A domination algorithm for {0,1}\{0,1\}-instances of the travelling salesman problem

We present an approximation algorithm for $\{0,1\}$-instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial-time algorithm which has domination ratio $1-n^{-1/29}$. In other words, given a $\{0,1\}$-edge-weighting of the complete graph $K_n$ on $n$ vertices, our algorithm outputs a Hamilton cycle $H^*$ of $K_n$ with the following property: the proportion of Hamilton cycles of $K_n$ whose weight is smaller than that of $H^*$ is at most $n^{-1/29}$. Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial-time algorithm with domination ratio $1/2-o(1)$ for arbitrary edge-weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant $C$ such that $n^{-1/29}$ cannot be replaced by $\exp(-(\log n)^C)$ in the result above.Comment: 29 pages (final version to appear in Random Structures and Algorithms

Parameterized TSP: Beating the Average

In the Travelling Salesman Problem (TSP), we are given a complete graph $K_n$ together with an integer weighting $w$ on the edges of $K_n$, and we are asked to find a Hamilton cycle of $K_n$ of minimum weight. Let $h(w)$ denote the average weight of a Hamilton cycle of $K_n$ for the weighting $w$. Vizing (1973) asked whether there is a polynomial-time algorithm which always finds a Hamilton cycle of weight at most $h(w)$. He answered this question in the affirmative and subsequently Rublineckii (1973) and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalisation of Vizing's result: for each fixed $k$, we give an algorithm that decides whether, for any input edge weighting $w$ of $K_n$, there is a Hamilton cycle of $K_n$ of weight at most $h(w)-k$ (and constructs such a cycle if it exists). For $k$ fixed, the running time of the algorithm is polynomial in $n$, where the degree of the polynomial does not depend on $k$ (i.e., the generalised Vizing problem is fixed-parameter tractable with respect to the parameter $k$)

CES-478 Comparison between MOEA/D and NSGA-II on the Multiobjective Travelling Salesman Problem

Most multiobjective evolutionary algorithms are based on Pareto dom- inance for measuring the quality of solutions during their search, among them NSGA-II is well-known. A very few algorithms are based on de- composition and implicitly or explicitly try to optimize aggregations of the objectives. MOEA/D is a very recent such an algorithm. One of the major advantages of MOEA/D is that it is very easy to use well-developed single optimization local search within it. This paper compares the perfor- mance of MOEA/D and NSGA-II on the multiobjective travelling sales- man problem and studies the e®ect of local search on the performance of MOEA/D

Genetic Algorithm Performance with Different Selection Strategies in Solving TSP

A genetic algorithm (GA) has several genetic operators that can be modified to improve the performance of particular implementations. These operators include parent selection, crossover and mutation. Selection is one of the important operations in the GA process. There are several ways for selection. This paper presents the comparison of GA performance in solving travelling salesman problem (TSP) using different parent selection strategy. Several TSP instances were tested and the results show that tournament selection strategy outperformed proportional roulette wheel and rank-based roulette wheel selections, achieving best solution quality with low computing times. Results also reveal that tournament and proportional roulette wheel can be superior to the rank-based roulette wheel selection for smaller problems only and become susceptible to premature convergence as problem size increases

O vizinho mais próximo e mais solitário de dois lados - uma variação da heurística do vizinho mais próximo para o problema do caixeiro viajante

Este artigo apresenta uma nova heurística para o problema do caixeiro viajante que introduz o conceito de solidão de uma cidade - calculada como a distância média dessa cidade a todas as outras - e o combina com ideias de outras variações de heurísticas do vizinho mais próximo. Tendo a mesma complexidade das heurísticas de vizinho mais próximo mais rápidas, o novo método conduz a melhores resultados que estas heurísticas, ultrapassando igualmente várias outras heurísticas reportadas na literatura. Uma característica interessante da heurística proposta é que dá prioridade a localizações mais isoladas na definição de rotas. A antecipação da distribuição de bens e serviços a localizações mais periféricas pode ser considerada uma externalidade social positiva, tornando a heurística passível de adopção por determinadas entidades por razões não meramente económicas mas também sociais.This paper presents a new tour construction heuristic for the travelling salesman problem that introduces the concept of loneliness of a city computed from the average distance of that city to all others and combines it with ideas from other nearest neighbour heuristics. Having the same time complexity of the faster nearest neighbour heuristics, the new method clearly leads to better tours, outperforming them as well as several other tour construction heuristics reported in the literature. A promising feature of the proposed heuristic is that it gives some priority to more isolated locations in travel route definitions. The earlier distribution of goods and services to loneliest sites might be considered a positive social externality that is appealing to the application of heuristics by public or private institutions that are engaged in acts of social responsibility

Learning-Augmented Online TSP on Rings, Trees, Flowers and (Almost) Everywhere Else

We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line [Gouleakis et al., 2023], in which the predictions correspond to the locations of the requests and extend it to more general metric spaces. We first propose an oracle-based algorithmic framework, inspired by previous work [Bampis et al., 2023]. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness). Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results