A backbone based TSP heuristic for large instances

We introduce a reduction technique for large instances of the traveling salesman problem (TSP). This approach is based on the observation that tours with good quality are likely to share many edges. We exploit this observation by neglecting the less important tour space defined by the shared edges, and searching the important tour subspace in more depth. More precisely, by using a basic TSP heuristic, we obtain a set of starting tours. We call the set of edges which are contained in each of these starting tours as pseudo-backbone edges. Then we compute the maximal paths consisting only of pseudo-backbone edges, and transform the TSP instance to another one with smaller size by contracting each such path to a single edge. This reduced TSP instance can be investigated more intensively, and each tour of the reduced instance can be expanded to a tour of the original instance. Combining our reduction technique with the currently leading TSP heuristic of Helsgaun, we experimentally investigate 32 difficult VLSI instances from the well-known TSP homepage. In our experimental results we set world records for seven VLSI instances, i.e., find better tours than the best tours known so far (two of these world records have since been improved upon by Keld Helsgaun and Yuichi Nagata, respectively). For the remaining instances we find tours that are equally good or only slightly worse than the world record tours.     

Multiobjective traveling salesperson problem on Halin graphs

In this paper, we study traveling salesperson (TSP) and bottleneck traveling salesperson (BTSP) problems on special graphs called Halin graphs. Although both problems are NP-Hard on general graphs, they are polynomially solvable on Halin graphs. We address the multiobjective versions of these problems. We show computational complexities of finding a single nondominated point as well as finding all nondominated points for different objective function combinations. We develop algorithms for the polynomially solvable combinations.     

A restricted dynamic programming heuristic algorithm for the time dependent traveling salesman problem

Dynamic programming (DP) algorithms for the traveling salesman problem (TSP) can easily incorporate time dependent travel times, time windows, and precedence relationships which present difficulties for algorithms based on linear or nonlinear programming formulations and for many TSP heuristics. However, exact DP algorithms for the TSP have exponential storage and computational time requirements and can solve only very small problems. We present a restricted DP heuristic (a generalization of the nearest neighbor heuristic) that can include all the above considerations but solves much larger problems. The heuristic cannot guarantee optimality because only a userspecified specified number of partial tours is retained at each stage. In this paper, the heuristic is implemented for the time dependent traveling salesman problem and is tested on a personal computer on randomly generated problems. The quality of the solution improves, on average, as more computational time is permitted.     

Min-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio

The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e. the minimum-weight double-tree shortcutting. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and in particular that the minimum-weight double-tree method provides a 1.622-approximation for planar Euclidean TSP.     

Four-point conditions for the TSP: The complete complexity classification

The combinatorial optimization literature contains a multitude of polynomially solvable special cases of the traveling salesman problem (TSP) which result from imposing certain combinatorial restrictions on the underlying distance matrices. Many of these special cases have the form of so-called four-point conditions: inequalities that involve the distances between four arbitrary cities.In this paper we classify all possible four-point conditions for the TSP with respect to computational complexity, and we determine for each of them whether the resulting special case of the TSP can be solved in polynomial time or whether it remains NP-hard.     

On the capacitated vehicle routing problem

 We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY. Received: October 30, 2000 / Accepted: December 19, 2001 Published online: September 5, 2002 Key words. vehicle routing problem – integer programming – decomposition algorithm – separation algorithm – branch and cut Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

An asymmetric analog of van der Veen conditions and the traveling salesman problem (II)

J.A.A. van der Veen [A new class of pyramidally solvable symmetric traveling salesman problems, SIAM J. Discrete Math. 7 (1994) 585–592] proved that for the traveling salesman problem (TSP) which satisfies some symmetric conditions (called van der Veen conditions), a shortest pyramidal tour is optimal, that is, an optimal tour can be computed in polynomial time. In this paper, we prove that a class satisfying an asymmetric analog of van der Veen conditions is polynomially solvable. An optimal tour of the instance in this class forms a tour which is an extension of pyramidal ones. Moreover, this class properly includes some known polynomially solvable classes.     

Monte Carlo Algorithms for Identifying Densely Connected Subgraphs

The problem of finding densely connected subgraphs in a network has attracted a lot of recent interest. Such subgraphs are sometimes referred to as communities in social networks or molecular modules in protein networks. In this article, we propose two Monte Carlo optimization algorithms for identifying the densest subgraphs with a fixed size or with size in a given range. The new algorithms combine the idea of simulated annealing and efficient moves for the Markov chain, and both algorithms are shown to converge to the set of optimal states (densest subgraphs) with probability 1. When applied to a yeast protein interaction network and a stock market graph, the algorithms identify interesting new densely connected subgraphs. Supplementary materials for the article are available online.     

Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem

Evolutionary algorithms are applied to problems that are not well understood as well as to problems in combinatorial optimization. The analysis of these search heuristics has been started for some well-known polynomial solvable problems. Such analyses are starting points for the analysis of evolutionary algorithms on difficult problems. We present the first runtime analysis of a multi-objective evolutionary algorithm on a NP-hard problem. The subject of our analysis is the multi-objective minimum spanning tree problem for which we give upper bounds on the expected time until a simple evolutionary algorithm has produced a population including for each extremal point of the Pareto front a corresponding spanning tree. These points are of particular interest as they give a 2-approximation of the Pareto front. We show that in expected pseudopolynomial time a population is produced that includes for each extremal point a corresponding spanning tree.     

A study of exponential neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem

This paper deals with exponential neighborhoods for combinatorial optimization problems. Exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods.?First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NP-complete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved in polynomial time, the corresponding situation for the QAP looks pretty hopeless: Every exponential neighborhood that is considered in this paper provably leads to an NP-complete optimization problem for the QAP. Received: September 5, 1997 / Accepted: November 15, 1999?Published online February 23, 2000     

Equivalent cyclic polygon of a euclidean travelling salesman problem tour and modified formulation

Central European Journal of Operations Research - We define a geometric transformation of Euclidean Travelling Salesman Problem (TSP) tours that leads to a new formulation of the TSP. For every...     

A class of polynomially solvable 0-1 programming problems and an application

It is well known that general 0-1 programming problems are NP-Complete and their optimal solutions cannot be found with polynomial-time algorithms unless P=NP. In this paper, we identify a specific class of 0-1 programming problems that is polynomially solvable, and propose two polynomial-time algorithms to find its optimal solutions. This class of 0-1 programming problems commits to a wide range of real-world industrial applications. We provide an instance of representative in the field of supply chain man...     

Quadratic bottleneck knapsack problems

In this paper we study the quadratic bottleneck knapsack problem (QBKP) from an algorithmic point of view. QBKP is shown to be NP-hard and it does not admit polynomial time ?-approximation algorithms for any ?>0 (unless P=NP). We then provide exact and heuristic algorithms to solve the problem and also identify polynomially solvable special cases. Results of extensive computational experiments are reported which show that our algorithms can solve QBKP of reasonably large size and produce good quality solutions very quickly. Several variations of QBKP are also discussed.     

Single machine scheduling with exponential learning functions

In this paper we consider the single machine scheduling problem with exponential learning functions. By the exponential learning functions, we mean that the actual job processing time is a function of the total normal processing times of the jobs already processed. We prove that the shortest processing time (SPT) rule is optimal for the total lateness minimization problem. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms. It also shows that the problems of minimizing the total tardiness and discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.     

Seeking global edges for traveling salesman problem in multi-start search

This study investigates the properties of the edges in a set of locally optimal tours found by multi-start search algorithm for the traveling salesman problem (TSP). A matrix data structure is used to collect global information about edges from the set of locally optimal tours and to identify globally superior edges for the problem. The properties of these edges are analyzed. Based on these globally superior edges, a solution attractor is formed in the data matrix. The solution attractor is a small region of the solution space, which contains the most promising solutions. Then an exhausted enumeration process searches the solution attractor and outputs all solutions in the attractor, including the globally optimal solution. Using this strategy, this study develops a procedure to tackler a multi-objective TSP. This procedure not only generates a set of Pareto-optimal solutions, but also be able to provide the structural information about each of the solutions that will allow a decision-maker to choose the best compromise solution.     

More on quasi-random graphs,subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases.The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.     

Partitioning Perfect Graphs into Stars

The partition of graphs into “nice” subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same‐size stars, a problem known to be NP‐complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial‐time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP‐complete cases, for example, on grid graphs and chordal graphs.     

A review of TSP based approaches for flowshop scheduling

Contemporary flowshops that are variants of the classical flowshop frequently pose challenging scheduling problems. Such flowshops include no-wait, blocking, and robotic flowshops. These may sometimes be modeled as traveling salesman problems (TSP) and then solved using efficient algorithms available for the TSP. Encountered in auto, electronic, chemical, steel and even modern service industries, such problems are reviewed in this paper. We show that the TSP based approach is quite effective over a broad range. It tackles no-wait flowshops, blocking flowshops, group scheduling of parts in a flowshop using a generalized extension of the TSP, lot streaming and scheduling problems, and as recently done, scheduling of parts and robot movements in automated production cells. In this review paper, we describe several well-documented applications of no-wait and blocking scheduling models and illustrate some ways in which the increasingly used modern manufacturing systems such as robotic cells may be modeled as TSP. We also review the computational complexity of a wide variety of flowshop models. Finally, we suggest some fruitful directions for future research.     

Solvable Cases of the No-Wait Flow-Shop Scheduling Problem

The no-wait flow-shop scheduling problem (NWFSSP) with a makespan objective function is considered. As is well known, this problem is ????-hard for three or more machines. Therefore, it is interesting to consider special cases, i.e. special structured processing time matrices, that allow polynomial time solution algorithms. Furthermore, it is well known that the NWFSSP with a makespan objective function can be formulated as a travelling salesman problem (TSP). It is observed that special structured processing time matrices for the NWFSSP lead to special structured distance matrices for which the TSP is polynomially solvable. Using this observation, it is shown that some NWFSSPs with fixed processing times on all except two machines are well solvable while the others are conjectured to be ????-hard. Also, it is shown that NWFSSPs with a mean completion time objective function restricted to semi-ordered processing time matrices are easily solvable.     

具有CON/SLK交货期指派的一类单机排序问题

研究工件的实际加工时间既具有指数学习效应,又依赖所消耗资源的准时制排序问题.在模型中,探讨了共同交货期(CON)和松弛交货期(SLK)两种情形.管理者的目标是确定最优序、最优资源分配方案和最佳工期(共同交货期或松弛交货期)以便极小化工件的总延误、总提前、总工期和资源消耗费用的总和.对于工件的实际加工时间是资源消耗量的线性函数的排序问题,通过将其转化为指派模型,给出了时间复杂性为O(n~3)的算法,从而证明该类排序问题是多项式时间可求解的.针对工件的实际加工时间是资源消耗量的凸函数的排序问题,也给出了多项式算法.