Maximum Satisfiability problem is the central problem in theoretical computer science. Local search has been testified to be effective in the practice to solve these problems. This paper presents new local search methods to solve the maximum satisfiability problems and analyzes the performance of the methods. We focus on the sub problem with each clause containing at least k literals, Max-(k)-Sat briefly. The central issue is to discuss the local search algorithms as well as their performance to solve Max-(2)-Sat and Max-(3)-Sat. We first propose a local search algorithm to solve Max-(2)-Sat. This algorithm can achieve the performance ratio not larger than 4/3 and can be extended to solve Max-(k)-Sat with performance ratio not larger than k+2/k+1 for k ≥ 3. We then propose a local search algorithm for Max-(3)-Sat and show that the algorithm can achieve the performance ratio not larger than 8/7. This algorithm can be extended to solve Max-(k)-Sat with performance ratio not larger than 2k+2/2k+1 for k ≥ 4. We can give examples to show that the aforementioned bounds are all tight. The algorithm for Max-(3)-Sat can naturally derive the local search algorithms for Not-2Sat and Not-3Sat with performance ratio 2 and 4/3 respectively.

合取范式最大可满足问题是理论计算机科学的核心问题.局部搜索被许多求解实践证明是解答合取范式最大可满足问题十分有效的方法,但未见关于局部搜索算法解答该问题性能分析的结果.文中讨论最大3可满足问题(Max-(3)-Sat)的局部搜索算法并分析算法性能.证明Max-(3)-Sat问题的一位跳变局部搜索算法的近似性能比为4/3;证明一位跳变局部搜索后跟有条件全体跳变算法,解答Max-(3)-Sat问题的近似性能比为5/4.设计一位跳变加全体跳变的新局部搜索算法,证明新算法解答Max-(3)-Sat问题的近似性能比为8/7.将8/7-近似局部搜索算法推广为解答Max-(k)-Sat问题的局部搜索算法,证明算法的近似性能比为(2k+2)/(2k+1),k≥4.设计解答Max-(k)-Sat问题的两位跳变局部搜索算法,证明两位跳变局部搜索算法的近似性能比为1+1/(2k+1+k(k-1)/(n-k)),k≥4.局部搜索算法经多次运行可进一步提高求解性能.文中结果显示,局部搜索算法在合取范式最大可满足问题求解实践中表现出高性能,有其必然性.

计算具有较小度的生成树是算法与复杂性研究的一个基本问题,同时在网络设计等领域具有重要应用.给定具有n个顶点的有向无环图G=(V,E)和根顶点r∈ V,最小度生成树问题欲求一棵以r为根的生成树T,使得在G的所有以r为根的生成树中T的最大度最小.给出该问题的一种迭代的多项式时间近似算法.该算法所求树的度不超过△*+1,其中△*为某一最优树的度.算法的时间复杂度为O(n2logn),其中n为顶点数目.算法没有运用过多的枚举,其实际运行时间要快得多.

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of O(logn/logOPT) of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ*/logΔ*) of the optimum in O(mnlogn)-time, where Δ* is the minimum k such that there is a spanning tree of the graph with maximum degree k.

This paper proves the binary fingerprints clustering problem for 2 missing values per fingerprint is NP-Hard, and improves the Figueroa's heuristic algorithm. The new algorithm improves the implementation method for the original algorithm. Firstly, the linked list is used to store the sets of compatible vertices. The linked list can be produced by scanning the fingerprint vectors bit by bit. Thus the time complexity for producing the sets of compatible vertices is reduced from O(m⋅n⋅2 p ) to O(m⋅(n−p+1)⋅2 p ), and the the running time of finding a unique maximal clique or a maximal clique is improved from O(m⋅p⋅2 p ) to O(m⋅2 p ). The real testing displays that the improved algorithm takes 49% or lower space complexity of the original algorithm on the average for the computation of the same instance. It can use 20% time of the original algorithm for solving the same instance. Particularly, the new algorithm can almost always use not more than 11% time of the original algorithm to solve the instance with more than 6 missing values per fingerprint.

An efficient and quick method based on 2-D pca and hierarchical clustering method is proposed. The coordinates are achieved by projecting the high dimensional celestial objects spectra data to the 2-D space after the principle component space is built, every principle component is the linear combination of the original variables and is irrelevant to each other. The coordinates can be clustered by hierarchical clustering method. An automatic celestial objects spectra classification software is designed based on the method to realize the quick and accurate classification of large numbers of spectra.

A randomized approximation algorithm for the set cover problem was proposed. Given a collection S of subsets of E = {e1, e2, ⋯, en} with weights on the subsets, the goal of the weighted set cover problem is to cover all the elements of E by picking a subset with the minimum total cost from S. For each uncovered element of the set E, a subset which contains the element was chosen with a certain elaborately designed probability distribution. The algorithm will not terminate until all the elements have been covered. The proposed algorithm computes a cover whose expected cost is at most B·opt where opt is the cost of some optimal cover for the problem, where B = maxe∈E{|L(e)|} and L(e) = {s|e∈s, s∈S}. The running time of the algorithm is O(n) where n is the cardinality of E.

设施定位问题即UFL问题是NP-hard的组合优化问题,是聚类问题领域的热点问题之一,在数据挖掘和分类识别方面有着重要应用.多年来其近似算法研究一直是计算机科学工作者关注的焦点,然而现有研究结果大多关于Metric空间,一般距离空间中该问题结果始终未见.针对最大连接费用至多是最小连接费用ω＞1倍的一般距离空间中设施定位问题,简称一般设施定位问题,借助集合覆盖问题,利用问题归约方法证明其不存在近似性能比小于1.243+0.316ln(ω-1)的多项式时间近似算法,除非NPDTIME(nO(log log n));设计了一般设施定位问题的局部搜索算法,证明算法近似性能比是(1+ω)/α,ω＞1,1≤α≤2.仿真实验表明,一般设施定位问题局部搜索算法的求解质量极高;通过实验进一步研究了该算法并给出了改进方法.

With the development of fast sequencing techniques, large-scale DNA molecules are investigated with respect to the relative order of genes in them. Genome rearrangement is an important area of computational biology, and bases the comparison analysis of genomes at the level of genes. The goal is to find the shortest sequence of genome arrangements operations that transform one genome into another. Such sequence is helpful to estimate the evolutionary events between different genomes, which is proved by many tests of molecule biology. Although the evolutionary process between genomes is very complicated, there are three basic rearrangement operations: reversal, translocation, transposition.In this paper, we discuss the rearrangement algorithms for these operations, and the methods to compute various types of sorting distances.

The minimum degree spanning tree problem has been studied extensively. In this paper, we present a polynomial time algorithm for the minimum degree spanning tree problem on directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the number of edges and vertices respectively.

Genome rearrangement is an important area in computational biology. There are three basic operations, reversal, translocation, and transposition. Here we study the translocation operations. Multichromosomal genomes frequently evolve by translocation events that exchange genetic material between two chromosomes. We focus on the signed case, where the direction of each gene is known. The signed translocation problem asks to find the minimum number of translocation operations as well as the sequence of translocation operations to transform one genome into the other. A lineartime algorithm that computes the the minimum number of translocation operations was given in Li et al., 2004 14 However, that algorithm cannot give the optimum sequence of translocation operations. The best known algorithm that can give the optimum sequence of translocation operations for signed translocation problem runs in O(n 2 log n) time. In this paper, we design an O(n 2) algorithm.

Research on the approximated algorithms for k-Median problem has been a focus of computer scientists, and most of the existing results are based on the Euclidean and Metric k-Median problem. However, results for general distance space k-Median has not been found for many years. In general distance space, let dmax/dmin denote the maximum value of the length of the longest edge divided by the length of the shortest edge for one client point. In this paper, it is first proved that there are no polynomial algorithms of approximation ratio 1+ ω ?1 e for k-Median with the condition dmax/dmin≤ω+?, unless NP ? DTIME(nO (loglog n)) . This result implies there are no polynomial algorithms of approximation ratio 1+ 2 (loglog n) e for Metric k-Median unless NP ? DTIME(nO ) . Then a local search algorithm for k-Median is presented. New analysis shows that the local search can achieve a ratio of 1+ ω ?1 2 . This result can also be extended to the Metric k-Median, and if ω≤5, the local search algorithm can achieve a ratio less than 3 for the Metric k-Median, which is better than the existing best ratio 3+ 2 . Finally, p computer verification is used to study the real computational effect and the improved method of the local search algorithm.

This paper discusses approximation local search algorithms for uncapacitated facility location problems (UFLP) and its new property inactual computation. This paper firstly presents an analysis result of approximation algorithm about UFLP by using local search, then it discusses therunning time and the quality of solutions through computer verification of local search algorithm.

In this short extended abstract, we give two notes on low-density subset sum algorithm. One is, by extending the variables range from {0, 1}to {− 1,0,1} and allowing the weight be negative, to prove that almost all extended subset sum problems of density <0.488... would be solved in polynomial time with a single call to a lattice oracle. Another is, by only allowing the weights be negative, to point out that almost all corresponding subset sum problems whose density is smaller than the same density bound 0.9408... as [CLOS] would be solved in polynomial time by calling lattice oracle. These two extened subset sum problems have clear significance in cryptanalysis, since breaking some cryptosystems can be reduced to solving one of them. As a example, we give a application of our notes in cryptanalysis of Idempotent Element cryptosystem proposed by Pieprzyk and Rutkowski [PR].

In this paper, a general conclusion for verifying the convergency of topology preserving neural networks is presented, by which the networks are proven to produce convergent feature maps for uniformly distributed inputs. As a special example, the Kohonen's self organizing networks are also proven to be convergent. This paper revises and extends the products in existance and provids a new method for further studying the convergence properties of self organizing neural networks.

本文重点介绍了组合优化问题反问题的研究进展.具体内容包括:线性规划问题反问题、最短路问题反问题、最小费用流问题反问题和网络容量扩充问题反问题的提出背景、研究成果、应用前景及一些可能的研究方向.

A special case of thebottleneck Steiner tree problem in the Euclidean plane was considered in this paper. The problem has applications in the design of wireless communication networks, multifacility location, VLSI routing and network routing. For the special case which requires that there should be no edge connecting any two Steiner points in the optimal solution, a 3-restricted Steiner tree can be found indicating the existence of the performance ratio √2. In this paper, the special case of the problem is proved to beNP-hard and cannot be approximated within ratio √2. First a simple polynomial time approximation algorithm with performance ratio √2 is presented. Then based on this algorithm and the existence of the 3-restricted Steiner tree, a polynomial time approximation algorithm with performance ratio—√2+∈ is proposed, for any ∈>0.

Time-dependent networks are more practical or immediate significance compared with traditional networks models. There are instances to prove the famous Dijkstra's algorithm cannot be effectively used to solve the shortest path problems.A new kind of neural networks for continuously computing the shortest path on a time-dependent network is presented and the stability of the network is proved. This makes a new study on solving the optimization, but not NP-hard problems by neural networks.