Gregory Z. Gutin

Traveling Salesman Should not be Greedy: Domination Analysis of Greedy-Type Heuristics for the TSP

Abstract Computational experiments show that the greedy algorithm (GR) and the nearest neighbor algorithm (NN), popular choices for tour construction heuristics, work at acceptable level for the Euclidean TSP, but produce very poor results for the general Symmetric and Asymmetric TSP (STSP and ATSP). We prove that for every n⩾2 there is an instance of ATSP (STSP) on n vertices for which GR finds the worst tour. The same result holds for NN. We also analyze the repetitive NN (RNN) that starts NN from every vertex and chooses the best tour obtained. We prove that, for the ATSP, RNN always produces a tour, which is not worse than at least n/2−1 other tours, but for some instance it finds a tour, which is not worse than at most n−2 other tours, n⩾4. We also show that, for some instance of the STSP on n⩾4 vertices, RNN produces a tour not worse than at most 2n−3 tours. These results are in sharp contrast to earlier results by Gutin and Yeo, and Punnen and Kabadi, who proved that, for the ATSP, there are tour construction heuristics, including some popular ones, that always build a tour not worse than at least (n−2)! tours.

When the greedy algorithm fails

We provide a characterization of the cases when the greedy algorithm may produce the unique worst possible solution for the problem of finding a minimum weight base in an independence system when the weights are taken from a finite range. We apply this theorem to TSP and the minimum bisection problem. The practical message of this paper is that the greedy algorithm should be used with great care, since for many optimization problems its usage seems impractical even for generating a starting solution (that will be improved by a local search or another heuristic).

Construction heuristics for the asymmetric TSP

Abstract Non-Euclidean traveling salesman problem (TSP) construction heuristics, and especially asymmetric TSP construction heuristics, have been neglected in the literature by comparison with the extensive efforts devoted to studying Euclidean TSP construction heuristics. This state of affairs is at odds with the fact that asymmetric models are relevant to a wider range of applications, and indeed are uniformly more general that symmetric models. Moreover, common construction approaches for the Euclidean TSP have been shown to produce poor quality solutions for non-Euclidean instances. Motivation for remedying this gap in the study of construction approaches is increased by the fact that such methods are a great deal faster than other TSP heuristics, which can be important for real time problems requiring continuously updated response. The purpose of this paper is to describe two new construction heuristics for the asymmetric TSP and a third heuristic based on combining the other two. Extensive computational experiments are performed for several different families of TSP instances, disclosing that our combined heuristic clearly outperforms well-known TSP construction methods and proves significantly more robust in obtaining (relatively) high quality solutions over a wide range of problems.

A memetic algorithm for the generalized traveling salesman problem

The generalized traveling salesman problem (GTSP) is an extension of the well-known traveling salesman problem. In GTSP, we are given a partition of cities into groups and we are required to find a minimum length tour that includes exactly one city from each group. The recent studies on this subject consider different variations of a memetic algorithm approach to the GTSP. The aim of this paper is to present a new memetic algorithm for GTSP with a powerful local search procedure. The experiments show that the proposed algorithm clearly outperforms all of the known heuristics with respect to both solution quality and running time. While the other memetic algorithms were designed only for the symmetric GTSP, our algorithm can solve both symmetric and asymmetric instances.

Cycles and paths in semicomplete multipartite digraphs, theorems, and algorithms: a survey

A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete m- partite digraphs including some recent results concerning tournaments.

Level of repair analysis and minimum cost homomorphisms of graphs

Level of repair analysis (LORA) is a prescribed procedure for defense logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORA-BR, Barros [The optimisation of repair decisions using life-cycle cost parameters. IMA J. Management Math. 9 (1998) 403-413] and Barros and Riley [A combinatorial approach to level of repair analysis, European J. Oper. Res. 129 (2001) 242-251] developed certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NP-hard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that our result for LORA-BR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable.

Lin–Kernighan heuristic adaptations for the generalized traveling salesman problem

The Lin-Kernighan heuristic is known to be one of the most successful heuristics for the Traveling Salesman Problem (TSP). It has also proven its efficiency in application to some other problems. In this paper, we discuss possible adaptations of TSP heuristics for the generalized traveling salesman problem (GTSP) and focus on the case of the Lin-Kernighan algorithm. At first, we provide an easy-to-understand description of the original Lin-Kernighan heuristic. Then we propose several adaptations, both trivial and complicated. Finally, we conduct a fair competition between all the variations of the Lin-Kernighan adaptation and some other GTSP heuristics. It appears that our adaptation of the Lin-Kernighan algorithm for the GTSP reproduces the success of the original heuristic. Different variations of our adaptation outperform all other heuristics in a wide range of trade-offs between solution quality and running time, making Lin-Kernighan the state-of-the-art GTSP local search.

Algorithms with large domination ratio

Let P be an optimization problem, and let A be an approximation algorithm for P. The domination ratio domr(A, n) is the maximum real q such that the solution x(I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial-time algorithm with domination ratio 1 - o(1) for the partition problem, and a deterministic, polynomial-time algoritiun with domination ratio Ω(1) for the MaxCut problem and for some far-reaching extensions of it, including Max-r-Sat, for each fixed r. The techniques combine combinatorial and probabilistic methods with tools from harmonic analysis.

Generalizations of tournaments: A survey

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly nontrivial, even for these “tournament-like” digraphs.

Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number

Abstract Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n !/ p ( n ) tours for some polynomial p ( n ) for every TSP instance on n cities. They conjectured that, unless P=NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadratic assignment problem, is also considered with respect to the analogous question. Probabilistic, graph-theoretical, group-theoretical and number-theoretical methods and results are used.