The travelling salesman problem asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.
The nearest neighbour algorithm was one of the first algorithms used to solve the travelling salesman problem approximately. In that problem, the salesman starts at a random city and repeatedly visits the nearest city until all have been visited. The algorithm quickly yields a short tour, but usually not the optimal one.
In computational complexity theory, NP-hardness is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem.
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time.
Leonard Adleman is an American computer scientist. He is one of the creators of the RSA encryption algorithm, for which he received the 2002 Turing Award, often called the Nobel prize of Computer science. He is also known for the creation of the field of DNA computing.
Richard Manning Karp is an American computer scientist and computational theorist at the University of California, Berkeley. He is most notable for his research in the theory of algorithms, for which he received a Turing Award in 1985, The Benjamin Franklin Medal in Computer and Cognitive Science in 2004, and the Kyoto Prize in 2008.
The Bottleneck traveling salesman problem is a problem in discrete or combinatorial optimization. The problem is to find the Hamiltonian cycle in a weighted graph which minimizes the weight of the highest-weight edge of the cycle. It was first formulated by Gilmore & Gomory (1964) with some additional constraints, and in its full generality by Garfinkel & Gilbert (1978).
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for scheduling on unrelated parallel machines.
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.
Václav (Vašek) Chvátal is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada and a Visiting Professor at Charles University in Prague. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization.
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such that
In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices, such that any vertical line crosses the chain at most twice.
The Concorde TSP Solver is a program for solving the travelling salesman problem. It was written by David Applegate, Robert E. Bixby, Vašek Chvátal, and William J. Cook, in ANSI C, and is freely available for academic use.
Alexander Hendrik George Rinnooy Kan is a Dutch politician, businessman and mathematician who served as Chairman of the Social and Economic Council from 2006 to 2012. A member of the Democrats 66 (D66) party, he was a member of the Senate from 2015 to 2019 and is a distinguished professor of Economics and Business Studies at the University of Amsterdam since 1 September 2012. He has also been president of the supervisory board of EYE Film Institute Netherlands since 2008 and of Museum Boerhaave since 2018.
Eugene Leighton (Gene) Lawler was an American computer scientist and a professor of computer science at the University of California, Berkeley.
William John Cook is an American operations researcher and mathematician, and Professor of Combinatorics and Optimization at the University of Waterloo.
David L. Applegate is an American computer scientist known for his research on the traveling salesperson problem.
Descriptive Complexity is a book in mathematical logic and computational complexity theory by Neil Immerman. It concerns descriptive complexity theory, an area in which the expressibility of mathematical properties using different types of logic is shown to be equivalent to their computability in different types of resource-bounded models of computation. It was published in 1999 by Springer-Verlag in their book series Graduate Texts in Computer Science.
Combinatorics: The Rota Way is a mathematics textbook on algebraic combinatorics, based on the lectures and lecture notes of Gian-Carlo Rota in his courses at the Massachusetts Institute of Technology. It was put into book form by Joseph P. S. Kung and Catherine Yan, two of Rota's students, and published in 2009 by the Cambridge University Press in their Cambridge Mathematical Library book series, listing Kung, Rota, and Yan as its authors. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
