Traveling Salesman Problem

Traveling Salesman Problem
Name	Traveling Salesman Problem
Field	Computer science, Operations research
Formulated	1930s
Notable	Dantzig, Held, Karp, Cook, Christofides

Contents

Traveling Salesman Problem

The Traveling Salesman Problem is a classical combinatorial optimization problem asking for a shortest route through a set of cities that visits each city exactly once and returns to the start. Originating in early 20th century logistics and formalized in twentieth-century mathematics, it has driven research across Princeton University, Bell Labs, IBM, Stanford University, Massachusetts Institute of Technology and influenced results involving George Dantzig, Richard Bellman, Jack Edmonds, Michael Held, Richard Karp, William Cook and Christos Papadimitriou.

Problem statement

The formal statement considers a weighted complete graph on n nodes with distances satisfying triangle inequality or arbitrary weights, seeking a Hamiltonian cycle of minimum total weight; foundations trace to work at RAND Corporation, Bell Laboratories, University of California, Berkeley, Harvard University, Cornell University and problems posed at Princeton University. Early computational explorations ran on machines at ENIAC, Whirlwind I, IBM 701 and researchers in the tradition of John von Neumann, Alan Turing, Norbert Wiener contributed to algorithmic thinking that framed instances studied by Hermann Minkowski and Kurt Gödel. Canonical formulations connect to linear programming relaxations used at AT&T Bell Labs, branch-and-bound strategies developed at General Electric research groups, and cutting-plane methods inspired by studies at École Polytechnique and University of Cambridge.

The problem is NP-hard with decision version NP-complete, a complexity classification formalized by Stephen Cook, Leonid Levin, Richard Karp and discussed in seminars at MIT, Stanford University, Carnegie Mellon University and University of Illinois Urbana-Champaign. Key hardness reductions involve transformations from Hamiltonian cycle and 3-SAT instances studied in conferences at ACM and IEEE symposia, linking to PCP theorem results from researchers at Princeton University and Rutgers University. The inapproximability landscape ties to work by Umesh Vazirani, Sanjeev Arora, Avi Wigderson, Mihalis Yannakakis and David Zuckerman presented at FOCS and STOC conferences, and to complexity classes discussed at Institute for Advanced Study lectures by Donald Knuth and Leslie Valiant.

Exact methods include branch-and-bound, branch-and-cut and dynamic programming such as the Held–Karp algorithm developed by Michael Held and Richard M. Karp in collaborations linked to University of California, Berkeley and reports circulated at RAND Corporation. Cutting-plane procedures built on integer programming were advanced by researchers at AT&T Bell Labs, IBM Research, University of Waterloo and École Polytechnique Fédérale de Lausanne with milestones by William Cook, David Applegate, Robert Bixby and Vijay Vazirani. Methods exploiting concorde and LKH implementations were influenced by work at Cornell University, University of Tennessee, University of Bologna and practical exact solvers used in projects at NASA, European Space Agency, Siemens and General Motors.

Approximation theory produced polynomial-time approximation algorithms such as Christofides' algorithm credited to Nicos Christofides and built upon metric insights developed at University of Cambridge and Imperial College London. Landmark approximation hardness results were influenced by research from Arora, Sanjeev Arora at Princeton University and Avi Wigderson at Institute for Advanced Study, with practical heuristics like nearest neighbor, genetic algorithms and simulated annealing explored at Bell Labs, IBM Research, MIT Lincoln Laboratory and Los Alamos National Laboratory. Heuristic implementations appear in software projects originating from Microsoft Research, Google, Amazon, Uber Technologies and research prototypes at ETH Zurich, University of Tokyo and National University of Singapore.

Many variants have distinct names and histories: asymmetric TSP studied at Bell Labs and AT&T, Euclidean TSP linked to computational geometry results at Stanford University and ETH Zurich, prize-collecting versions explored at University of Bonn and University of British Columbia, vehicle routing problems developed at Daimler, UPS and FedEx research groups, and time-dependent TSP addressed at Toyota and Nissan research labs. Connections to matching and spanning tree problems reference classics by Kruskal and Edmonds and link to polyhedral studies at Université Paris-Saclay and Princeton University. Special-case solvability leverages planar graph results from University of Waterloo and Hamiltonian properties examined in work associated with Cambridge University Press publications by Paul Erdős and Ronald Graham.

Applications span logistics and routing for UPS, FedEx, DHL, Amazon Logistics and USPS; circuit board drilling and machining practiced in firms like Siemens and Bosch; DNA sequencing and bioinformatics projects at Broad Institute, Cold Spring Harbor Laboratory and European Bioinformatics Institute; and touring route planning used by BBC and National Geographic production teams. Industrial implementations appear in commercial solvers from IBM, Gurobi, CPLEX at IBM Watson Research Center and open-source tools developed at Cornell University and École Polytechnique Fédérale de Lausanne; research deployments occur in projects at NASA Jet Propulsion Laboratory, European Space Agency mission planning, Tesla logistics and autonomous vehicle routing at Waymo. Ongoing collaborations involve academic hubs such as MIT, Stanford University, University of Oxford, University of Cambridge, Princeton University, ETH Zurich and industry labs at Google DeepMind and Microsoft Research.