Hardness and approximation of minimum maximal matchings

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2?1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2?23/18k) in k-regular graphs admitting a perfect matching.     

Hardness and approximation results for packing steiner trees

We study approximation algorithms and hardness of approximation for several versions of the problem of packing Steiner trees. For packing edge-disjoint Steiner trees of undirected graphs, we show APX-hardness for four terminals. For packing Steiner-node-disjoint Steiner trees of undirected graphs, we show a logarithmic hardness result, and give an approximation guarantee ofO (√n logn), wheren denotes the number of nodes. For the directed setting (packing edge-disjoint Steiner trees of directed graphs), we show a hardness result of Θ(m ^1/3/−ɛ) and give an approximation guarantee ofO(m ^1/2/+ɛ), wherem denotes the number of edges. We have similar results for packing Steiner-node-disjoint priority Steiner trees of undirected graphs. Supported by NSERC Grant No. OGP0138432. Supported by an NSERC postdoctoral fellowship, Department of Combinatorics and Optimization at University of Waterloo, and a University start-up fund at University of Alberta.     

The hardness of approximation: Gap location

We refine the complexity analysis of approximation problems by relating it to a new parameter calledgap location. Many of the results obtained so far for approximations yield satisfactory analysis with respect to this refined parameter, but some known results (e.g.,max-k-colorability, max 3-dimensional matching andmax not-all-equal 3sat) fall short of doing so. As a second contribution, our work fills the gap in these cases by presenting new reductions.Next, we present definitions and hardness results of new approximation versions of some NP-complete optimization problems. The problems we treat arevertex cover (for which we define a different optimization problem from the one treated in Papadimitriou & Yannakakis 1991),k-edge coloring, andset splitting.     

On the Hardness of Approximating Spanners

A k -spanner of a connected graph G=(V,E) is a subgraph G' consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G' is larger than the distance in G by no more than a factor of k . This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k , approximating the spanner problem is at least as hard as approximating the set-cover problem. We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k=2 and the case k\geq 5 . Received October 30, 1998; revised March 4, 1999, and April 17, 2000.     

Hardness results for covering arrays avoiding forbidden edges and error-locating arrays

Covering arrays avoiding forbidden edges (CAFEs) are used in testing applications (software, networks, circuits, drug interaction, material mixtures, etc.) where certain combinations of parameter values are forbidden. Danziger et al. (2009) [8] have studied this problem and shown some computational complexity results. Around the same time, Martinez et al. (2009) [19] defined and studied error-locating arrays (ELAs), which are closely related to CAFEs. Both papers left some computational complexity questions. In particular, these papers showed polynomial-time solvability of the existence of CAFEs and ELAs for binary alphabets (g=2), and the NP-hardness of these problems for g≥5. In this paper, we prove that optimizing CAFEs and ELAs is indeed NP-hard even when restricted to the case of binary alphabets, using a reduction from edge clique covers of graphs (ECCs). We also provide a hardness of approximation result. We explore important relationships between ECCs and CAFEs and give some new bounds for uniform ECCs and CAFEs.     

Random parallel algorithms for finding exact branchings,perfect matchings,and cycles

In this paper we devise randomized parallel algorithms which given a unary weighted (di)graphG=(V, E)construct in time O(log²¦ V¦) branchings, perfect matchings, and disjoint cycles of weightexactly k belonging toG. These problems have been studied by Papadimitriou and Yannakakis [PY1], by Barahona and Pulleyblank [BP], by Cameriniet al [CGM], and by Mulmuleyet al. [MVV]. Our algorithms improve previous solutions. Moreover, we give an NC² algorithm for computing the absolute value of the pfaffian of a skew-symmetric matrix.Supported in part by MURST 40%.     

Uniquely Restricted Matchings

A matching in a graph is a set of edges no two of which share a common vertex. In this paper we introduce a new, specialized type of matching which we call uniquely restricted matchings, originally motivated by the problem of determining a lower bound on the rank of a matrix having a specified zero/ non-zero pattern. A uniquely restricted matching is defined to be a matching M whose saturated vertices induce a subgraph which has only one perfect matching, namely M itself. We introduce the two problems of recognizing a uniquely restricted matching and of finding a maximum uniquely restricted matching in a given graph, and present algorithms and complexity results for certain special classes of graphs. We demonstrate that testing whether a given matching M is uniquely restricted can be done in O(|M||E|) time for an arbitrary graph G=(V,E) and in linear time for cacti, interval graphs, bipartite graphs, split graphs and threshold graphs. The maximum uniquely restricted matching problem is shown to be NP-complete for bipartite graphs, split graphs, and hence for chordal graphs and comparability graphs, but can be solved in linear time for threshold graphs, proper interval graphs, cacti and block graphs. Received April 12, 1998; revised June 21, 1999.     

Simultaneous inner and outer approximation of shapes

For compact Euclidean bodiesP, Q, we define (P, Q) to be the smallest ratior/s wherer > 0,s > 0 satisfy . HeresQ denotes a scaling ofQ by the factors, andQ,Q are some translates ofQ. This function gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies arehomothetic if one can be obtained from the other by scaling and translation.)For integerk 3, define (k) to be the minimum value such that for each convex polygonP there exists a convexk-gonQ with (P, Q) (k). Among other results, we prove that 2.118 ... <-(3) 2.25 and (k) = 1 + (k ^–2). We give anO(n ² log² n)-time algorithm which, for any input convexn-gonP, finds a triangleT that minimizes (T, P) among triangles. However, in linear time we can find a trianglet with (t, P)<-2.25.Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion-planning problem. In each case we describe algorithms which run faster when certain implicitslackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.Work of all authors was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM). Rudolf Fleischer and Kurt Mehlhorn acknowledge also DFG (Grant SPP Me 620/6). Chee Yap acknowledges also DFG (Grant Be 142/46-1) and NSF (Grants DCR-84-01898 and CCR-87-03458). This research was performed when Günter Rote and Chee Yap were at the Freie Universität Berlin.     

Hardness and methods to solve CLIQUE

The paper briefly reviews NP-hard optimization problems and their inapproximability.The hardness of solving CLIQUE problem is specifically discussed.A dynamic-programming algorithm and its improved version for CLIQUE are reviewed and some additional analysis is presented.The analysis implies that the improved algorithm.HEWN(hierarchical edge-weighted network),only provides a heuristic or useful method,but cannot be called a polynomial algorithm.     

关于同步部分规约的有限自动机的优化问题的近似难度

自动机是可同步的是指它具有满足以下性质的同步字:不论自动机当前所处的状态,以同步字为输入执行后它一定会到达某个特定状态。同步自动机问题的核心是计算最短同步字。聚焦于这一核心问题,文中就一类称为部分规约的确定的有限自动机的最短同步字问题,研究了近似计算这类自动机的最短同步字的复杂性,即近似计算它的难度,该工作有助于其近似算法的分析与设计。通过建立由两个优化问题(MAX SAT问题以及MAX FA-INT问题)到最短同步字长度计算这一问题(即Shortest-Syn)的归约,利用与概率可检验证明(Probabilistically Checkable Proofs,PCP)定理和概率可检验辩论(Probabilistically Checkable Debate,PCD)定理有关的若干结果证明了文中的主要结论:对于部分规约的确定的有限自动机,在某个近似因子内Shortest-Syn的近似难度是NP-难的和PSPACE-难的,除非NP和PSPACE分别坍塌到P。     

Comparing accuracy of second-order approximation and dynamic programming

The accuracy of the solution of dynamic general equilibrium models has become a major issue. Recent papers, in which second-order approximations have been substituted for first-order, indicate that this change may yield a significant improvement in accuracy. Second order approximations have been used with considerable success when solving for the decision variables in both small and large-scale models. Additionally, the issue of accuracy is relevant for the approximate solution of value functions. In numerous dynamic decision problems, welfare is usually computed via this same approximation procedure. However, Kim and Kim (Journal of International Economics, 60, 471–500, 2003) have found a reversal of welfare ordering when they moved from first- to second-order approximations. Other researchers, studying the impact of monetary and fiscal policy on welfare, have faced similar challenges with respect to the accuracy of approximations of the value function. Employing a base-line stochastic growth model, this paper compares the accuracy of second-order approximations and dynamic programming solutions for both the decision variable and the value function as well. We find that, in a neighborhood of the equilibrium, the second-order approximation method performs satisfactorily; however, on larger regions, dynamic programming performs significantly better with respect to both the decision variable and the value function. We want to thank Jinill Kim and Harald Uhlig for discussions and a referee of the Journal for extensive comments on a previous version of our paper.     

模糊系统万能逼近理论研究综述

模糊系统通用逼近理论是模糊理论研究的一个重要方向．目前，对模糊系统通用逼近性的研究已经取得了很大的进展．对模糊系统的通用逼近性、模糊系统作为通用逼近器的充分条件和必要条件以及模糊系统的逼近精度等方面的研究进行了较为详尽的综述，分析了各种分析方法的主要成果及其特点（包括优点和局限性），并指出了今后模糊系统通用逼近理论研究中有待解决的许多问题．     

Parameterized Computation and Complexity： A New Approach Dealing with NP-Hardness

The theory of parameterized computation and complexity is a recently developed subarea in theoretical computer science. The theory is aimed at practically solving a large number of computational problems that are theoretically intractable.The theory is based on the observation that many intractable computational problems in practice are associated with a parameter that varies within a small or moderate range. Therefore, by taking the advantages of the small parameters, many theoretically intractable problems can be solved effectively and practically. On the other hand, the theory of parameterized computation and complexity has also offered powerful techniques that enable us to derive strong computational lower bounds for many computational problems, thus explaining why certain theoretically tractable problems cannot be solved effectively and practically. The theory of parameterized computation and complexity has found wide applications in areas such as database systems, programming languages, networks, VLSI design, parallel and distributed computing, computational biology, and robotics. This survey gives an overview on the fundamentals, algorithms, techniques, and applications developed in the research of parameterized computation and complexity. We will also report the most recent advances and excitements, and discuss further research directions in the area.     

Branching and pruning: An optimal temporal POCL planner based on constraint programming

A key feature of modern optimal planners such as graphplan and blackbox is their ability to prune large parts of the search space. Previous Partial Order Causal Link (POCL) planners provide an alternative branching scheme but lacking comparable pruning mechanisms do not perform as well. In this paper, a domain-independent formulation of temporal planning based on Constraint Programming is introduced that successfully combines a POCL branching scheme with powerful and sound pruning rules. The key novelty in the formulation is the ability to reason about supports, precedences, and causal links involving actions that are not in the plan. Experiments over a wide range of benchmarks show that the resulting optimal temporal planner is much faster than current ones and is competitive with the best parallel planners in the special case in which actions have all the same duration.¹     

Rigorous cubical approximation and persistent homology of continuous functions

The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.     

Spline functions in chemistry: approximation of surfaces over triangle domains

Chemists often come across triangle domains – usually with the basic simplex in ?². A smooth surface is needed for approximating the chemical properties between the measured data for solving some model (differential) equations numerically.

Our research group has been working on approximating ternary chemical surfaces of two special fields by smooth functions (vapour – Liquid equilibrium data and explosion-limit surfaces of ternary gas systems).

A mathematical solution was given in both fields by special spline surfaces, and for visualization, our own software (TRIGON) was used. In this paper the summarized chemical background of each problem is provided, the mathematical solutions, the newest theoretical developments and their results are discussed.     

Dynamical low-rank approximation: applications and numerical experiments

Dynamical low-rank approximation is a differential-equation-based approach to efficiently compute low-rank approximations to time-dependent large data matrices or to solutions of large matrix differential equations. We illustrate its use in the following application areas: as an updating procedure in latent semantic indexing for information retrieval, in the compression of series of images, and in the solution of time-dependent partial differential equations, specifically on a blow-up problem of a reaction-diffusion equation in two and three spatial dimensions. In 3D and higher dimensions, space discretization yields a tensor differential equation whose solution is approximated by low-rank tensors, effectively solving a system of discretized partial differential equations in one spatial dimension.     

Discrete contours in multiple views: approximation and recognition

Recognition of discrete planar contours under similarity transformations has received a lot of attention but little work has been reported on recognizing them under more general transformations. Planar object boundaries undergo projective or affine transformations across multiple views. We present two methods to recognize discrete curves in this paper. The first method computes a piecewise parametric approximation of the discrete curve that is projectively invariant. A polygon approximation scheme and a piecewise conic approximation scheme are presented here. The second method computes an invariant sequence directly from the sequence of discrete points on the curve in a Fourier transform space. The sequence is shown to be identical up to a scale factor in all affine related views of the curve. We present the theory and demonstrate its applications to several problems including numeral recognition, aircraft recognition, and homography computation.