Journal ArticleDOI
The traveling salesman problem in bounded degree graphs
TLDR
It is shown that the traveling salesman problem in bounded-degree graphs can be solved in time <i>O</i>((2-ε)<i><sup>n</sup></i>), where ε > 0 depends only on the degree bound but not on the number of cities.Abstract:
We show that the traveling salesman problem in bounded-degree graphs can be solved in time O((2-e)n), where e > 0 depends only on the degree bound but not on the number of cities, n. The algorithm is a variant of the classical dynamic programming solution due to Bellman, and, independently, Held and Karp. In the case of bounded integer weights on the edges, we also give a polynomial-space algorithm with running time O((2-e)n) on bounded-degree graphs. In addition, we present an analogous analysis of Ryser's algorithm for the permanent of matrices with a bounded number of nonzero entries in each column.read more
Citations
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Posted Content
Solving connectivity problems parameterized by treewidth in single exponential time
Marek Cygan,Jesper Nederlof,Marcin Pilipczuk,Michał Pilipczuk,Joham M.M. van Rooij,Jakub Onufry Wojtaszczyk +5 more
TL;DR: It is shown that the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing, and in several cases it is able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail.
Proceedings ArticleDOI
Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time
Marek Cygan,Jesper Nederlof,Marcin Pilipczuk,Michał Pilipczuk,Joham M.M. van Rooij,Jakub Onufry Wojtaszczyk +5 more
TL;DR: Cut&Count as mentioned in this paper is a Monte Carlo algorithm that runs in O(1) time for most connectivity-type problems, including Hamiltonian Path, Steiner Tree, Feedback Vertex Set and Connected Dominating Set.
Proceedings ArticleDOI
Determinant Sums for Undirected Hamiltonicity
TL;DR: A Monte Carlo algorithm for Hamilton city detection in an $n$-vertex undirected graph running in $O*(1.657^{n})$ time is presented, which is the first super polynomial improvement on the worst case runtime for the problem since the O^*(2^n) bound established for TSP almost fifty years ago.
Proceedings ArticleDOI
Computing the Tutte Polynomial in Vertex-Exponential Time
TL;DR: The fastest known general-purpose algorithm for computing a host of fundamental graph invariants, such as the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin-Kasteleyn in statistical physics, runs in time roughly proportional to the number of spanning trees in the input graph.
Book ChapterDOI
The Travelling Salesman Problem in Bounded Degree Graphs
TL;DR: It is shown that the travelling salesman problem in bounded-degreegraphs can be solved in time $O\bigl((2-\epsilon)^n\bigr)$, wheree> 0 depends only on the degree bound but not on the number of cities, n.
References
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Book
The Traveling Salesman Problem: A Computational Study
TL;DR: Open Library features a library with books from the Internet Archive and lists them in the open library and gives you access to over 1 million free e-Books and the ability to search using subject, title and author.
Book
The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization
TL;DR: In this paper, Johnson and Papadimitriou proposed a performance guarantee for heuristics, based on the notion of well-solved special cases (P. Gilmore, et al.).
BookDOI
The traveling salesman problem and its variations
Gregory Gutin,Abraham P. Punnen +1 more
TL;DR: This paper presents Polyhedral Theory and Branch-and-Cut Algorithms for the Symmetric TSP, a model for solving the Asymmetric Traveling Salesman Problem, and some examples of how this model was applied to the Geometric TSP.
Journal ArticleDOI
A Dynamic Programming Approach to Sequencing Problems
Michael Held,Richard M. Karp +1 more
TL;DR: In this paper, a dynamic programming approach to the solution of three sequencing problems, namely, a scheduling problem involving arbitrary cost functions, the traveling-salesman problem, and an assembly line balancing problem, is presented.
Journal ArticleDOI
THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization
TL;DR: Combinatorial OptimizationAdvanced Intelligent Computing Theories and ApplicationsOpt ArtHybrid Feedback ControlAdvances in Multi-Objective Nature Inspired ComputingThe Traveling Salesman ProblemIntelligent Computational Optimization in EngineeringThe Traveled SalesmanFundamental Problems in ComputingSpecial Cases of the Traveling salesman Problem