The traveling salesman problem on grids with forbidden neighborhoods

We introduce the traveling salesman problem with forbidden neighborhoods (TSPFN). This is an extension of the Euclidean TSP in the plane where direct connections between points that are too close are forbidden. The TSPFN is motivated by an application in laser beam melting. In the production of a workpiece in several layers using this method one hopes to reduce the internal stresses of the workpiece by excluding the heating of positions that are too close. The points in this application are often arranged in some regular (grid) structure. In this paper we study optimal solutions of TSPFN instances where the points in the Euclidean plane are the points of a regular grid. Indeed, we explicitly determine the optimal values for the TSPFN and its associated path version on rectangular regular grids for different minimal distances of the points visited consecutively. For establishing lower bounds on the optimal values we use combinatorial counting arguments depending on the parities of the grid dimensions. Furthermore we provide construction schemes for optimal TSPFN tours for the considered cases.     

Online traveling salesman problem with time cost and non-zealous server

Journal of Combinatorial Optimization - Considering that the time of meeting the demands is very important for emergency vehicle and emergency vehicle can’t reject any request, we introduce a...     

The multiple traveling salesman problem: an overview of formulations and solution procedures

The multiple traveling salesman problem (mTSP) is a generalization of the well-known traveling salesman problem (TSP), where more than one salesman is allowed to be used in the solution. Moreover, the characteristics of the mTSP seem more appropriate for real-life applications, and it is also possible to extend the problem to a wide variety of vehicle routing problems (VRPs) by incorporating some additional side constraints. Although there exists a wide body of the literature for the TSP and the VRP, the mTSP has not received the same amount of attention. The purpose of this survey is to review the problem and its practical applications, to highlight some formulations and to describe exact and heuristic solution procedures proposed for this problem.     

Angular bisector insertion algorithm for solving small-scale symmetric and asymmetric traveling salesman problem

Journal of Combinatorial Optimization - Different algorithmic performances are required in different engineering fields for solving both the symmetric and asymmetric traveling salesman problem...     

Bounds for the traveling salesman paths of two-dimensional modular lattices

We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi–Perron type algorithms.     

Online covering salesman problem

Given a graph \(G=(V,E,D,W)\), the generalized covering salesman problem (CSP) is to find a shortest tour in G such that each vertex \(i\in D\) is either on the tour or within a predetermined distance L to an arbitrary vertex \(j\in W\) on the tour, where \(D\subset V\),\(W\subset V\). In this paper, we propose the online CSP, where the salesman will encounter at most k blocked edges during the traversal. The edge blockages are real-time, meaning that the salesman knows about a blocked edge when it occurs. We present a lower bound \(\frac{1}{1 + (k + 2)L}k+1\) and a CoverTreeTraversal algorithm for online CSP which is proved to be \(k+\alpha \)-competitive, where \(\alpha =0.5+\frac{(4k+2)L}{OPT}+2\gamma \rho \), \(\gamma \) is the approximation ratio for Steiner tree problem and \(\rho \) is the maximal number of locations that a customer can be served. When \(\frac{L}{\texttt {OPT}}\rightarrow 0\), our algorithm is near optimal. The problem is also extended to the version with service cost, and similar results are derived.     

Polyhedral combinatorics of the cardinality constrained quadratic knapsack problem and the quadratic selective travelling salesman problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets. Author now works at Motorola. (2005 onwards)     

Local search algorithms for multiple-depot vehicle routing and for multiple traveling salesman problems with proved performance guarantees

We consider two related problems: the multiple-depot vehicle routing problem (MDVRP) and the Multiple traveling salesman problem (mTSP). In both of them, given is the complete graph on n vertices \(G = (V,E)\) with nonnegative edge lengths that form a metric on V. Also given is a positive integer k. In typical applications, V represents locations of customers and k represents the number of available vehicles. In MDVPR, we are also given a set of k depots \(\{O_1,\ldots ,O_k\} \subseteq V\) , and the goal is to find a minimum-length cycle cover of G of size k, that is, a collection of k (possibly empty) cycles such that each \(v \in V\) is in exactly one cycle, and each cycle in the cover contains exactly one depot. In mTSP, no depots are given, so the goal is to find (any) minimum-length cycle cover of G of size k. We present local search algorithms for both problems, and we prove that their approximation ratio is 2.     

Two new robust genetic algorithms for the flowshop scheduling problem

The flowshop scheduling problem (FSP) has been widely studied in the literature and many techniques for its solution have been proposed. Some authors have concluded that genetic algorithms are not suitable for this hard, combinatorial problem unless hybridization is used. This work proposes new genetic algorithms for solving the permutation FSP that prove to be competitive when compared to many other well known algorithms. The optimization criterion considered is the minimization of the total completion time or makespan (_Cmax$C_{\max }$). We show a robust genetic algorithm and a fast hybrid implementation. These algorithms use new genetic operators, advanced techniques like hybridization with local search and an efficient population initialization as well as a new generational scheme. A complete evaluation of the different parameters and operators of the algorithms by means of a Design of Experiments approach is also given. The algorithm's effectiveness is compared against 11 other methods, including genetic algorithms, tabu search, simulated annealing and other advanced and recent techniques. For the evaluations we use Taillard's well known standard benchmark. The results show that the proposed algorithms are very effective and at the same time are easy to implement.     

Two approaches to solving the multi-depot vehicle routing problem with time windows in a time-based logistics environment

This paper presents a two-phase heuristic method that can be used to efficiently solve the intractable multi-depot vehicle routing problem with time windows. The waiting time that was ignored by previous researchers is considered in this study. The necessity of this consideration is verified through an initial experiment. The results indicate that the waiting time has a significant impact on the total distribution time and the number of vehicles used when solving test problems with narrow time windows. In addition, to fairly evaluate the performance of the proposed heuristic method, a meta-heuristic method, which extends the unified tabu search of Cordeau et al., is proposed. The results of a second experiment reveal that the proposed heuristic method can obtain a better solution in the case of narrow time windows and a low capacity ratio, while the proposed meta-heuristic method outperforms the proposed heuristic method, provided that wide time windows and a high capacity ratio are assumed. Finally, a well-known logistics company in Taiwan is used to demonstrate the method, and a comparison is made, which shows that the proposed heuristic method is superior to the current method adopted by the case company.     

On the Bandpass problem

The complexity of the Bandpass problem is re-investigated. Specifically, we show that the problem with any fixed bandpass number B≥2 is NP-hard. Next, a row stacking algorithm is proposed for the problem with three columns, which produces a solution that is at most 1 less than the optimum. For the special case B=2, the row stacking algorithm guarantees an optimal solution. On approximation, for the general problem, we present an O(B ²)-algorithm, which reduces to a 2-approximation algorithm for the special case B=2.     

On the personnel assignment problem

Recent papers have developed methods for personnel assignment under risk of failure. In this paper, a two-parameter model is given. It uses both mean and standard deviation in the stochastic model to bring the risk of failure under control.     

On the fractionality of the path packing problem

Given an undirected graph G=(N,E), a subset T of its nodes and an undirected graph (T,S), G and (T,S) together are often called a network. A?collection of paths in G whose end-pairs lie in S is called an integer multiflow. When these paths are allowed to have fractional weight, under the constraint that the total weight of the paths traversing a single edge does not exceed 1, we have a fractional multiflow in G. The problems of finding the maximum weight of paths with end-pairs in S over all fractional multiflows in G is called the fractional path packing problem. In 1989, A. Karzanov had defined the fractionality of the fractional path packing problem for a class of networks {G,(T,S)} as the smallest natural D such that for any network from the class, the fractional path packing problem has a solution which becomes integer-valued when multiplied by D (see A.?Karzanov in Linear Algebra Appl. 114–115:293–328, 1989). He proved that the fractional path packing problem has infinite fractionality outside a very specific class of networks, and conjectured that within this class, the fractionality does not exceed 4. A.?Karzanov also proved that the fractionality of the path packing problem is at most 8 by studying the fractionality of the dual problem. Special cases of Karzanov’s conjecture were proved in or are implied by the works of L.R.?Ford and D.R.?Fulkerson, Y.?Dinitz, T.C.?Hu, B.V.?Cherkassky, L.?Lov?sz and H.?Hirai. We prove Karzanov’s conjecture by showing that the fractionality of the path packing problem is at most 4. Our proof is stand-alone and does not rely on Karzanov’s results.     

Scheduling and the problem of computational complexity

Why is it that the problem of scheduling is so computationally difficult to solve? At last recent developments in modern mathematical complexity theory are providing some insights. The paper describes in essentially non-mathematical terms the computational technique known as the ‘Branch and Bound Method’. This, the best general optimising technique available for scheduling, is also shown to have its limitations. It now appears that efficient computational and optimising algorithms are unlikely ever to be found for all except special cases of the general industrial scheduling problem. It seems that heuristic (rule-of-thumb) methods leading to approximate solutions are likely to offer the only real promise for the future.     

Approximability of the subset sum reconfiguration problem

The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme, while the problem is APX-hard if we are given a conflict graph.     

On the max-weight edge coloring problem

We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.     

IRR computation and the multi-asset problem

The IRR model for individual assets has been the target of serious criticism over the years. The purpose of this paper is to provide a partial rationale for interpreting aggregate firm data based on IRR calculations of individual assets. More specifically, it will be argued that the value of accounting information will be enhanced as a result of depreciation and income recognition schedules being determined by an individual asset's IRR model.