Fast minimal triangulation algorithm using minimum degree criterion

We propose an algorithm for minimal triangulation which, using simple and efficient strategy, subdivides the input graph in different, almost non-overlapping, subgraphs. Using the technique of matrix multiplication for saturating the minimal separators, we show that the partition of the graph can be computed in time O(n^α) where n^α is the time required by the binary matrix multiplication. After saturating the minimal separators, the same procedure is recursively applied on each subgraphs. We also present a variant of the algorithm in which the minimum degree criterion is used. In this way, we obtain an algorithm that uses minimum degree criterion and at the same time produces a minimal triangulation, thus shedding new light on the effectiveness of the minimum degree heuristics.     

Maximum Induced Matchings for Chordal Graphs in Linear Time

The Maximum Induced Matching (MIM) Problem asks for a largest set of pairwise vertex-disjoint edges in a graph which are pairwise of distance at least two. It is well-known that the MIM problem is NP-complete even on particular bipartite graphs and on line graphs. On the other hand, it is solvable in polynomial time for various classes of graphs (such as chordal, weakly chordal, interval, circular-arc graphs and others) since the MIM problem on graph G corresponds to the Maximum Independent Set problem on the square G ^*=L(G)² of the line graph L(G) of G, and in some cases, G ^* is in the same graph class; for example, for chordal graphs G, G ^* is chordal. The construction of G ^*, however, requires time, where m is the number of edges in G. Is has been an open problem whether there is a linear-time algorithm for the MIM problem on chordal graphs. We give such an algorithm which is based on perfect elimination order and LexBFS.     

Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints

Much research in the area of constraint processing has recently been focused on extracting small unsatisfiable “cores” from unsatisfiable constraint systems with the goal of finding minimal unsatisfiable subsets (MUSes). While most techniques have provided ways to find an approximation of an MUS (not necessarily minimal), we have developed a sound and complete algorithm for producing all MUSes of an unsatisfiable constraint system. In this paper, we describe a relationship between satisfiable and unsatisfiable subsets of constraints that we subsequently use as the foundation for MUS extraction algorithms, implemented for Boolean satisfiability constraints. The algorithms provide a framework with which many related subproblems can be solved, including relaxations of completeness to handle intractable instances, and we develop several variations of the basic algorithms to illustrate this. Experimental results demonstrate the performance of our algorithms, showing how the base algorithms run quickly on many instances, while the variations are valuable for producing results on instances whose complete results are intractably large. Furthermore, our algorithms are shown to perform better than the existing algorithms for solving either of the two distinct phases of our approach.     

Sequential and parallel triangulating algorithms for Elimination Game and new insights on Minimum Degree

Elimination Game is a well-known algorithm that simulates Gaussian elimination of matrices on graphs, and it computes a triangulation of the input graph. The number of fill edges in the computed triangulation is highly dependent on the order in which Elimination Game processes the vertices, and in general the produced triangulations are neither minimum nor minimal. In order to obtain a triangulation which is close to minimum, the Minimum Degree heuristic is widely used in practice, but until now little was known on the theoretical mechanisms involved.     

Algorithms for Maximum Independent Set in Convex Bipartite Graphs

A bipartite graph G=(V,W,E) is convex if there exists an ordering of the vertices of W such that, for each v∈V, the neighbors of v are consecutive in W. We describe both a sequential and a BSP/CGM algorithm to find a maximum independent set in a convex bipartite graph. The sequential algorithm improves over the running time of the previously known algorithm and the BSP/CGM algorithm is a parallel version of the sequential one. The complexity of the algorithms does not depend on |W|. This work was supported by FAPESP (Proc. 98/06327-0). The first author was also supported by FAPESP (Proc. 96/04505–2), and CNPq/MCT/FINEP (PRONEX project 107/97).     

Parallel Algorithms for Maximum Matching in Complements of Interval Graphs and Related Problems

Given a set of n intervals representing an interval graph, the problem of finding a maximum matching between pairs of disjoint (nonintersecting) intervals has been considered in the sequential model. In this paper we present parallel algorithms for computing maximum cardinality matchings among pairs of disjoint intervals in interval graphs in the EREW PRAM and hypercube models. For the general case of the problem, our algorithms compute a maximum matching in O( log ³ n) time using O(n/ log ² n) processors on the EREW PRAM and using n processors on the hypercubes. For the case of proper interval graphs, our algorithm runs in O( log n ) time using O(n) processors if the input intervals are not given already sorted and using O(n/ log n ) processors otherwise, on the EREW PRAM. On n -processor hypercubes, our algorithm for the proper interval case takes O( log n log log n ) time for unsorted input and O( log n ) time for sorted input. Our parallel results also lead to optimal sequential algorithms for computing maximum matchings among disjoint intervals. In addition, we present an improved parallel algorithm for maximum matching between overlapping intervals in proper interval graphs. Received November 20, 1995; revised September 3, 1998.     

A parallel algorithm for generating bicompatible elimination orderings of proper interval graphs

In this paper, we first show how a certain ordering of vertices, called bicompatible elimination ordering (BCO), of a proper interval graph can be used to solve efficiently several problems in proper interval graphs. We, then, propose an NC parallel algorithm (i.e., polylogarithmic-time employing a polynomial number of processors) in SIMD CRCW PRAM (Single Instruction Stream Multiple Data Stream Concurrent Read Concurrent Write Parallel Random Access Machine) model of parallel computation to compute a BCO of a proper interval graph. To the best of our knowledge, this is the first NC parallel algorithm to compute a BCO of a proper interval graph.     

在基于模型诊断中计算最小碰集算法

介绍了基于模型诊断中的计算碰集的算法 ,并分析比较了各算法的效率和计算结果。其中的逻辑型数组算法、递归算法、BHS 树算法、布尔代数算法、GA算法均是笔者近年来研究的结果。     

Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions

We study graph properties that admit an increasing, or equivalently decreasing, sequence of graphs on the same vertex set such that for any two consecutive graphs in the sequence their difference is a single edge. This is useful for characterizing and computing minimal completions and deletions of arbitrary graphs into having these properties. We prove that threshold graphs and chain graphs admit such sequences. Based on this characterization and other structural properties, we present linear-time algorithms both for computing minimal completions and deletions into threshold, chain, and bipartite graphs, and for extracting a minimal completion or deletion from a given completion or deletion. Minimum completions and deletions into these classes are NP-hard to compute.     

Planning as Propositional CSP: From Walksat to Local Search Techniques for Action Graphs

Graphplan-style of planning can be formulated as an incremental propositional CSP where the (boolean) variables correspond to operator instantiations (actions) that are or are not scheduled at certain time steps. In this paper we present a framework for solving this class of propositional CSPs using local search in planning graphs. The search space consists of particular subgraphs of a planning graph corresponding to (complete) variable assignments, and representing partial plans. The operators for moving from one search state to the next one are graph modifications corresponding to revisions of the current variable assignment (partial plan), or to an extension of the represented CSP.Our techniques are implemented in a planner called LPG using various types of heuristics based on a parametrized objective function, where the parameters weight different constraint violations, and are dynamically evaluated using Lagrange multipliers. LPG's basic heuristic was inspired by Walksat, which in Kautz and Selman's Blackbox can be used to solve the SAT-encoding of a planning graph. An advantage of LPG is that its heuristics exploit the structure of the planning graph, while Blackbox relies on general heuristics for SAT-problems, and requires the translation of the planning graph into propositional clauses. Another major difference is that LPG can handle action execution costs to produce good quality plans. This is achieved by an anytime process minimizing an objective function based on the number of constraint violations in a plan and on its overall cost. Experimental results illustrate the efficiency of our approach, showing, in particular, that LPG is significantly faster than Blackbox and other planners based on planning graphs.     

Finding Maximum Induced Matchings in Subclasses of Claw-Free and <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs,and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P ₅,D _m )-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.     

多部图的匹配算法研究

本文给出了一个多部图的商匹配问题的定义,提出了求解多部图商匹配问题的一个算法。该算法使用圈与割集中偶图的交相结合的方法,利用求二部图的最大匹配算法,求解多部图的最大商匹配问题。     

An Algorithm for Straight-Line Drawing of Planar Graphs

We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs ``nicely,' as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n-4) x (n-2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Received September 21, 1995; revised March 15, 1996.     

右边正则纠删码中二部图的构造算法

通过深入研究右边正则度序列的分析性质,设计了右边正则纠删码度序列的参数优化算法.基于此算法,提出了右边正则纠删码设计中随机二部图的连边构造算法.数值结果证明了所给的度序列参数优化算法的有效性.仿真结果表明基于右边正则度序列的级联型纠删码的性能优于Tornado码.随机二部图的连边构造算法和度序列的参数优化算法有助于右边正则纠删码的设计及其工程应用.     

Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k ^O(dk) n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain K _h as a topological minor, we give an improved algorithm for the problem with running time (O(h)) ^hk n. For graphs which are K _h -minor-free, the running time is further reduced to (O(log h)) ^hk/2 n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(nlog n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs. A preliminary version of this paper appeared in the Proceedings of the 13th Annual International Computing and Combinatorics Conference (COCOON), Banff, Alberta, Canada (2007), pp. 394–405. N. Alon research supported in part by a grant from the Israel Science Foundation, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. This paper forms part of a Ph.D. thesis written by S. Gutner under the supervision of Prof. N. Alon and Prof. Y. Azar in Tel Aviv University.     

随机图点覆盖1度顶点核化算法分析

将随机图引入参数计算领域,利用随机图统计和概率分布等特性,从全局和整体上研究参数化点覆盖问题1度点核化过程中问题的核及度分布演变的内在机制和变化规律,并得出关于随机图1度点核化强度与顶点平均度关系及随机图点覆盖问题的决策与度分布关系的两个重要推论.最后分别从MIPS和BIND提取数据进行1度核化实验和分析.初步结果表明,对随机图点覆盖问题的分析方法不仅具有理论上的意义,而且随着问题随机度的大小而对问题有不同程度的把握能力.     

Minimal cycle bases for analysis of frames with semi-rigid joints

For an efficient force method of frame analysis, special cycle bases should be generated for the formation of localized self-equilibrating systems, leading to sparse flexibility matrices. In this paper, an algorithm is presented using a fundamental cycle basis, where the selected cycles are improved via an algebraic exchange method. Optimal analysis is performed for frames with semi-rigid joints. In this method, flexibility matrices are generated which are highly sparse. An ordering algorithm is also used for profile reduction to acquire an efficient method for the solution of the corresponding equations. Thus a complete force method analysis of semi-rigid frames is formulated and a computer code is developed. Examples are analyzed with the present approach and the results are compared to those of a stiffness method program with semi-rigid joints.     

云计算下相关性缺失大数据分块填补仿真

传统数据填补手段填补规模受限,存在运行不稳定、内存占比较大以及填补精度较低等缺点,为此提出一种云计算下相关性缺失大数据分块填补。根据数据填补原理,可通过较小的区间代替缺失数据,计算大数据集信息熵与指标之间的相关性系数,将数据集填充于原始大数据中,计算新得到的数据集信息熵,利用新旧信息熵的相似性关系扩大区间范围。随后对相关性缺失大数据做分块处理,分成已知分块和未知分块,已知分块可以直接对其进行填补,未知分块需要利用基于稀疏性的K-means算法约束目标函数中变量权重,并划分其聚类结果获得未知分块数据集,最后利用宿主法实现填补。仿真结果证明,所提方法相比其它方法,精准度较高、填补效果良好且运行稳定。     

Minimal model generation for refined answering of generalized queries in disjunctive deductive databases

Generalized queries are defined as sets of clauses in implication form. They cover several tasks of practical importance for database maintenance such as answering positive queries, computing database completions and integrity constraints checking. We address the issue of answering generalized queries under the minimal model semantics for the class of disjunctive deductive databases (DDDBs). The advanced approach is based on having the query induce an order on the models returned by a sound and complete minimal model generating procedure. We consider answers that are true in all and those that are true in some minimal models of the theory. We address the issue of answering positive queries through the construction of the minimal model state of the DDDB, using a minimal model generating procedure. The refinements allowed by the procedure include isolating a minimal component of a disjunctive answer, the specification of possible updates to the theory to enable the derivability of certain queries and deciding the monotonicity properties of answers to different classes of queries.     

Discrete‐Time Consensus Filters for Average Tracking of Time‐Varying Inputs on Directed Switching Graphs

In this paper, we consider discrete‐time distributed estimation on a directed graph with switching topologies. Motivated by a recent PI consensus filter, we modify the protocol and remove the limitation of undirected graph and gain conditions with requirement for global topological information. The protocol is then extended to switching topologies. Convergence analysis is conducted for constant inputs under both balanced directed and general directed graphs with topological switching. Then, the result is extended to all cases for time‐varying inputs with an analytical derivation of the error bound. Satisfactory simulation results are shown to validate theoretical claims.