Efficiently solvable special cases of bottleneck travelling salesman problems

AbstractThe paper investigates bottleneck travelling salesman problems (BTSP) which can be solved in polynomial time. At first a BTSP whose cost matrix is a circulant is treated. It is shown that in the symmetric case such a BTSP can be solved in O(n log n) time. Secondly conditions are derived which guarantee that an optimal solution is a pyramidal tour. Thus this problem can be solved in O(n2) time. Finally it is shown that a BTSP with cost matrix C = (cij, where cij = aibj with a1 ≤ … ≤ an and b1 ≥ … ≥ bn can be solved in O(n2) time

The multi-stripe travelling salesman problem

In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q ≥ 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP

TSP and its variants : use of solvable cases in heuristics

This thesis proposes heuristics motivated by solvable cases for the travelling salesman problem (TSP) and the cumulative travelling salesman path problem (CTSPP). The solvable cases are investigated in three aspects: specially structured matrices, special neighbourhoods and small-size problems. This thesis demonstrates how to use solvable cases in heuristics for the TSP and the CTSPP and presents their promising performance in theoretical research and empirical research. Firstly, we prove that the three classical heuristics, nearest neighbour, double-ended nearest neighbour and GREEDY, have the theoretical property of obtaining the permutation for permuted strong anti-Robinson matrices for the TSP such that the renumbered matrices satisfy the anti-Robinson conditions. Inspired by specially structured matrices, we propose Kalmanson heuristics, which not only have the theoretical property of solving permuted strong Kalmanson matrices to optimality for the TSP, but also outperform their classical counterparts for general cases. Secondly, we propose three heuristics for the CTSPP. The pyramidal heuristic is motivated by the special pyramidal neighbourhood. The chains heuristic and the sliding window heuristic are motivated by solvable small-size problems. The experiments suggest the proposed heuristics outperform the classical GRASP-2-opt on general cases for the CTSPP. Thirdly, we conduct both theoretical and empirical research on specially structured cases for the CTSPP. Theoretically, we prove the solvability of Line- CTSPP on more general cases and the time complexity of the CTSPP on SUM matrices. We also conjecture that the CTSPP on two rays is NP-hard. Empirically, we propose three heuristics, which perform well on specially structured cases. The Line heuristic, based on Line-CTSPP, performs better than GRASP-2-opt when nodes are distributed on two close parallel lines. The Up-Down heuristic is inspired by the Up-Down structure in solvable Path TSP and outperforms GRASP-2-opt in convex-hull cases and close-to-convex-hull cases. The Two-Ray heuristic combines the path structures in the first two heuristics and obtains high-quality solutions when nodes are along two rays

Traveling Salesman Problem for Surveillance Mission Using Particle Swarm Optimization

The surveillance mission requires aircraft to fly from a starting point through defended terrain to targets and return to a safe destination (usually the starting point). The process of selecting such a flight path is known as the Mission Route Planning (MRP) Problem and is a three-dimensional, multi-criteria (fuel expenditure, time required, risk taken, priority targeting, goals met, etc.) path search. Planning aircraft routes involves an elaborate search through numerous possibilities, which can severely task the resources of the system being used to compute the routes. Operational systems can take up to a day to arrive at a solution due to the combinatoric nature of the problem. This delay is not acceptable because timeliness of obtaining surveillance information is critical in many surveillance missions. Also, the information that the software uses to solve the MRP may become invalid during computation. An effective and efficient way of solving the MRP with multiple aircraft and multiple targets is desired. One approach to finding solutions is to simplify and view the problem as a two-dimensional, minimum path problem. This approach also minimizes fuel expenditure, time required, and even risk taken. The simplified problem is then the Traveling Salesman Problem (TSP)

The maximum Travelling Salesman Problem on symmetric Demidenko matrices

It is well-known that the Travelling Salesman Problem (TSP) is solvable in polynomial time, if the distance matrix fulfills the so-called Demidenko conditions. This paper investigates the closely related Maximum Travelling Salesman Problem (MaxTSP) on symmetric Demidenko matrices. Somewhat surprisingly, we show that — in strong contrast to the minimization problem — the maximization problem is NP-hard to solve. Moreover, we identify several special cases that are solvable in polynomial time. These special cases contain and generalize several predecessor results by Quintas and Supnick and by Kalmanson

Particle Physics Reference Library

This second open access volume of the handbook series deals with detectors, large experimental facilities and data handling, both for accelerator and non-accelerator based experiments. It also covers applications in medicine and life sciences. A joint CERN-Springer initiative, the “Particle Physics Reference Library” provides revised and updated contributions based on previously published material in the well-known Landolt-Boernstein series on particle physics, accelerators and detectors (volumes 21A,B1,B2,C), which took stock of the field approximately one decade ago. Central to this new initiative is publication under full open access