示例学习的扩张矩阵理论

本文提出示例学习的一种计算理论,扩张矩阵论.根据这个理论,示例学习中一些主要最优化问题被证明是NP难题,并给出这些难题的近似解法及下界的估计.     

基于动作空间的求解三维矩形装箱问题的穴度算法

基于拟人途径求解三维矩形装箱问题。在穴度算法的基础之上,通过定义当前格局下的极大空闲矩形空间即动作空间,使得穴度的定义既能反映其本质,同时又大能幅度地缩减计算量,从而使算法能在较短的时间内得出空间利用率较高的布局图案。试算了OR-Library中无方向约束的全部47个算例。实验结果表明,改进后的穴度算法得到的平均空间利用率为95. 24 %,将目前的最好结果提高了0.32%,且花费了更少的计算时间。     

Walrasian Equilibrium: Hardness,Approximations and Tractable Instances

We study the complexity issues for Walrasian equilibrium in a special case of combinatorial auction, called single-minded auction, in which every participant is interested in only one subset of commodities. Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004) showed that it is NP-hard to decide the existence of a Walrasian equilibrium for a single-minded auction and proposed a notion of approximate Walrasian equilibrium called relaxed Walrasian equilibrium. We show that every single-minded auction has a relaxed Walrasian equilibrium that satisfies at least two-thirds of the participants, proving a conjecture posed in Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004). Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show NP-completeness and hardness of approximation results for weak Walrasian equilibria. In search of positive results, we restrict our attention to the tollbooth problem (Guruswami et al. in Proceedings of the Symposium on Discrete Algorithms (SODA), pp. 1164–1173, 2005), where every participant is interested in a single path in some underlying graph. We give a polynomial time algorithm to determine the existence of a Walrasian equilibrium and compute one (if it exists), when the graph is a tree. However, the problem is still NP-hard for general graphs.     

一种在元素与颜色规模相近时的有效着色算法及其应用

着色算法(color-coding)是求解NP难问题的重要手段之一.而在应用着色算法时,着色算法所产生的着色方案的规模极大地影响着问题的求解性能,故构造一个尽可能小的着色方案是着色算法所寻求的目标.目前存在的着色算法均基于完全散列函数,并要求元素数目n远大于颜色数目k,且k比较小,这个限制条件使得这些着色算法在一些实际情况下无法应用.该文主要研究在元素与颜色规模相近时(n2k)的有效着色算法,并着重分析在n2k情况下着色算法的性能.该文提出了一种基于划分思想的着色方案构造算法PBCC,证明了由PBCC产生的着色方案确实可以覆盖到所有的子集,并具体给出了可应用于(l,d)-(20,16)Motif查找问题的403种着色的构造方法.文章进一步分析了PBCC产生的着色方案规模,并证明了在n2k且n-k2的情况下,任何着色算法所产生的着色方案的规模|S(n,k)|都不小于[n/2 n-k] [[n n-k]-n/2 n-k]2~(n-k)]/(2~(n-k)-2).此外,文中也采用了渐进分析技术,证明了PBCC算法生成着色方案规模为O(e2Rootof(ex-eμx 1)(n-k)),在n=2k的情况下结果是O(2.62n-k);同时,文中也证明了n2k情况下着色方案规模的下界为2n-k.     

A pure quasi-human algorithm for solving the cuboid packing problem

We excavate the wisdom from an old Chinese proverb “gold corner, silver side and strawy void”, and further improve it into “maximum value in diamond cave” for solving the NP-hard cuboid packing problem. We extract, integrate and formalize the idea by west modern mathematical tools, and propose a pure quasi-human algorithm. The performance of the algorithm is evaluated on two sets of public benchmarks. For 100 strongly heterogeneous difficult benchmarks, experiments show an average packing utilization of 87.31%, which surpasses current best record reported in the literature by 1.83%. For 47 difficult benchmarks without orientation constraint, experiments show an average volume utilization of 92.05%, which improves current best record reported in the literature by 1.05%. Supported by the National Natural Science Foundation of China (Grant No. 60773194), the National Basic Research Program of China (Grant No. 2004CB318000), and Postdoctoral Science Foundation of China (Grant No. 20070420174)     

最大简单共享问题的快速近似算法

介绍了一种基于复制结点的消除线路交叉的模型.该模型提出了一个优化问题,就是最小化结点复制的数量.同时提出一个自定义问题——"最大简单共享问题",并证明最小化结点复制的数量与最大共享问题是等价的.证明了最大简单共享问题是NP-hard的,给出了一种简单的贪心算法,并证明该贪心算法的近似度为3.引入一个"最大互斥简单共享问题",该问题是最大简单共享问题的2-近似.将其转化为在一系列图上的完美匹配问题,使该问题可以在多项式时间内得到完美解决.最后,在最大互斥简单共享的基础上,用局部搜索的方法将近似度提高到12/7.     

TSP问题的脂肪计算复杂性与启发式算法设计

旅行商问题(traveling salesman problem,简称TSP)是经典的NP-难解组合优化问题之一,求解它的高效启发式算法一直是计算机科学研究的热点.脂肪作为描述TSP结构特征的新工具,对启发式算法设计具有重要意义.目前,TSP问题的脂肪研究还处于初始阶段,缺乏理论分析结果及相关的启发式算法.首先分析了TSP问题的脂肪计算复杂性,通过构造偏移实例的技巧,证明了获取TSP的脂肪是NP-难解的,即在P(NP的假设下,不存在算法可以在多项式时间内获得完整脂肪.在此基础上,通过分析TSP问题局部最优解与脂肪的关系,给出了求解TSP问题的元启发式算法--动态候选集搜索(dynamic candidate set search,简称DCSS)算法.利用该算法,改进了目前求解TSP问题最好算法之一的LKH.TSPLIB典型实例的实验结果表明,改进后的算法在解的质量上有较为明显的提高.新的基于脂肪的启发式算法对于求解其他NP-难解问题具有一定的参考价值.     

单体型组装问题计算模型的比较与分析

单体型检测在遗传病基因的定位、药理反应的研究、个体识别等方面有极其广阔的应用前景。单体型组装问题指如何利用个体的基因测序片断数据,根据不同的优化准则确定该个体单体型的计算问题。对MSR,MFR,MEC,WMLF,MEC／GI等单体型组装模型做了详细的分析比较,得出了如下结论：在没有引入测序误差情况下,上述模型的重构精度基本一致。随着测序误差的增加,MEC／GI模型的容错性最好,重构精度最高;MSR模型受测序误差的影响最大,只适用于测序误差极小的情形。     

Cops and Robbers from a distance

Cops and Robbers is a pursuit and evasion game played on graphs that has received much attention. We consider an extension of Cops and Robbers, distance k Cops and Robbers, where the cops win if at least one of them is of distance at most k from the robber in G. The cop number of a graph G is the minimum number of cops needed to capture the robber in G. The distance k analogue of the cop number, written c_k(G), equals the minimum number of cops needed to win at a given distance k. We study the parameter c_k from algorithmic, structural, and probabilistic perspectives. We supply a classification result for graphs with bounded c_k(G) values and develop an O(n^2s+3) algorithm for determining if c_k(G)≤s for s fixed. We prove that if s is not fixed, then computing c_k(G) is NP-hard. Upper and lower bounds are found for c_k(G) in terms of the order of G. We prove that

     

Backbone analysis and algorithm design for the quadratic assignment problem

As the hot line in NP-hard problems research in recent years, backbone analysis is crucial for phase transition, hardness, and algorithm design. Whereas theoretical analysis of backbone and its applications in algorithm design are still at a begin- ning state yet, this paper took the quadratic assignment problem （QAP） as a case study and proved by theoretical analysis that it is NP-hard to find the backbone, i.e., no algorithm exists to obtain the backbone of a QAP in polynomial time. Results of this paper showed that it is reasonable to acquire approximate backbone by inter- section of local optimal solutions. Furthermore, with the method of constructing biased instances, this paper proposed a new meta-heuristic -- biased instance based approximate backbone （BI-AB）, whose basic idea is as follows： firstly, construct a new biased instance for every QAP instance （the optimal solution of the new instance is also optimal for the original one）; secondly, the approximate backbone is obtained by intersection of multiple local optimal solutions computed by some existing algorithm; finally, search for the optimal solutions in the reduced space by fixing the approximate backbone. Work of the paper enhanced the research area of theoretical analysis of backbone. The meta-heuristic proposed in this paper provided a new way for general algorithm design of NP-hard problems as well.