TSP Heuristics: Domination Analysis and Complexity

We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph Kn produce a solution no worse than the average cost of a tour in Kn in a polynomial number of iterations. As a consequence, we get that the domination numbers of the 2- Opt , 3- Opt , Carlier—Villon, Shortest Path Ejection Chain, and Lin—Kernighan heuristics are all at least (n-2)! / 2 . The domination number of the Christofides heuristic is shown to be no more than $\lceil{n}/{2}\rceil !$ , and for the Double Tree heuristic and a variation of the Christofides heuristic the domination numbers are shown to be one (even if the edge costs satisfy the triangle inequality). Further, unless P = NP, no polynomial time approximation algorithm exists for the TSP on the complete digraph $\vec{K}_n$ with domination number at least (n-1)!-k for any constant k or with domination number at least (n-1)! - (( k /(k+1))(n+r))!-1 for any non-negative constants r and k such that (n+r) $\equiv$ 0 mod (k+1). The complexities of finding the median value of costs of all the tours in $\vec{K}_n$ and of similar problems are also studied.     

Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem

Let G=(V,E) be a complete undirected graph with vertex set V , edge set E , and edge weights l(e) satisfying triangle inequality. The vertex set V is partitioned into clusters V ₁ , . . ., V _k . The clustered traveling salesman problem is to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper we consider several variants of this basic problem and provide polynomial time approximation algorithms for them. Received February 13, 1998; revised July 8, 1998.     

Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of and a randomized approximation algorithm that achieves a ratio of . In particular, we obtain a 2+ε approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2). A preliminary version of this work has been presented at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006) (Lecture Notes in Computer Science, vol. 4368, pp. 302–315, 2007). B. Manthey is supported by the Postdoc-Program of the German Academic Exchange Service (DAAD). He is on leave from Saarland University and has done part of the work at the Institute for Theoretical Computer Science of the University of Lübeck supported by DFG research grant RE 672/3 and at the Department of Computer Science at Saarland University.     

Improved Algorithms and Complexity Results for Power Domination in Graphs

The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G=(V,E), find a minimum-size set P?V such that all vertices in V are “observed” by the vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” For treewidth being upper-bounded by a constant, we achieve a linear-time algorithm. In particular, we present a simplified linear-time algorithm for Power Dominating Set in trees. Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P| is W[2]-hard and it cannot be better approximated than Dominating Set.     

On the Complexity of Approximating Euclidean Traveling Salesman Tours and Minimum Spanning Trees

We consider the problems of computing r-approximate traveling salesman tours and r-approximate minimum spanning trees for a set of n points in , where d≥ 1 is a constant. In the algebraic computation tree model, the complexities of both these problems are shown to be , for all n and r such that r < n and r is larger than some constant. In the more powerful model of computation that additionally uses the floor function and random access, both problems can be solved in O(n) time if . Received January 8, 1996; revised July 15, 1996.     

一种求解TSP问题的多种群并行遗传算法

遗传算法是一种基于自然群体遗传机制的有效搜索算法,由于它在搜索空间中同时考虑许多点.减少了收敛于局部极值的可能,也增加了处理的并行性.因此可以利用并行遗传算法研究典型的TSP问题的求解.提出一种有效的多种群并行算法求解旅行商(TSP)问题,应用多种群遗传并行进化的思想,并在种群之间进行遗传信息交流,以解决经典遗传的收敛到局部最优值问题.仿真实验结果表明,方法在解的精度上以及解的质量上优于经典的遗传算法.     

Particle swarm optimization-based algorithms for TSP and generalized TSP

A novel particle swarm optimization (PSO)-based algorithm for the traveling salesman problem (TSP) is presented. An uncertain searching strategy and a crossover eliminated technique are used to accelerate the convergence speed. Compared with the existing algorithms for solving TSP using swarm intelligence, it has been shown that the size of the solved problems could be increased by using the proposed algorithm.Another PSO-based algorithm is proposed and applied to solve the generalized traveling salesman problem by employing the generalized chromosome. Two local search techniques are used to speed up the convergence. Numerical results show the effectiveness of the proposed algorithms.     

旅行商问题(TSP)的一种改进遗传算法

传统的序号编码遗传算法(GA)使用PMX、CX和OX等特殊的交叉算子，这些算子实施起来很麻烦。针对TSP问题的求解，提出了一种新的改进遗传算法：单亲进化遗传算法(PEGA)，PEGA是利用父体所提供的有效边的信息，使用保留最小边的方法进行个体的进化。与传统的遗传算法相比，PEGA算法弥补了它们的不足之处，简化了遗传算法。给出了PEGA算法的数值算例，仿真实验表明了该算法对于对称的TSP和非对称的TSP问题，都具有收敛速度快的特点，证明了该算法的有效性。     

Approximating the Maximum Agreement Forest on k trees

The Maximum Agreement Forest problem (MAF) asks for the largest common subforest of a set of binary trees. This problem is known to be MAXSNP-complete for instances consisting of 2 trees. We show that it remains MAXSNP-complete for k?2 trees.     

On the hardness of approximating Max-Satisfy

Max-Satisfy is the problem of finding an assignment that satisfies the maximum number of equations in a system of linear equations over Q. We prove that unless NP⊂BPP Max-Satisfy cannot be efficiently approximated within an approximation ratio of 1/n1−?, if we consider systems of n linear equations with at most n variables and ?>0 is an arbitrarily small constant. Previously, it was known that the problem is NP-hard to approximate within a ratio of 1/nα, but 0<α<1 was some specific constant that could not be taken to be arbitrarily close to 1.     

Approximating Maximum Weight Cycle Covers in Directed Graphs with Weights Zero and One

A cycle cover of a graph is a spanning subgraph, each node of which is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. Given a complete directed graph with edge weights zero and one, Max-k-DDC(0,1) is the problem of finding a k-cycle cover with maximum weight. We present a 2/3 approximation algorithm for Max-k-DDC(0,1) with running time O(n ^5/2). This algorithm yields a 4/3 approximation algorithm for Max-k-DDC(1,2) as well. Instances of the latter problem are complete directed graphs with edge weights one and two. The goal is to find a k-cycle cover with minimum weight. We particularly obtain a 2/3 approximation algorithm for the asymmetric maximum traveling salesman problem with distances zero and one and a 4/3 approximation algorithm for the asymmetric minimum traveling salesman problem with distances one and two. As a lower bound, we prove that Max-k-DDC(0,1) for k 3 and Max-k-UCC(0,1) (finding maximum weight cycle covers in undirected graphs) for k 7 are \APX-complete.     

TSP问题分层求解算法的复杂度研究

１ＴＳＰ问题及其区域划分求解算法ＴＳＰ（ｔｒａｖｅｌｉｎｇｓａｌｅｓｍａｎｐｒｏｂｌｅｍ）问题已被证明是ＮＰ问题,用现有的优化算法,如分支定界、动态规划等求最优解,需要问题规模的指数阶时间［１,２］．在问题规模增大时,往往由于计算时间的限制而丧失可行...     

Finding optimal paths in MREP routing

Maximum Residual Energy Path (MREP) routing has been shown an effective routing scheme for energy conservation in battery powered wireless networks. Past studies on MREP routing are based on the assumption that the transmitting node consumes power, but the receiving node does not. This assumption is false if acknowledgment is required as occurs, for example, in some Bluetooth applications.If the receiving node does not consume power then the MREP routing problem for a single message is easily solvable in polynomial time using a simple Dijkstra-like algorithm. We further show in that when the receiving node does consume power the problem becomes NP-complete and is even impossible to approximate with an exponential approximation factor in polynomial time unless P=NP.     

基于遗传算法求解TSP问题的一种算法

主要研究了用遗传算法求解TSP问题。阐述了简单遗传算法的设计方法、基本原理和基本步骤。描述了简单遗传算法在TSP问题中的应用现状。根据种群个体的多样性和分布情况,提出了判定遗传算法的截止代数。简单遗传算法具有易于陷入局部最优解、收敛速度慢的特点,针对这些特点,通过改进交叉算子,加入初始化启发信息,提高了遗传算法解的精度和收敛性。     

引入基因簇求解TSP的遗传算法

在用遗传算法求解TSP时,极易破坏已经发现的较短线路片段,从而使遗传算法的收敛变慢.为了保护较短的线路片段,遗传操作以基因和基因簇为单位进行,优良基因簇可完整地遗传到下一代.在获得第一个近似最优解后,粉碎已发现的基因簇并继续寻优,以期能够获得全局最优解.使用CHN144及TSPLIB中的数据进行试验,找到了CHN144问题的当前最优路径.通过对TSP225的实验获得了最短路径3859,优于目前已经公布的最短路径3916.实验表明,基于基因簇的算法具备3000个城市左右的寻优能力.     

一种基于免疫原理求解TSP问题的模型

基于人工免疫原理,建立了一个基于免疫机制求解TSP问题的数学模型。在该模型中,定义了TSP问题中的抗原和抗体,描述了记忆细胞动态进化过程,并借鉴遗传算法中基因变异思想,提出了优势基因进化的GFE算法,结合生物免疫系统抗体浓度稳定原理,在克隆选择过程中实现了抗体集合的进化计算,快速有效地求解出问题的全局近似最优解。实验结果表明该算法对解决组合优化问题不仅可行,而且有较快的收敛速度和较强的全局搜索能力。