This note aims to give an application of entropy to the analysis of algorithms, using Shearer’s entropy lemma to bound the running time of Bellman’s dynamic programming solution to the traveling salesman problem. The original venue for this material was my course Randommised algorithms (2008) at Lund University. 1. The Traveling Salesman Problem Given a weighted graph like

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...

Theoretical computer science has given birth to a very intriguing and fascinating class of problems known as Nondeterministic Polynomial-time (NP) complete problems. Among their captivating properties are their computational complexity despite often deceptively simple formulations. An example of this is the ubiquitous Traveling Salesman Problem (TSP). This problem has had an uncontested popular...

by Howard Kleiman

Traveling salesman problem is a NP-hard problem. Until now, researchers have not found a polynomial time algorithm for traveling salesman problem. Among the existing algorithms, dynamic programming algorithm can solve the problem in time O(n2·2n) where n is the number of nodes in the graph. The branch-and-cut algorithm has been applied to solve the problem with a large number of nodes. However,...

Consider a truck running along a road. It picks up a load Li at point βi and delivers it at αi, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L1, . . . , Ln is equivalent to solving the Traveling Salesman Problem (TSP) where the c...

The Traveling Salesman Problem(TSP) is one of the most intensively studied problem in computational mathematics. To solve this problem a number of algorithms have been developed using genetic algorithms. But these algorithms are not so suitable for solving large-scale TSP. This paper proposes a new solution for TSP using hierarchical clustering and genetic algorithm. Keywords— Traveling Salesma...