are computed using multi-objective concepts, a novel approach in the field of game artificial intelligence competitions. The winning entry of the contest is also explained in detail. This controller is based on the Monte Carlo Tree Search algorithm, and employed Covariance Matrix Adaptation Evolution Strategy (CMA-ES) for parameter tuning

A new genetic algorithm for the asymmetric traveling salesman problem

The asymmetric traveling salesman problem (ATSP) is one of the most important combinatorial optimization problems. It allows us to solve, either directly or through a transformation, many real-world problems. We present in this paper a new competitive genetic algorithm to solve this problem. This algorithm has been checked on a set of 153 benchmark instances with known optimal solution and it outperforms the results obtained with previous ATSP heuristic methods. © 2012 Elsevier Ltd. All rights reserved.This work has been partially supported by the Ministerio de Educacion y Ciencia of Spain (Project No. TIN2008-06441-C02-01).Yuichi Nagata; Soler Fernández, D. (2012). A new genetic algorithm for the asymmetric traveling salesman problem. Expert Systems with Applications. 39(10):8947-8953. https://doi.org/10.1016/j.eswa.2012.02.029S89478953391

An Iterated Local Search Approach for Finding Provably Good Solutions for Very Large TSP Instances

Abstract. Meta-heuristics usually lack any kind of performance guar-antee and therefore one cannot be certain whether the resulting solutions are (near) optimum solutions or not without relying on additional algo-rithms for providing lower bounds (in case of minimization). In this paper, we present a highly effective hybrid evolutionary local search algorithm based on the iterated Lin-Kernighan heuristic combined with a lower bound heuristic utilizing 1-trees. Since both upper and lower bounds are improved over time, the gap between the two bounds is minimized by means of effective heuristics. In experiments, we show that the proposed approach is capable of finding short tours with a gap of 0.8 % or less for TSP instances up to 10 million cities. Hence, to the best of our knowledge, we present the first evolutionary algorithm and meta-heuristic in general that delivers provably good solutions and is highly scalable with the problem size. We show that our approach outperforms all existing heuristics for very large TSP instances.

New Applications of Nearest-Neighbor Chains: Euclidean TSP and Motorcycle Graphs

We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We use it to construct the greedy multi-fragment tour for Euclidean TSP in O(n log n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n sqrt(n)log n) time; we compute motorcycle graphs, a central step in straight skeleton algorithms, in O(n^(4/3+epsilon)) time for any epsilon>0

Fine-Grained Complexity Analysis of Two Classic TSP Variants

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in $O(n^2)$ time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in $O(n \log^2 n)$ time and its bottleneck version in $O(n \log^3 n)$ time. Our second set of results concerns the popular $k$-OPT heuristic for TSP in the graph setting. More precisely, we study the $k$-OPT decision problem, which asks whether a given tour can be improved by a $k$-OPT move that replaces $k$ edges in the tour by $k$ new edges. A simple algorithm solves $k$-OPT in $O(n^k)$ time for fixed $k$. For 2-OPT, this is easily seen to be optimal. For $k=3$ we prove that an algorithm with a runtime of the form $\tilde{O}(n^{3-\epsilon})$ exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. The results for $k=2,3$ may suggest that the actual time complexity of $k$-OPT is $\Theta(n^k)$. We show that this is not the case, by presenting an algorithm that finds the best $k$-move in $O(n^{\lfloor 2k/3 \rfloor + 1})$ time for fixed $k \geq 3$. This implies that 4-OPT can be solved in $O(n^3)$ time, matching the best-known algorithm for 3-OPT. Finally, we show how to beat the quadratic barrier for $k=2$ in two important settings, namely for points in the plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016

Solving the Physical Traveling Salesman Problem: Tree Search and Macro Actions

This paper presents a number of approaches for solving a real-time game consisting of a ship that must visit a number of waypoints scattered around a 2-D maze full of obstacles. The game, the Physical Traveling Salesman Problem (PTSP), which featured in two IEEE conference competitions during 2012, provides a good balance between long-term planning (finding the optimal sequence of waypoints to visit), and short-term planning (driving the ship in the maze). This paper focuses on the algorithm that won both PTSP competitions: it takes advantage of the physics of the game to calculate the optimal order of waypoints, and it employs Monte Carlo tree search (MCTS) to drive the ship. The algorithm uses repetitions of actions (macro actions) to reduce the search space for navigation. Variations of this algorithm are presented and analyzed, in order to understand the strength of each one of its constituents and to comprehend what makes such an approach the best controller found so far for the PTSP. © 2009-2012 IEEE

Developing Programming Tools to Handle Traveling Salesman Problem by the Three Object-Oriented Languages

The traveling salesman problem (TSP) is one of the most famous problems. Many applications and programming tools have been developed to handle TSP. However, it seems to be essential to provide easy programming tools according to state-of-theart algorithms. Therefore, we have collected and programmed new easy tools by the three object-oriented languages. In this paper, we present ADT (abstract data type) of developed tools at first; then we analyze their performance by experiments. We also design a hybrid genetic algorithm (HGA) by developed tools. Experimental results show that the proposed HGA is comparable with the recent state-of-the-art applications