New Approximation Algorithms for Minimum Cycle Bases of Graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over \mathbbF₂\mathbb{F}_{2} generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G.     

Minimum Cycle Bases for Network Graphs

The minimum cycle basis problem in a graph G = (V,E) is the task to construct a minimum length basis of its cycle vector space. A well-known algorithm by Horton of 1987 needs running time O(|V||E|^2.376). We present a new combinatorial approach which generates minimum cycle bases in time O(\max{|E|³,|E||V|²log |V|}) with a space requirement of (|E|²). This method is especially suitable for large sparse graphs of electric engineering applications since there, typically, |E| is close to linear in |V|.     

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X|=2 . We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log ² |X|) . The second algorithm achieves an approximation factor of O(min{log τ ^* log log τ ^* , log n log log n)} , where τ ^* is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution. Received May 31, 1995; revised June 11, 1996, and October 9, 1996.     

大图数据上顶点驱动的并行最小生成树算法

最小生成树(minimum spanning tree, MST)是图论中最为经典算法之一.基于MST结构的聚类、分类和最短路径查询等复杂图算法,在效率和结果质量方面均有显著提高.然而,随着互联网的迅猛发展,图数据规模也变得越来越大,包含千万甚至上亿个顶点的大图数据越发常见.因此,如何在大图数据上实现查询处理和数据挖掘算法已成为亟待解决的问题之一.除此之外,由于大图数据的动态性特征,如何动态地维护算法结果也势必成为最受关注的问题之一.针对目前集中式的最小生成树算法无法解决海量和动态图数据的问题,首先提出了分区Prim(partition Prim, PP)算法,基于此提出了顶点驱动的并行MST算法——PB(PP Boruvka)算法,并论证了PB算法的正确性.另外,基于MapReduce和BSP框架实现了PB算法.针对只删除动态图特征,提出了MST维护算法,以实现高效的增量计算.对提出的计算和维护算法进行了代价分析和比较.最后,使用真实和模拟数据集,验证了PB算法和维护算法的有效性、高效性和可扩展性.     

Irredundant Algorithms for Traversing Directed Graphs: The Deterministic Case

Problems of testing program systems modeled by deterministic finite automata are considered. The necessary (and, sometimes, sufficient) component of such testing is a traversal of the graph of the automaton state transitions. The main attention is given to the so-called irredundant traversal algorithms (algorithms for traversing unknown graphs, or on-line algorithms), which do not require an a priori knowledge of the total graph structure.     

Optimal Sequential And Parallel Algorithms To Compute A Steiner Tree On Permutation Graphs

This paper presents an optimal sequential and an optimal parallel algorithm to compute a minimum cardinality Steiner set and a Steiner tree. The sequential algorithm takes O ( n ) time and parallel algorithm takes O (log n ) time and O ( n /log n ) processors on an EREW PRAM model.     

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.     

Join-Reachability Problems in Directed Graphs

For a given collection \(\mathcal{G}\) of directed graphs we define the join-reachability graph of \(\mathcal{G}\) , denoted by \(\mathcal{J}(\mathcal{G})\) , as the directed graph that, for any pair of vertices u and v, contains a path from u to v if and only if such a path exists in all graphs of  \(\mathcal{G}\) . Our goal is to compute an efficient representation of  \(\mathcal{J}(\mathcal{G})\) . In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for  \(\mathcal{G}\) . In the implicit version we wish to build an efficient data structure, in terms of space and query time, such that we can report fast the set of vertices that reach a query vertex in all graphs of  \(\mathcal{G}\) . This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problems. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.     

Directed Graphs and Substitutions

A substitution naturally determines a directed graph with an ordering of the edges incident at each vertex. We describe a simple method by which any primitive substitution can be modified (without materially changing the bi-infinite fixed points of the substitution) so that points in the substitution minimal shift are in bijective correspondence with one-sided infinite paths on its associated directed graph. Using this correspondence, we show that primitive substitutive sequences in the substitution minimal shift are precisely those sequences represented by eventually periodic paths. We use directed graphs to show that all measures of cylinders in a substitution minimal shift lie in a finite union of geometric sequences, confirming a conjecture of Boshernitzan. Our methods also yield sufficient conditions for a geometric realization of a primitive substitution to be ``almost injective.' Received January 24, 2000, and in final form August 9, 2001. Online publication November 23, 2001.     

关系数据库模式中候选码的求解算法

在设计和应用关系数据库时,都需要使用候选码。本文介绍了求解一个关系模式的候选码的两种算法。算法一可计算出一个候选码,算法二可计算出全部候选码。     

A Browser for Large Directed Graphs

The paper describes an experimental graph browsing tool which provides fast interactive browsing on very large graphs. The particular graph for which the program was developed is based on the distributed tile system of a group of workstations but there is no reason why the same techniques should not be used on graphs from other sources.     

A Comparison of Two Approaches to Ranking Algorithms Used to Compute Hill Slopes

The calculation of slope (downhill gradient) for a point in a digital elevation model (DEM) is a common procedure in the hydrological, environmental and remote sensing sciences. The most commonly used slope calculation algorithms employed on DEM topography data make use of a three-by-three search window or kernel centered on the grid point (grid cell) in question in order to calculate the slope at that point. A comparison of eight such slope calculation algorithms has been carried out using an artificial DEM, consisting of a smooth synthetic test surface with various amounts of added Gaussian noise. Morrison's Surface III, a trigonometrically defined surface, was used as the synthetic test surface. Residual slope grids were calculated by subtracting the slope grids produced by the algorithms on test from true/reference slope grids derived by analytic partial differentiation of the synthetic surface. The resulting residual slope grids were used to calculate root-mean-square (RMS) residual error estimates that were used to rank the slope algorithms from best (lowest value of RMS residual error) to worst (largest value of RMS residual error). Fleming and Hoffers method gave the best results for slope estimation when values of added Gaussian noise were very small compared to the mean smooth elevation difference (MSED) measured within three-by-three elevation point windows on the synthetic surface. Horns method (used in ArcInfo GRID) performed better than Fleming and Hoffers as a slope estimator when the noise amplitude was very much larger than the MSED. For the large noise amplitude situation the best overall performing method was that of Sharpnack and Akin. The popular Maximum Downward Gradient Method (MDG) performed poorly coming close to last in the rankings, for both situations, as did the Simple Method. A nonogram was produced in terms of standard deviation of the Gaussian noise and MSED values that gave the locus of the trade-off point between Fleming and Hoffers and Horns methods.     

Relaxed Spanners for Directed Disk Graphs

Let (V,δ) be a finite metric space, where V is a set of n points and δ is a distance function defined for these points. Assume that (V,δ) has a doubling dimension d and assume that each point p∈V has a disk of radius r(p) around it. The disk graph that corresponds to V and r(?) is a directed graph I(V,E,r), whose vertices are the points of V and whose edge set includes a directed edge from p to q if δ(p,q)≤r(p). In Peleg and Roditty (Proc. 7th Int. Conf. on Ad-Hoc Networks and Wireless (AdHoc-NOW), pp. 622–633, 2008) we presented an algorithm for constructing a (1+?)-spanner of size O(n? ^?d logM), where M is the maximal radius r(p). The current paper presents two results. The first shows that the spanner of Peleg and Roditty (in Proc. 7th Int. Conf. on Ad-Hoc Networks and Wireless (AdHoc-NOW), pp. 622–633, 2008) is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of M. The second result shows that by slightly relaxing the requirements and allowing a small augmentation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph I(V,E,r _1+? ), where r _1+? (p)=(1+?)?r(p) for every p∈V, then it is possible to get a (1+?)-spanner of size O(n/? ^d ) for I(V,E,r). Our algorithm is simple and can be implemented efficiently.     

基于OBDD的有向图的存储与操作研究

本文讨论了一种基于OBDD的有向图的存储结构,给出了基于OBDD的有向图的操作方法及搜索算法.实验结果表明,该存储结构与传统的邻接表的存储结构相比,在处理大规模的有向图时,具有较高的存储效率.     

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.