A note on ‘Algorithms for connected set cover problem and fault-tolerant connected set cover problem’

A flaw in the greedy approximation algorithm proposed by Zhang et al. (2009) [1] for the minimum connected set cover problem is corrected, and a stronger result on the approximation ratio of the modified greedy algorithm is established. The results are now consistent with the existing results on the connected dominating set problem which is a special case of the minimum connected set cover problem.     

基于模拟植物生长算法的求解MCCS问题的研究

为了降低耗能和减少花费,提出了对无线传感网络设计中的最小连通集合划分的方法.采用对网络进行Voronoi划分成近似覆盖集合,对不满足连通的情况采用一种基于模拟植物生长算法生成Steiner最优树的连通算法来实现网络连通的方法.通过对算法的时间复杂度分析及算例实验,验证了该算法不但可获得最优解,同时精度和性能也有提高,明显优于其它方法.     

Parallel and serial heuristics for the minimum set cover problem

We present a theoretical analysis and an experimental evaluation of four serial heuristics and four parallel heuristics for the minimum set cover problem. The serial heuristics trade off run time with the quality of the solution. The parallel heuristics are derived from one of the serial heuristics. These heuristics show considerable speedup when the number of processors is increased. The quality of the solution computed by the heuristics does not degrade with an increase in the number of processors.Research of both authors was supported by NSF Grant No. MIP-8807540.     

Solving the set cover problem on a supercomputer

Supercomputers, such as CRAY-1, CRAY X-MP, CYBER 205, ETA10, … etc, have been regularly used for solving numerical problems. It is very rare that supercomputers are used to solve combinatorial problems. In this paper, we present an efficient vectorized algorithm to solve the set cover problem, which was proved to be NP-complete, on a supercomputer, ETA10-Q108. This algorithm fully utilizes vector instructions. Experiments are performed both on ETA10-Q108 and VAX/8550 for comparison. It takes VAX/8550 1174.5 seconds to solve a set of problem instances while it takes ETA10-Q108 only 26.6 seconds to solve the same set of problems. For a problem instance involving 7000 elements in a set, it takes 47.74 seconds for the supercomputer to solve it. If VAX/8550 is used, it will need roughly 15 hours. Thus we conclude that it is quite feasible to solve the set cover problem by using supercomputers.     

A 2-approximation NC algorithm for connected vertex cover and tree cover

The connected vertex cover problem is a variant of the vertex cover problem, in which a vertex cover is additional required to induce a connected subgraph in a given connected graph. The problem is known to be NP-hard and to be at least as hard to approximate as the vertex cover problem is. While several 2-approximation NC algorithms are known for vertex cover, whether unweighted or weighted, no parallel algorithm with guaranteed approximation is known for connected vertex cover. Moreover, converting the existing sequential 2-approximation algorithms for connected vertex cover to parallel ones results in RNC algorithms of rather high complexity at best.In this paper we present a 2-approximation NC (and RNC) algorithm for connected vertex cover (and tree cover). The NC algorithm runs in O(log²n) time using O(Δ²(m+n)/logn) processors on an EREW-PRAM, while the RNC algorithm runs in O(logn) expected time using O(m+n) processors on a CRCW-PRAM, when a given graph has n vertices and m edges with maximum vertex degree of Δ.     

The fault-tolerant extension problem for complete multipartitenetworks

We develop a characterization for m-fault-tolerant extensions, and for optimal m-fault-tolerant extensions, of a complete multipartite graph. Our formulation shows that this problem is equivalent to an interesting combinatorial problem on the partitioning of integers. This characterization leads to a new procedure for constructing an optimal m-fault-tolerant extension of any complete multipartite graph, for any m⩾0. The proposed procedure is mainly useful when the size of the graph is relatively small, because the search time required is exponential. This exponential search, however, is not always necessary. We prove several necessary conditions that help us, in several cases, to identify some optimal m-fault-tolerant extensions without performing any search     

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

Using quantum annealing to solve an optimization problem requires minor embedding a logic graph into a known hardware graph. In an effort to reduce the complexity of the minor embedding problem, we introduce the minor set cover (MSC) of a known graph \({\mathcal {G}}\): a subset of graph minors which contain any remaining minor of the graph as a subgraph. Any graph that can be embedded into \({\mathcal {G}}\) will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete bipartite graph. We show that the complete bipartite graph \(K_{N,N}\) has a MSC of N minors, from which \(K_{N+1}\) is identified as the largest clique minor of \(K_{N,N}\). The case of determining the largest clique minor of hardware with faults is briefly discussed but remains an open question.     

Fast Algorithms for max independent set

We first propose a method, called “bottom-up method” that, informally, “propagates” improvement of the worst-case complexity for “sparse” instances to “denser” ones and we show an easy though non-trivial application of it to the min set cover problem. We then tackle max independent set. Here, we propagate improvements of worst-case complexity from graphs of average degree?d to graphs of average degree greater than?d. Indeed, using algorithms for max independent set in graphs of average degree 3, we successively solve max independent set in graphs of average degree 4, 5 and?6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree?5 and?6 but also for general graphs. The computation bounds obtained for max independent set are?O ^?(1.1571 ⁿ ), O ^?(1.1895 ⁿ ) and?O ^?(1.2050 ⁿ ), for graphs of maximum (or more generally average) degree?4, 5 and?6 respectively, and?O ^?(1.2114 ⁿ ) for general graphs. These results improve upon the best known results for these cases for polynomial space algorithms.     

PTAS for connected vertex cover in unit disk graphs

This paper gives the first polynomial time approximation scheme for the connected vertex cover problem in unit disk graphs.     

Adjacency stable connected operators and set levelings

This paper studies morphological connected operators. Particularly, it focuses on an adjacency constraint, as well as on the so-called set levelings. Some important findings are reported in this work. First, the relationships between the so-called adjacency stable operators and set levelings are investigated, and an equivalence is established. This permits to apply some properties to the related operator class. Second, the implications and limits of a property presented elsewhere that states that certain connected operators can be expressed as a sequential composition of an opening and a closing (and vice-versa) based on markers are treated. Then, a commutative property that involves alternated attribute filters is presented. In addition, other related expressions are discussed.     

A comparison of algorithms for connected set openings and closings

The implementation of morphological connected set operators for image filtering and pattern recognition is discussed. Two earlier algorithms based on priority queues and hierarchical queues, respectively, are compared to a more recent union-find approach. Unlike the earlier algorithms which process regional extrema in the image sequentially, the union-find method allows simultaneous processing of extrema. In the context of area openings, closings, and pattern spectra, the union-find algorithm outperforms the previous methods on almost all natural and synthetic images tested. Finally, extensions to pattern spectra and the more general class of attribute operators are presented for all three algorithms, and memory usages are compared     

A linear kernel for a planar connected dominating set

We provide polynomial time data reduction rules for Connected Dominating Set on planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm analyzes the input graph and either finds an appropriate reduction rule that can be applied, or zooms in on a region of the graph which is more amenable to reduction. We find this method of independent interest and believe that it will be useful for obtaining linear kernels for other problems on planar graphs.     

Algorithms for solving minimax scheduling problem

A series of exact and approximate algorithms are developed for the problem of time-optimal scheduling without interruptions and switchings in a multiprocessor system. The efficiency of these algorithms is determined and the comparative analysis is performed.     

Algorithms for the maximum satisfiability problem

Old and new algorithms for the Maximum Satisfiability problem are studied. We first summarize the different heuristics previously proposed, i.e., the approximation algorithms of Johnson and of Lieberherr for the general Maximum Satisfiability problem, and the heuristics of Lieberherr and Specker, Poljak and Turzik for the Maximum 2-Satisfiability problem. We then consider two recent local search algorithmic schemes, the Simulated Annealing method of Kirkpatrick, Gelatt and Vecchi and the Steepest Ascent Mildest Descent method, and adapt them to the Maximum Satisfiability problem. The resulting algorithms, which avoid being blocked as soon as a local optimum has been found, are shown empirically to be more efficient than the heuristics previously proposed in the literature.     

Algorithms for the coalitional manipulation problem

We investigate the problem of coalitional manipulation in elections, which is known to be hard in a variety of voting rules. We put forward efficient algorithms for the problem in Borda, Maximin and Plurality with Runoff, and analyze their windows of error. Specifically, given an instance on which an algorithm fails, we bound the additional power the manipulators need in order to succeed. We finally discuss the implications of our results with respect to the popular approach of employing computational hardness to preclude manipulation.     

Algorithms for the boundary selection problem

User-oriented clustering schemes enable the classification of documents based upon the user's perception of the similarity between documents, rather than on some similarity function presumed by the designer to represent the user's criteria. In an earlier paper it was shown that such a classification scheme can be developed in two stages. The first stage involves the accumulation of relevance judgements provided by users,vis-à-vis past query instances, into a suitable structure. The second stage consists of cluster identification. When the structure chosen, in the first stage, for the accumulation of corelevance characteristics of documents is a straight line, the second stage can be formulated as a function optimization problem termed the Boundary Selection Problem (BSP). A branch-and-bound algorithm with a good bounding function is developed for the BSP. Although significant pruning is achieved due to the bounding function, the complexity is still high for a problem of a large size. For such a problem a heuristic that divides it into a number of subproblems, each being solved by a branch-and-bound approach, is developed. Then the overall problem is mapped to an integer knapsack problem and solved by the use of dynamic programming. The tradeoff between accuracy and complexity can be controlled, giving the user a preference of one over the other. Assuming that the heuristic which divides the overall problem introduces no errors and is given sufficient time, the branch and bound with dynamic programming (BBDP) approach will converge to the optimal solution. Two other heuristic approaches, one with the application of a polynomial dynamic programming algorithm and the other which works in a greedy way, are also proposed for the BSP and an experimental comparison of all these approaches is provided. Experimental results indicate that all proposed algorithms show better performance compared with the existing algorithm.     

一种参考能量的最小连通支配集近似算法

在无线传感器网络中,能量效率问题至关重要,构造精简的虚拟骨干网可以节约有限资源,这等同于在图论中求解最小连通支配集(MCDS)问题.由此,提出一种构造MCDS的启发式算法.首先根据均值公式为顶点建立次序表,其次构造极大独立集(MIS),再次连接MIS节点,最后优化.仿真实验表明:该算法能够在短时间内找到规模较小的连通支配集(CDS),并且有效地均衡了各节点能量,延长了网络生命周期.     

Exponential-time approximation of weighted set cover

The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponential-time algorithms for problems of that kind. The goal is getting the time complexity still of order O(cn), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation.In fact, we describe a kind of approximation schemes—trade-offs between approximation factor and the time complexity. We use general transformations from exponential-time exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate r, one can transform any O^∗(cn)-time¹ algorithm for Set Cover into a (1+lnr)-approximation algorithm running in time O^∗(cn/r). We believe that results of that kind extend the applicability of exact algorithms for NP-hard problems.