TSP Heuristics: Domination Analysis and Complexity

   Abstract. We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph K _n produce a solution no worse than the average cost of a tour in K _n 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

关于TSP的计算复杂性

本文研究TSP的计算复杂性,指出了判定型TSP与最优型TSP具有不同的计算复杂性,同时指出了文献[1]-[3]中的一些错误。     

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.     

TSP问题的脂肪计算复杂性与启发式算法设计

旅行商问题(traveling salesman problem,简称TSP)是经典的NP-难解组合优化问题之一,求解它的高效启发式算法一直是计算机科学研究的热点.脂肪作为描述TSP结构特征的新工具,对启发式算法设计具有重要意义.目前,TSP问题的脂肪研究还处于初始阶段,缺乏理论分析结果及相关的启发式算法.首先分析了TSP问题的脂肪计算复杂性,通过构造偏移实例的技巧,证明了获取TSP的脂肪是NP-难解的,即在P(NP的假设下,不存在算法可以在多项式时间内获得完整脂肪.在此基础上,通过分析TSP问题局部最优解与脂肪的关系,给出了求解TSP问题的元启发式算法--动态候选集搜索(dynamic candidate set search,简称DCSS)算法.利用该算法,改进了目前求解TSP问题最好算法之一的LKH.TSPLIB典型实例的实验结果表明,改进后的算法在解的质量上有较为明显的提高.新的基于脂肪的启发式算法对于求解其他NP-难解问题具有一定的参考价值.     

The Parameterized Complexity of Local Search for TSP, More Refined

We extend previous work on the parameterized complexity of local search for the Traveling Salesperson Problem (TSP). So far, its parameterized complexity has been investigated with respect to the distance measures (defining the local search area) “Edge Exchange” and “Max-Shift”. We perform studies with respect to the distance measures “Swap” and “r-Swap”, “Reversal” and “r-Reversal”, and “Edit”, achieving both fixed-parameter tractability and W[1]-hardness results. In particular, from the parameterized reduction showing W[1]-hardness we infer running time lower bounds (based on the exponential time hypothesis) for all corresponding distance measures. Moreover, we provide non-existence results for polynomial-size problem kernels and we show that some in general W[1]-hard problems turn fixed-parameter tractable when restricted to planar graphs.     

A型变尺寸装箱问题之模型及算法研究

作为对经典一维装箱问题的推广,提出一种A型变尺寸装箱问题（A-shaped Variable-sized BinPacking Problem,简称A SVBP）,即在物品的装箱过程中,每样物品有高度和横截面积两个参数,并且箱子的大小不一。该问题在文件系统管理和日常生活中的运输等问题中有着广泛的应用背景。把装箱问题的经典算法以及遗传算法推广到A型变尺寸装箱问题,实验结果表明：按照本文提出的求解模式,离线情况下求解A型变尺寸装箱问题最终结果的质量取决于预先求解其退化为经典装箱问题时的算法,求解物品装箱序列时用首次适应混合遗传算法比用Next Fit算法、First Fit算法、Best Fit算法最终得到的结果要好。     

Analysis of Heuristics for Number Partitioning

We illustrate the use of phase transition behavior in the study of heuristics. Using an "annealed" theory, we define a parameter that measures the "constrainedness" of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show that constrainedness can be used to analyze and compare algorithms and heuristics for number partitioning in a precise and quantitative manner. For example, we demonstrate that on uniform random problems both the Karmarkar–Karp and greedy heuristics minimize the constrainedness, but that the decisions made by the Karmarkar–Karp heuristic are superior at reducing constrainedness. This supports the better performance observed experimentally for the Karmarkar–Karp heuristic. Our results refute a conjecture of Fu that phase transition behavior does not occur in number partitioning. Additionally, they demonstrate that phase transition behavior is useful for more than just simple benchmarking. It can, for instance, be used to analyze heuristics, and to compare the quality of heuristic solutions.     

The Kernelization Complexity of Connected Domination in Graphs with (no) Small Cycles

In the Parameterized Connected Dominating Set problem the input consists of a graph G and a positive integer k, and the question is whether there is a set S of at most k vertices in G—a connected dominating set of G—such that (i) S is a dominating set of G, and (ii) the subgraph G[S] induced by S is connected; the parameter is k. The underlying decision problem is a basic connectivity problem which is long known to be NP-complete, and it has been extensively studied using several algorithmic approaches. Parameterized Connected Dominating Set is W[2]-hard, and therefore it is unlikely (Downey and Fellows, Parameterized Complexity, Springer, 1999) that the problem has fixed-parameter tractable (FPT) algorithms or polynomial kernels in graphs in general. We investigate the effect of excluding short cycles, as subgraphs, on the kernelization complexity of Parameterized Connected Dominating Set. The girth of a graph G is the length of a shortest cycle in G. It turns out that the Parameterized Connected Dominating Set problem is hard on graphs with small cycles, and becomes progressively easier as the girth increases. More precisely, we obtain the following kernelization landscape: Parameterized Connected Dominating Set

• does not have a kernel of any size on graphs of girth three or four (since the problem is W[2]-hard);
• admits a kernel of size 2 ^k k ^3k on graphs of girth at least five;
• has no polynomial kernel (unless the Polynomial Hierarchy collapses to the third level) on graphs of girth at most six, and,
• has a cubic ( $\mathcal {O}(k^{3})$ ) vertex kernel on graphs of girth at least seven.

While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.     

Heuristics and pedagogy

In its focus on heuristics as opposed to hierarchically structured general principles, expert systems technology suggests a pedagogic strategy with affinities to the approaches of some of the creative philosophers of East and West, and a challenge to the reliance on presentation of general principles found in academic tradition. A tutoring approach to classroom presentation may be seen to relate to the point that non-trivial general principles cannot be verbally expressed without substantial loss of meaning.     

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.     

System Project Failure: The Heuristics of Risk

Abstract

System project failures are a well-known part of systems development; however, all the potential risks of planning and executing a project effort are not This article offers heuristic guidelines to help IS managers assess these inherent risk factors before initiating and while executing a project. Case examples illustrate how a particular risk factor can result in a failed systems development effort.     

蚁群算法的收敛性分析及其在TSP上的求解

根据基本蚁群算法的特点对其收敛性进行分析,给出寻找最短路径的蚁群算法收敛的充分条件.并把算法运用到旅行商问题上,试验结果表明该算法在求解TSP问题上解的精度优于组合优化算法以及遗传算法且收敛速度比较快.     

PASS Approximation: A Framework for Analyzing and Designing Heuristics

We introduce a new framework for designing and analyzing algorithms. Our framework applies best to problems that are inapproximable according to the standard worst-case analysis. We circumvent such negative results by designing guarantees for classes of instances, parameterized according to properties of the optimal solution. Given our parameterized approximation, called PArametrized by the Signature of the Solution (PASS) approximation, we design algorithms with optimal approximation ratios for problems with additive and submodular objective functions such as the capacitated maximum facility location problems. We consider two types of algorithms for these problems. For greedy algorithms, our framework provides a justification for preferring a certain natural greedy rule over some alternative greedy rules that have been used in similar contexts. For LP-based algorithms, we show that the natural LP relaxation for these problems is not optimal in our framework. We design a new LP relaxation and show that this LP relaxation coupled with a new randomized rounding technique is optimal in our framework. In passing, we note that our results strictly improve over previous results of Kleinberg et al. (J. ACM 51(2):263–280, 2004) concerning the approximation ratio of the greedy algorithm.     

Guest Editors' Introduction: Advanced Heuristics in Transportation and Logistics

Transportation and logistics organizations often face large-scale combinatorial problems on both operational and strategic levels. By exploiting problem-specific characteristics, classical heuristic methods--such as constructive and iterative local search methods--aim at a relatively limited exploration of the search space, thereby producing acceptable-quality solutions in modest computing times. In a major departure from a classical heuristic, a metaheuristic method exploits not only the problem characteristics but also ideas based on artificial intelligence methodologies, such as different types of memory structures and learning mechanisms, as well as analogies with optimization methods found in nature. Solutions produced by metaheuristics typically are of a much higher quality than those obtained with classical heuristic approaches.This article is part of a special issue on advanced heuristics in transportation and logistics.     

System Project Failure: The Heuristics of Risk

System project failures are a well-known part of systems development; however, all the potential risks of planning and executing a project effort are not This article offers heuristic guidelines to help IS managers assess these inherent risk factors before initiating and while executing a project. Case examples illustrate how a particular risk factor can result in a failed systems development effort.