Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

We present a framework for an automated generation of exact search tree algorithms for NP-hard problems. The purpose of our approach is twofold—rapid development and improved upper bounds. Many search tree algorithms for various problems in the literature are based on complicated case distinctions. Our approach may lead to a much simpler process of developing and analyzing these algorithms. Moreover, using the sheer computing power of machines it may also lead to improved upper bounds on search tree sizes (i.e., faster exact solving algorithms) in comparison with previously developed hand-made search trees. Among others, such an example is given with the NP-complete Cluster Editing problem (also known as Correlation Clustering on complete unweighted graphs), which asks for the minimum number of edge additions and deletions to create a graph which is a disjoint union of cliques. The hand-made search tree for Cluster Editing had worst-case size O(2.27^k), which now is improved to O(1.92^k) due to our new method. (Herein, k denotes the number of edge modifications allowed.)     

(Incremental) Priority Algorithms

We study the question of which optimization problems can be optimally or approximately solved by greedy or greedy-like algorithms. For definiteness, we limit the present discussion to some well-studied scheduling problems although the underlying issues apply in a much more general setting. Of course, the main benefit of greedy algorithms lies in both their conceptual simplicity and their computational efficiency. Based on the experience from online competitive analysis, it seems plausible that we should be able to derive approximation bounds for greedy-like algorithms exploiting only the conceptual simplicity of these algorithms. To this end, we need (and will provide) a precise definition of what we mean by greedy and greedy-like.     

Quantum Algorithms for Matching Problems

We present quantum algorithms for the following matching problems in unweighted and weighted graphs with n vertices and m edges:

Polynomial Time Algorithms for 2-Edge-Connectivity Augmentation Problems

Abstract. Given a 2-edge-connected, real weighted graph G with n vertices and m edges, the 2-edge-connectivity augmentation problem is that of finding a minimum weight set of edges of G to be added to a spanning subgraph H of G to make it 2-edge-connected. While the general problem is NP-hard and 2 -approximable, in this paper we prove that it becomes polynomial time solvable if H is a depth-first search tree of G . More precisely, we provide an efficient algorithm for solving this special case which runs in O (M · α(M,n)) time, where α is the classic inverse of Ackermann's function and M=m · α(m,n) . This algorithm has two main consequences: first, it provides a faster 2 -approximation algorithm for the general 2 -edge-connectivity augmentation problem; second, it solves in O (m · α(m,n)) time the problem of restoring, by means of a minimum weight set of replacement edges, the 2 -edge-connectivity of a 2-edge-connected communication network undergoing a link failure.     

基于固定优先级和贪心法的航班时隙分配算法

航班时隙分配在空中交通管理领域中有着重要应用,考虑到在相同的延误时间情况下,不同类型的航班和不同的载客人数造成的综合损失差异,提出一种基于贪心法的航班分配算法。该算法在对航班进行排序的时候,在考虑到航空公司公平性的基础上,根据航班类型和载客数量,计算每架航班的优先级,然后根据当前可用时隙,以贪心法的规则找出优先级最高的航班,若有多个航班满足条件,则根据先来先服务原则进行选择,从而使经济损失和人员延误损失二者构成的综合损失最小化。算法仿真结果显示：该算法在很大程度上改进机场的运营效率,确保航空公司航班分配的公平性,维护航空公司及其服务对象的利益,具有一定的实用性和有效性。     

Efficient Parallel Graph Algorithms for Coarse-Grained Multicomputers and BSP

In this paper we present deterministic parallel algorithms for the coarse-grained multicomputer (CGM) and bulk synchronous parallel (BSP) models for solving the following well-known graph problems: (1) list ranking, (2) Euler tour construction in a tree, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, and (7) 2-edge connectivity and biconnectivity (testing and component computation). The algorithms require O(log p) communication rounds with linear sequential work per round (p = no. processors, N = total input size). Each processor creates, during the entire algorithm, messages of total size O(log (p) (N/p)) . The algorithms assume that the local memory per processor (i.e., N/p ) is larger than p ^ε , for some fixed ε > 0 . Our results imply BSP algorithms with O(log p) supersteps, O(g log (p) (N/p)) communication time, and O(log (p) (N/p)) local computation time. It is important to observe that the number of communication rounds/ supersteps obtained in this paper is independent of the problem size, and grows only logarithmically with respect to p . With growing problem size, only the sizes of the messages grow but the total number of messages remains unchanged. Due to the considerable protocol overhead associated with each message transmission, this is an important property. The result for Problem (1) is a considerable improvement over those previously reported. The algorithms for Problems (2)—(7) are the first practically relevant parallel algorithms for these standard graph problems. Received July 5, 2000; revised April 16, 2001.     

多目标进化算法测试问题的设计与分析

为了有效检测多目标优化进化算法的性能,从3个方面进行多目标优化测试问题的设计,即约束条件、最优解分布的均匀性、算法逼近Pareto最优前沿的难度,采用NSGA-Ⅱ算法对这些测试问题进行仿真实验,并将算法求得的最优解可视化。结果显示,测试问题能够有效检测算法在上述3方面的性能。     

On Certificates and Lookahead in Dynamic Graph Problems

Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closure, as well as some undirected graph problems such as maximum matchings and cuts. We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate complexity for the problems which have efficient dynamic algorithms. In many applications of dynamic (di)graph problems, a certain form of lookahead is available. Specifically, we consider the problems of assembly planning in robotics and the maintenance of relations in databases. These give rise to dynamic strong connectivity and dynamic transitive closure problems, respectively. We explain why it is reasonable, and indeed natural and desirable, to assume that lookahead is available in these two applications. Exploiting lookahead to circumvent their inherent complexity, we obtain efficient dynamic algorithms for strong connectivity and transitive closure. Received August 11, 1995; revised August 23, 1996.     

求解多维背包问题的贪心粒子群算法

将多维背包问题的贪心变换和两种求解算法,用于求解具有重量和体积两个约束的背包问题,分别将物品按价值/重量、价值/体积比的凸组合和无穷范数的定义获得两组混合"性价比"权值向量,再以该混合"性价比"权值为依据构造两种贪心粒子群算法(wPSO,infPSO)。数值试验表明,算法wPSO、infPSO不仅大大优于现有粒子群算法,而且表现出优秀、稳定的搜索能力和快速定位最优解的搜索能力。     

Greedy Algorithms for the Shortest Common Superstring That Are Asymptotically Optimal

There has recently been a resurgence of interest in the shortest common superstring problem due to its important applications in molecular biology (e.g., recombination of DNA) and data compression. The problem is NP-hard, but it has been known for some time that greedy algorithms work well for this problem. More precisely, it was proved in a recent sequence of papers that in the worst case a greedy algorithm produces a superstring that is at most β times (2≤β≤ 4 ) worse than optimal. We analyze the problem in a probabilistic framework, and consider the optimal total overlap O _n ^opt and the overlap O _n ^rg produced by various greedy algorithms. These turn out to be asymptotically equivalent. We show that with high probability , where n is the number of original strings and H is the entropy of the underlying alphabet. Our results hold under a condition that the lengths of all strings are not too short. Received November 1996; revised March 1997.     

贪心算法求解k-median问题

文章讨论了用贪心算法解k-m edian问题以及其试验结果。首先提出了一个解k-m edian问题的简单贪心算法,然后对求解质量和求解的近似性能比进行了探讨。主要讨论了公制空间和非公制空间初始解的产生,用贪心算法解k-m edian问题以及全局最优解的计算。试验结果表明:贪心算法解公制空间的k-m edian问题效果要好于解非公制空间的k-m edian问题;用贪心算法解公制空间和非公制空间k-m edian问题都能得到较好的结果。     

BubbleSearch: A simple heuristic for improving priority-based greedy algorithms

We introduce BubbleSearch, a general approach for extending priority-based greedy heuristics. Following the framework recently developed by Borodin et al., we consider priority algorithms, which sequentially assign values to elements in some fixed or adaptively determined order. BubbleSearch extends priority algorithms by selectively considering additional orders near an initial good ordering. While many notions of nearness are possible, we explore algorithms based on the Kendall-tau distance (also known as the BubbleSort distance) between permutations. Our contribution is to elucidate the BubbleSearch paradigm and experimentally demonstrate its effectiveness.     

A Functional Approach to External Graph Algorithms

Abstract. We present a new approach for designing external graph algorithms and use it to design simple, deterministic and randomized external algorithms for computing connected components, minimum spanning forests, bottleneck minimum spanning forests, maximal independent sets (randomized only), and maximal matchings in undirected graphs. Our I/ O bounds compete with those of previous approaches. We also introduce a semi-external model, in which the vertex set but not the edge set of a graph fits in main memory. In this model we give an improved connected components algorithm, using new results for external grouping and sorting with duplicates. Unlike previous approaches, ours is purely functional—without side effects—and is thus amenable to standard checkpointing and programming language optimization techniques. This is an important practical consideration for applications that may take hours to run.     

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。     

When Ants Attack: Ant Algorithms for Constraint Satisfaction Problems

We describe an ant algorithm for solving constraint problems (Solnon 2002, IEEE Transactions on Evolutionary Computation 6(4): 347–357). We devise a number of variants and carry out experiments. Our preliminary results suggest that the best way to deposit pheromone and the best heuristics for state transitions may differ from current practice     

Approximation Algorithms for Degree-Constrained Minimum-Cost Network-Design Problems

We study network-design problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degree-constrained node-weighted Steiner tree problem: We are given an undirected graph G(V,E) , with a nonnegative integral function d that specifies an upper bound d(v) on the degree of each vertex v ∈ V in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of k nodes called terminals. The goal is to construct a Steiner tree T containing all the terminals such that the degree of any node v in T is at most the specified upper bound d(v) and the total cost of the nodes in T is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria—the degree of any node v ∈ V in the output Steiner tree is O(d(v) log k) and the cost of the tree is O(log k) times that of a minimum-cost Steiner tree that obeys the degree bound d(v) for each node v . Our result extends to the more general problem of constructing one-connected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees. Received December 21, 1998; revised September 24, 1999.     

The Power of Priority Algorithms for Facility Location and Set Cover

We apply and extend the priority algorithm framework introduced by Borodin, Nielsen, and Rackoff to define greedy-like algorithms for the (uncapacitated) facility location problems and set cover problems. These problems have been the focus of extensive research from the point of view of approximation algorithms and for both problems greedy-like algorithms have been proposed and analyzed. The priority algorithm definitions are general enough to capture a broad class of algorithms that can be characterized as greedy-like while still possible to derive non-trivial lower bounds on the approximability of the problems by algorithms in such a class. Our results are orthogonal to complexity considerations, and hence apply to algorithms that are not necessarily polynomial time.     

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.     

Greedy algorithms for generalized k-rankings of paths

A k-ranking of a graph is a labeling of the vertices with positive integers 1,2,…,k so that every path connecting two vertices with the same label contains a vertex of larger label. An optimal ranking is one in which k is minimized. Let Pn be a path with n vertices. A greedy algorithm can be used to successively label each vertex with the smallest possible label that preserves the ranking property. We seek to show that when a greedy algorithm is used to label the vertices successively from left to right, we obtain an optimal ranking. A greedy algorithm of this type was given by Bodlaender et al. in 1998 [1] which generates an optimal k-ranking of Pn. In this paper we investigate two generalizations of rankings. We first consider bounded (k,s)-rankings in which the number of times a label can be used is bounded by a predetermined integer s. We then consider kt-rankings where any path connecting two vertices with the same label contains t vertices with larger labels. We show for both generalizations that when G is a path, the analogous greedy algorithms generate optimal k-rankings. We then proceed to quantify the minimum number of labels that can be used in these rankings. We define the bounded rank number to be the smallest number of labels that can be used in a (k,s)-ranking and show for n?2, where i=⌊log₂(s)⌋+1. We define the kt-rank number, to be the smallest number of labels that can be used in a kt-ranking. We present a recursive formula that gives the kt-rank numbers for paths, showing for all an−1<j?an where {an} is defined as follows: a₁=1 and an=⌊((t+1)/t)an−1⌋+1.     

Efficient Parallel Algorithms for Some Graph Theory Problems

In this paper,a sequential algorithm computing the aww vertex pair distance matrix D and the path matrix Pis given.On a PRAM EREW model with p,1≤p≤n^2,processors,a parallel version of the sequential algorithm is shown.This method can also be used to get a parallel algorithm to compute transitive closure array A^* of an undirected graph.The time complexity of the parallel algorithm is O(n^3/p).If D,P and A^* are known,it is shown that the problems to find all connected components,to compute the diameter of an undirected graph,to determine the center of a directed graph and to search for a directed cycle with the minimum(maximum)length in a directed graph can all be solved in O(n^2/p logp)time.