Path optimization for graph partitioning problems

This paper presents a new heuristic for graph partitioning called Path Optimization (PO), and the results of an extensive set of empirical comparisons of the new algorithm with two very well-known algorithms for partitioning: the Kernighan-Lin algorithm and simulated annealing. Our experiments are described in detail, and the results are presented in such a way as to reveal performance trends based on several variables. Sufficient trials are run to obtain 99% confidence intervals small enough to lead to a statistical ranking of the implementations for various circumstances. The results for geometric graphs, which have become a frequently used benchmark in the evaluation of partitioning algorithms, show that PO holds an advantage over the others.

In addition to the main test suite described above, comparisons of PO to more recent partitioning approaches are also given. We present the results of comparisons of PO with a parallelized implementation of Goemans' and Williamson's 0.878 approximation algorithm, a flow-based heuristic due to Lang and Rao, and the multilevel algorithm of Hendrickson and Leland.     

Between local and global logarithmic averages

We obtain an approximation for the logarithmic averages of I{k^1/2a(k) S(k) k^1/2b(k)}, where a(k) → 0, b(k) → 0 (k → ∞) and S(k) is partial sum of independent, identically distributed random variables.     

A practical map labeling algorithm

The map labeling problem is a classical problem of cartography. There is a theoretically optimal approximation algorithm A. Unfortunately A is useless in practice as it typically produces results that are intolerably far off the optimal size. On the other hand there are heuristics with good practical results.

In this paper we present an algorithm B that (1) guarantees the optimal approximation quality and runtime behaviour of A, and (2) yields results significantly closer to the optimum than the best heuristic known so far.

The sample data used in the experimental evaluation consists of three different classes of random problems and a selection of problems arising in the production of groundwater quality maps by the authorities of the City of Munich.     

Multi-step ahead nonlinear identification of Lorenz’s chaotic system using radial basis neural network with learning by clustering and particle swarm optimization

An important problem in engineering is the identification of nonlinear systems, among them radial basis function neural networks (RBF-NN) using Gaussian activation functions models, which have received particular attention due to their potential to approximate nonlinear behavior. Several design methods have been proposed for choosing the centers and spread of Gaussian functions and training the RBF-NN. The selection of RBF-NN parameters such as centers, spreads, and weights can be understood as a system identification problem. This paper presents a hybrid training approach based on clustering methods (k-means and c-means) to tune the centers of Gaussian functions used in the hidden layer of RBF-NNs. This design also uses particle swarm optimization (PSO) for centers (local clustering search method) and spread tuning, and the Penrose–Moore pseudoinverse for the adjustment of RBF-NN weight outputs. Simulations involving this RBF-NN design to identify Lorenz’s chaotic system indicate that the performance of the proposed method is superior to that of the conventional RBF-NN trained for k-means and the Penrose–Moore pseudoinverse for multi-step ahead forecasting.     

移动传感器网络中的最大价值路径扫描覆盖算法

扫描覆盖是当前移动传感器网络的一个重要覆盖技术,其主要通过规划移动传感器的巡逻路径对事件兴趣点（Points of Interest,POI）进行定期监测,从而以相对于普通覆盖方案更低廉的成本实现对POI监控．研究最大价值路径扫描覆盖,即使用移动传感器扫描覆盖分布在一条路径上的POI集合,使得被覆盖POI的价值总和达到最大．首先设计了一个基于线性规划随机取整的近似算法,通过将问题松弛并刻画为一个线性规划,然后对线性规划最优解取整得到一个扫描覆盖方案．该算法可在O（mn^3.5L）时间内求解,并具有可证明的近似比1-1/e．其次,通过扩展基于贪心策略的集合覆盖算法,设计了一个时间复杂度为O（m²n²）的贪心算法,其主要思想为循环选取一个单位巡逻范围覆盖POI价值最大的传感器．为优化运行时间,基于MVSCP问题的特殊结构将算法时间进一步改进至O（m log m+mn²）．最后,通过仿真实验分析所设计算法的实际性能.实验结果表明,线性规划随机取整算法运行时间低至整数规划算法的百分之一,但其所求解的质量只略低于整数规划算法;改进的贪心算法虽然不具有可证明的近似比,但其实际所求解的质量并不弱于线性规划随机取整算法,并且具有三者中最佳的运行时间．     

Bit-parallel approximate string matching algorithms with transposition

Using bit-parallelism has resulted in fast and practical algorithms for approximate string matching under Levenshtein edit distance, which permits a single edit operation to insert, delete or substitute a character. Depending on the parameters of the search, currently the fastest non-filtering algorithms in practice are the O(km/wn) algorithm of Wu and Manber, the O((k+2)(m−k)/wn) algorithm of Baeza-Yates and Navarro, and the O(m/wn) algorithm of Myers, where m is the pattern length, n is the text length, k is the error threshold and w is the computer word size. In this paper we discuss a uniform way of modifying each of these algorithms to permit also a fourth type of edit operation: transposing two adjacent characters in the pattern. This type of edit distance is also known as Damerau edit distance. In the end we also present an experimental comparison of the resulting algorithms.     

ContributionsAn algorithm for the Tutte polynomials of graphs of bounded treewidth

Let k be a fixed, positive integer. We give an algorithm which computes the Tutte polynomial of any graph G of treewidth at most k in time O(n^{2+7 log₂ c}), where c is twice the number of partitions of a set with 3k + 3 elements and n the number of vertices of G.     

基于启发式Q学习的汽车涂装车间作业排序优化

针对汽车涂装车间中的作业优化排序问题,提出一种基于启发式Q学习的优化算法。首先,建立包括满足总装车间生产顺序和最小化喷枪颜色切换次数的多目标整数规划模型。将涂装作业优化排序问题抽象为马尔可夫过程,建立基于启发式Q算法的求解方法。通过具体案例,对比分析了启发式Q学习、Q学习、遗传算法三种方案的优劣。结果表明：在大规模问题域中,启发式Q学习算法具有寻优效率更高、效果更好的优势。本研究为机器学习算法在汽车涂装作业优化排序问题的应用提出了新思路。     

An elementary algorithm for reporting intersections of red/blue curve segments

Let E_r and E_b be two sets of x-monotone and non-intersecting curve segments, E=E_rE_b and |E|=n. We give a new sweep-line algorithm that reports the k intersecting pairs of segments of E. Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is O(nlogn+kloglogn) and it requires O(n) space.     

On some properties of the series ∑k=0 kx and the Stirling numbers of the second kind

We partially characterize the rational numbers x and integers n 0 for which the sum ∑_k=0^∞ kⁿx^k assumes integers. We prove that if ∑_k=0^∞ kⁿx^k is an integer for x = 1 − a/b with a, b> 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain for all integers k, 2 k n, and a 3, provided a is odd or divisible by 4, where v_a(m) denotes the exponent of the highest power of a which divides m, for m and a> 1 integers.

New identities are also derived for the Stirling numbers, e.g., we show that ∑_k=0²ⁿk! S(2n, k) , for all integers n 1.     

Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering

The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is an NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph 1-Laplacian. In this paper, we first give new notations to describe the paths, among critical eigenvectors of the graph 1-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize m₃(G), a special eigenvalue for the graph 1-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant h₃(G), that is, h₃(G) 6 m₃(G). This is a first step for proving that the k-th Cheeger constant is the minimum of the 1-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous k - 1 Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute m₃(G), based on a generalized inverse power method.     

Finding all essential terms of a characteristic maxpolynomial

Let us denote ab=max(a,b) and ab=a+b for and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n²(m+n log n)) algorithm for finding all essential terms of the max-algebraic characteristic polynomial of an n×n matrix over with m finite elements. In the cases when all terms are essential, this algorithm also solves the following problem: Given an n×n matrix A and k{1,…,n}, find a k×k principal submatrix of A whose assignment problem value is maximum.     

Networks with small stretch number

In a previous work, the authors introduced the class of graphs with bounded induced distance of order k (BID(k) for short), to model non-reliable interconnection networks. A network modeled as a graph in BID(k) can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is at most k times the distance in the non-faulty graph. The smallest k such that GBID(k) is called stretch number of G. We show an odd characteristic of the stretch numbers: every rational number greater or equal 2 is a stretch number, but only discrete values are admissible for smaller stretch numbers. Moreover, we give a new characterization of classes BID(2−1/i), i1, based on forbidden induced subgraphs. By using this characterization, we provide a polynomial time recognition algorithm for graphs belonging to these classes, while the general recognition problem is Co-NP-complete.     

Triangular range counting query in 2D and its application in finding k nearest neighbors of a line segment

We present an efficient algorithm for finding k nearest neighbors of a query line segment among a set of points distributed arbitrarily on a two dimensional plane. Along the way to finding this algorithm, we have obtained an improved triangular range searching technique in 2D. Given a set of n points, we preprocess them to create a data structure using O(n²) time and space, such that given a triangular query region Δ, the number of points inside Δ can be reported in O(logn) time. Finally, this triangular range counting technique is used for finding k nearest neighbors of a query straight line segment in O(log²n+k) time.     

On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.     

Circular arrangements and cyclic broadcast scheduling

Motivated by a scheduling problem in multicast environments, we consider the problem of arranging a weighted graph around a circle so as to minimize the total weighted arc length. We describe the first polynomial-time approximation algorithms for this problem, and specifically an O(logn)-approximation algorithm for undirected circular arrangements and an -approximation algorithm for directed circular arrangements. We will also conduct an experimental evaluation of our algorithms and show that a simple heuristic has the best performance in simulations based on busy Web server logs.     

The reduction of graph families closed under contraction

Let be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in if and only if the contraction G/H is in . Examples of such an : graphs with a spanning closed trail; graphs with at least k edge-disjoint spanning trees; and k-edge-connected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in and to study its structure if it is not in . This reduction method generalizes known special cases.     

Reporting curve segment intersections using restricted predicates

We investigate how to report all k intersecting pairs among a collection of n x-monotone curve segments in the plane, using only predicates of the following forms: is an endpoint to the left of another? is an endpoint above a segment? do two segments intersect? By studying the intersection problem in an abstract setting that assumes the availability of certain “detection oracles”, we obtain a near-optimal randomized algorithm that runs in expected time. In the bichromatic case (where segments are colored red or blue with no red/red or blue/blue intersections), we find a better algorithm that runs in O((n+k)log_2+k/nn) worst-case time, by modifying a known segment-tree method. Two questions of Boissonnat and Snoeyink are thus answered to within logarithmic factors.     

On uniform k-partition problems

We study various uniform k-partition problems which consist in partitioning m sets, each of cardinality k, into k sets of cardinality m such that each of these sets contains exactly one element from every original set. The problems differ according to the particular measure of “set uniformity” to be optimized. Most problems are polynomial and corresponding solution algorithms are provided. A few of them are proved to be NP-hard. Examples of applications to scheduling and routing problems are also discussed.     

Spectral partitioning with multiple eigenvectors

The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy is in contrast to that of the widely used spectral bipartitioning (SB) heuristic (which uses only a single eigenvector) and several previous multi-way partitioning heuristics [8, 11, 17, 27, 38] (which use k eigenvectors to construct k-way partitionings). Our result motivates a simple ordering heuristic that is a multiple-eigenvector extension of SB. This heuristic not only significantly outperforms recursive SB, but can also yield excellent multi-way VLSI circuit partitionings as compared to [1, 11]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective partitioning heuristics. The present paper updates and improves a preliminary version of this work [5].