Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex trans...

We give a short proof that any comparison-based (n 1−ǫ)-approximation algorithm for the 1-dimensional Traveling Salesman Problem (TSP) requires Ω(n log n) comparisons.

We present two simple results for generalizations of the traveling salesman problem (TSP): For the universal TSP, we show that one can compute a tour that is universally optimal whenever the input is a tree metric. A (randomized) O(logn)-approximation algorithm for the a priori TSP follows as a corollary.

We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the l...

Biogeography-Based Optimization (BBO) algorithm has recently been of great interest to researchers for simplicity of implementation, efficiency, and the low number of parameters. The BBO Algorithm in optimization problems is one of the new algorithms which have been developed based on the biogeography concept. This algorithm uses the idea of animal migration to find suitable habitats for solvin...

We propose a generalized definition for the multi-objective traveling salesman problem which uses multigraphs and which allows multiple visits of cities. The definition has two benefits: it captures typical real-world scenarios and it contains the conventional definition (componentwise metric cost function) as a special case. We provide approximation algorithms for this general version of the t...

Owing to its complexity, the traveling salesman problem (TSP) is one of the most intensively studied problems in computational mathematics. The TSP is defined as the provision of minimization of total distance, cost, and duration by visiting the n number of points only once in order to arrive at the starting point. Various heuristic algorithms used in many fields have been developed to solve th...

The Traveling Salesman Problem or TSP is a fundamental andwell-known problem in combinatorial optimisation. At present, the mostsuccessful algorithms for solving large-scale instances of the TSP to proven(near-)optimality are based on integer programming. This entry introducesthe main theoretical and algorithmic tools involved. Topics covered include:formulations of the TSP,...

We describe a hybrid procedure for solving the traveling salesman problem (TSP) to provable optimality. We first sparsify the instance, and then use a hybrid algorithm that combines a branch-and-cut TSP solver with a Hamiltonian cycle problem solver. We demonstrate that this procedure enables us to solve difficult instances to optimality, including one which had remained unsolved since its cons...

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on...