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We consider a variation of the well-known traveling salesman problem in which there are multiple agents who all have to tour the whole set of nodes of the same graph, while obeying node- and edge-capacity constraints require that agents…

Discrete Mathematics · Computer Science 2020-12-02 Gyula Pap , József Varnyú

The Clustered Traveling Salesman Problem (CTSP) is a variant of the popular Traveling Salesman Problem (TSP) arising from a number of real-life applications. In this work, we explore a transformation approach that solves the CTSP by…

Artificial Intelligence · Computer Science 2022-04-15 Yongliang Lu , Jin-Kao Hao , Qinghua Wu

Most neural solvers for the Traveling Salesperson Problem (TSP) are trained to output a single solution, even though practitioners rarely stop there: at test time, they routinely spend extra compute on sampling or post-hoc search. This…

Machine Learning · Computer Science 2026-05-04 Andoni Irazusta Garmendia

Given a graph whose arc traversal times vary over time, the Time-Dependent Travelling Salesman Problem consists in finding a Hamiltonian tour of least total duration covering the vertices of the graph. The main goal of this work is to…

Artificial Intelligence · Computer Science 2023-01-05 Tommaso Adamo , Gianpaolo Ghiani , Pierpaolo Greco , Emanuela Guerriero

The multiple Stack Travelling Salesman Problem, STSP, deals with the collect and the deliverance of n commodities in two distinct cities. The two cities are represented by means of two edge-valued graphs (G1,d2) and (G2,d2). During the…

Computational Complexity · Computer Science 2010-09-28 Sophie Toulouse , Roberto Wolfler Calvo

This technical report provides additional results for the main paper ``Probabilistic bounds on the $k-$Traveling Salesman Problem ($k-$TSP) and the Traveling Repairman Problem (TRP)''. For the $k-$TSP, we extend the probabilistic bounds…

Discrete Mathematics · Computer Science 2022-11-22 Moïse Blanchard , Alexandre Jacquillat , Patrick Jaillet

The multiple Traveling Salesmen Problem (mTSP) is a general extension of the famous NP-hard Traveling Salesmen Problem (TSP), that there are m (m > 1) salesmen to visit the cities. In this paper, we address the mTSP with both the minsum…

Artificial Intelligence · Computer Science 2022-03-01 Jiongzhi Zheng , Yawei Hong , Wenchang Xu , Wentao Li , Yongfu Chen

Combinatorial optimization is the field devoted to the study and practice of algorithms that solve NP-hard problems. As Machine Learning (ML) and deep learning have popularized, several research groups have started to use ML to solve…

Artificial Intelligence · Computer Science 2019-10-01 Antoine François , Quentin Cappart , Louis-Martin Rousseau

This paper describes TSP exact solution of polynomial complexity. It is considered properties of proposed method. Effectiveness of proposed solution is illustrated by outcomes of computer modeling.

Computational Complexity · Computer Science 2007-05-23 Vadim Yatsenko

The $k$-opt algorithm is one of the simplest and most widely used heuristics for solving the traveling salesman problem. Starting from an arbitrary tour, the $k$-opt algorithm improves the current tour in each iteration by exchanging up to…

Data Structures and Algorithms · Computer Science 2025-07-17 Sophia Heimann , Hung P. Hoang , Stefan Hougardy

We show improved approximation guarantees for the traveling salesman problem on cubic graphs, and cubic bipartite graphs. For cubic bipartite graphs with n nodes, we improve on recent results of Karp and Ravi (2014) by giving a simple…

Data Structures and Algorithms · Computer Science 2016-06-27 Anke van Zuylen

The traveling salesman problem (TSP) is a cornerstone of combinatorial optimization and has deeply influenced the development of algorithmic techniques in both exact and approximate settings. Yet, improving on the decades-old bounds for…

Data Structures and Algorithms · Computer Science 2026-04-08 Justin Dallant , László Kozma

The Traveling-Salesperson-Problem (TSP) is arguably one of the best-known NP-hard combinatorial optimization problems. The two sophisticated heuristic solvers LKH and EAX and respective (restart) variants manage to calculate close-to…

Artificial Intelligence · Computer Science 2020-05-28 Jakob Bossek , Pascal Kerschke , Heike Trautmann

We show that certain ways of solving some combinatorial optimization problems can be understood as using query planes to divide the space of problem instances into polyhedra that could fit into those that characterize the problem's various…

Computational Complexity · Computer Science 2023-04-24 Jian Yang

In this article, we present a novel formulation for the load-dependent traveling salesman problem (LD-TSP), in which travel cost (or energy expended) depends on the vehicle's current load. This problem is relevant for package delivery and…

Optimization and Control · Mathematics 2026-05-08 Deepak Prakash Kumar , Saurabh Belgaonkar , Sivakumar Rathinam , Swaroop Darbha , David W. Casbeer

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…

Data Structures and Algorithms · Computer Science 2014-01-10 Stefan Hougardy , Mirko Wilde

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes $o_1, \ldots, o_k$. The TSP solution must have that $o_{i+1}$ is visited at…

Data Structures and Algorithms · Computer Science 2026-03-23 Martin Böhm , Zachary Friggstad , Tobias Mömke , Joachim Spoerhase

In this article, a heuristic is proposed for a min-max heterogeneous multi-vehicle multi-depot traveling salesman problem (TSP), wherein heterogeneous vehicles start from given depot positions and need to cover a given set of targets. The…

Optimization and Control · Mathematics 2024-01-01 Deepak Prakash Kumar , Sivakumar Rathinam , Swaroop Darbha , Trevor Bihl

In this paper, we study the approximability of the metric Traveling Salesman Problem (TSP) and prove new explicit inapproximability bounds for that problem. The best up to now known hardness of approximation bounds were 185/184 for the…

Computational Complexity · Computer Science 2013-06-12 Marek Karpinski , Michael Lampis , Richard Schmied

Given a traveling salesman problem (TSP) tour $H$ in graph $G$ a $k$-move is an operation which removes $k$ edges from $H$, and adds $k$ edges of $G$ so that a new tour $H'$ is formed. The popular $k$-OPT heuristics for TSP finds a local…

Data Structures and Algorithms · Computer Science 2017-08-02 Marek Cygan , Lukasz Kowalik , Arkadiusz Socala