Special Issue on Memetic Algorithms

The ten papers in this special section are devoted to memetic algorithms. The papers are loosely grouped into two categories: memetic algorithm methodologies and domain-specific memetic algorithms. Briefly summarizes the articles included in this section.     

Approximation Algorithms for Treewidth

This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in polynomial-time and finds a factor-O(log OPT) approximation, where OPT is the treewidth of the graph. This is the first polynomial-time algorithm that approximates the optimal by a factor that does not depend on n, the number of nodes in the input graph. As a result, we get an algorithm for finding pathwidth within a factor of O(log OPT⋅log n) from the optimal. We also present algorithms that approximate the treewidth of a graph by constant factors of 3.66, 4, and 4.5, respectively and take time that is exponential in the treewidth. These are more efficient than previously known algorithms by an exponential factor, and are of practical interest. Finding triangulations of minimum treewidth for graphs is central to many problems in computer science. Real-world problems in artificial intelligence, VLSI design and databases are efficiently solvable if we have an efficient approximation algorithm for them. Many of those applications rely on weighted graphs. We extend our results to weighted graphs and weighted treewidth, showing similar approximation results for this more general notion. We report on experimental results confirming the effectiveness of our algorithms for large graphs associated with real-world problems.     

Exact and Approximation Algorithms for Clustering

In this paper we present an n^ O(k ^1-1/d ) -time algorithm for solving the k -center problem in \reals ^d , under L _∈ fty - and L ₂ -metrics. The algorithm extends to other metrics, and to the discrete k -center problem. We also describe a simple (1+ɛ) -approximation algorithm for the k -center problem, with running time O(nlog k) + (k/ɛ)^ O(k ^1-1/d ) . Finally, we present an n^ O(k ^1-1/d ) -time algorithm for solving the L -capacitated k -center problem, provided that L=Ω(n/k ^1-1/d ) or L=O(1) . Received July 25, 2000; revised April 6, 2001.     

Approximation Algorithms for Aligning Points

We study the problem of aligning as many points as possible horizontally, vertically, or diagonally, when each point is allowed to be placed anywhere in its own given region. Different shapes of placement regions and different sets of alignment orientations are considered. More generally, we assume that a graph is given on the points, and only the alignments of points that are connected in the graph count. We show that for planar graphs the problem is NP-hard, and we provide an inapproximability result for general graphs. For the case of trees and planar graphs, we give approximation algorithms whose performance depends on the shape of the given regions and the set of orientations. When the orientations are the ones given by the axes and the regions are axis-parallel rectangles, we obtain a polynomial time approximation scheme.     

Approximation Algorithms for Intersection Graphs

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.     

Approximation Algorithms for Time-Constrained Scheduling on Line Networks

We consider the problem of time-constrained scheduling of packets in a communication network. Each packet has, in addition to its source and its destination, a release time and a deadline. The goal of an algorithm is to maximize the number of packets that arrive to their destinations by their respective deadlines, given the network constraints.     

关于Steiner网络设计问题的近似算法综述

随着因特网中应用的爆炸性增长与网络通讯技术的发展,无论在国防、财政和电源产业等传统领域,还是在新兴的可信计算和网络、云计算系统和下一代互联网等领域,网络的可靠性都得到越来越多的重视.如何在最小化占用网络资源的同时,通过网络的拓扑结构提高网络的可靠性,吸引了广大研究者的兴趣.著名的最小Steiner网络问题就是这个课题中最为引人关注的问题之一.在过去的十年里,作为可靠网络领域的基础问题之一,Steiner网络设计问题得到很好的研究.我们总结了关于Steiner网络设计问题当前最好的近似算法的近似比与时间复杂度,并简明的概述了这些算法的主要思想.     

Approximation Algorithms for a k-Line Center

Given a set P of n points in ℝd and an integer k ≥ 1, let w^* denote the minimum value so that P can be covered by k congruent cylinders of radius w^*. We describe a randomized algorithm that, given P and an ε > 0, computes k cylinders of radius (1 + ε) w^* that cover P. The expected running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and ε. We first show that there exists a small ”certificate” Q ⫅ P, whose size does not depend on n, such that for any k congruent cylinders that cover Q, an expansion of these cylinders by a factor of (1 + ε) covers P. We then use a well-known scheme based on sampling and iterated re-weighting for computing the cylinders.     

Approximation Algorithms for Connected Dominating Sets

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (c _n +1) \ln n where c _n ln k is the approximation factor for the node weighted Steiner tree problem (currently c _n = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ). Received June 22, 1996; revised February 28, 1997.     

Approximation Algorithms for Minimum K -Cut

Let G=(V,E) be a complete undirected graph, with node set V={v ₁ , . . ., v _n } and edge set E . The edges (v _i ,v _j ) ∈ E have nonnegative weights that satisfy the triangle inequality. Given a set of integers K = { k _i } _i=1 ^p , the minimum K-cut problem is to compute disjoint subsets with sizes { k _i } _i=1 ^p , minimizing the total weight of edges whose two ends are in different subsets. We demonstrate that for any fixed p it is possible to obtain in polynomial time an approximation of at most three times the optimal value. We also prove bounds on the ratio between the weights of maximum and minimum cuts. Received September 4, 1997; revised July 15, 1998.