15-854 Approximations Algorithms Lecturer Anupam Gupta Topic Max-cut, Hardness of Approxima

合集下载

stoerwagner-mincut.[Stoer-Wagner,Prim,连通性,无向图,最小边割集]

A Simple Min-Cut AlgorithmMECHTHILD STOERTeleverkets Forskningsinstitutt,Kjeller,NorwayANDFRANK WAGNERFreie Universita¨t Berlin,Berlin-Dahlem,GermanyAbstract.We present an algorithm for finding the minimum cut of an undirected edge-weighted graph.It is simple in every respect.It has a short and compact description,is easy to implement,and has a surprisingly simple proof of correctness.Its runtime matches that of the fastest algorithm known.The runtime analysis is straightforward.In contrast to nearly all approaches so far,the algorithm uses no flow techniques.Roughly speaking,the algorithm consists of about͉V͉nearly identical phases each of which is a maximum adjacency search.Categories and Subject Descriptors:G.L.2[Discrete Mathematics]:Graph Theory—graph algorithms General Terms:AlgorithmsAdditional Key Words and Phrases:Min-Cut1.IntroductionGraph connectivity is one of the classical subjects in graph theory,and has many practical applications,for example,in chip and circuit design,reliability of communication networks,transportation planning,and cluster analysis.Finding the minimum cut of an undirected edge-weighted graph is a fundamental algorithmical problem.Precisely,it consists in finding a nontrivial partition of the graphs vertex set V into two parts such that the cut weight,the sum of the weights of the edges connecting the two parts,is minimum.A preliminary version of this paper appeared in Proceedings of the2nd Annual European Symposium on Algorithms.Lecture Notes in Computer Science,vol.855,1994,pp.141–147.This work was supported by the ESPRIT BRA Project ALCOM II.Authors’addresses:M.Stoer,Televerkets Forskningsinstitutt,Postboks83,2007Kjeller,Norway; e-mail:mechthild.stoer@nta.no.;F.Wagner,Institut fu¨r Informatik,Fachbereich Mathematik und Informatik,Freie Universita¨t Berlin,Takustraße9,Berlin-Dahlem,Germany;e-mail:wagner@inf.fu-berlin.de.Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage,the copyright notice,the title of the publication,and its date appear,and notice is given that copying is by permission of the Association for Computing Machinery(ACM),Inc.To copy otherwise,to republish,to post on servers,or to redistribute to lists,requires prior specific permission and/or a fee.᭧1997ACM0004-5411/97/0700-0585$03.50Journal of the ACM,Vol.44,No.4,July1997,pp.585–591.586M.STOER AND F.WAGNER The usual approach to solve this problem is to use its close relationship to the maximum flow problem.The famous Max-Flow-Min-Cut-Theorem by Ford and Fulkerson[1956]showed the duality of the maximum flow and the so-called minimum s-t-cut.There,s and t are two vertices that are the source and the sink in the flow problem and have to be separated by the cut,that is,they have to lie in different parts of the partition.Until recently all cut algorithms were essentially flow algorithms using this duality.Finding a minimum cut without specified vertices to be separated can be done by finding minimum s-t-cuts for a fixed vertex s and all͉V͉Ϫ1possible choices of tʦVگ{s}and then selecting the lightest one.Recently Hao and Orlin[1992]showed how to use the maximum flow algorithm by Goldberg and Tarjan[1988]in order to solve the minimum cut problem in timeᏻ(͉VʈE͉log(͉V͉2/͉E͉),which is nearly as fast as the fastest maximum flow algorithms so far[Alon1990;Ahuja et al.1989;Cheriyan et al. 1990].Nagamochi and Ibaraki[1992a]published the first deterministic minimum cut algorithm that is not based on a flow algorithm,has the slightly better running time ofᏻ(͉VʈE͉ϩ͉V͉2log͉V͉),but is still rather complicated.In the unweighted case,they use a fast-search technique to decompose a graph’s edge set E into subsets E1,...,E␭such that the union of the first k E i’s is a k-edge-connected spanning subgraph of the given graph and has at most k͉V͉edges.They simulate this approach in the weighted case.Their work is one of a small number of papers treating questions of graph connectivity by non-flow-based methods [Nishizeki and Poljak1989;Nagamochi and Ibaraki1992a;Matula1992].Karger and Stein[1993]suggest a randomized algorithm that with high probability finds a minimum cut in timeᏻ(͉V͉2log͉V͉).In this context,we present in this paper a remarkably simple deterministic minimum cut algorithm with the fastest running time so far,established in Nagamochi and Ibaraki[1992b].We reduce the complexity of the algorithm of Nagamochi and Ibaraki by avoiding the unnecessary simulated decomposition of the edge set.This enables us to give a comparably straightforward proof of correctness avoiding,for example,the distinction between the unweighted, integer-,rational-,and real-weighted case.This algorithm was found independently by Frank[1994].Queyranne[1995]generalizes our simple approach to the minimization of submodular functions.The algorithm described in this paper was implemented by Kurt Mehlhorn from the Max-Planck-Institut,Saarbru¨cken and is part of the algorithms library LEDA[Mehlhorn and Na¨her1995].2.The AlgorithmThroughout the paper,we deal with an ordinary undirected graph G with vertex set V and edge set E.Every edge e has nonnegative real weight w(e).The simple key observation is that,if we know how to find two vertices s and t, and the weight of a minimum s-t-cut,we are nearly done:T HEOREM2.1.Let s and t be two vertices of a graph G.Let G/{s,t}be the graph obtained by merging s and t.Then a minimum cut of G can be obtained by taking the smaller of a minimum s-t-cut of G and a minimum cut of G/{s,t}.The theorem holds since either there is a minimum cut of G that separates s and t ,then a minimum s -t -cut of G is a minimum cut of G ;or there is none,then a minimum cut of G /{s ,t }does the job.So a procedure finding an arbitrary minimum s -t -cut can be used to construct a recursive algorithm to find a minimum cut of a graph.The following algorithm,known in the literature as maximum adjacency search or maximum cardinality search ,yields the desired s -t -cut.M INIMUM C UT P HASE (G ,w ,a )A 4{a }while A Vadd to A the most tightly connected vertexstore the cut-of-the-phase and shrink G by merging the two vertices added lastA subset A of the graphs vertices grows starting with an arbitrary single vertex until A is equal to V .In each step,the vertex outside of A most tightly connected with A is added.Formally,we add a vertexz ʦ͞A such that w ͑A ,z ͒ϭmax ͕w ͑A ,y ͉͒y ʦ͞A ͖,where w (A ,y )is the sum of the weights of all the edges between A and y .At the end of each such phase,the two vertices added last are merged ,that is,the two vertices are replaced by a new vertex,and any edges from the two vertices to a remaining vertex are replaced by an edge weighted by the sum of the weights of the previous two edges.Edges joining the merged nodes are removed.The cut of V that separates the vertex added last from the rest of the graph is called the cut-of-the-phase .The lightest of these cuts-of-the-phase is the result of the algorithm,the desired minimum cut:M INIMUM C UT (G ,w ,a )while ͉V ͉Ͼ1M INIMUM C UT P HASE (G ,w ,a )if the cut-of-the-phase is lighter than the current minimum cutthen store the cut-of-the-phase as the current minimum cutNotice that the starting vertex a stays the same throughout the whole algorithm.It can be selected arbitrarily in each phase instead.3.CorrectnessIn order to proof the correctness of our algorithms,we need to show the following somewhat surprising lemma.L EMMA 3.1.Each cut -of -the -phase is a minimum s -t -cut in the current graph ,where s and t are the two vertices added last in the phase .P ROOF .The run of a M INIMUM C UT P HASE orders the vertices of the current graph linearly,starting with a and ending with s and t ,according to their order of addition to A .Now we look at an arbitrary s -t -cut C of the current graph and show,that it is at least as heavy as the cut-of-the-phase.587A Simple Min-Cut Algorithm588M.STOER AND F.WAGNER We call a vertex v a active(with respect to C)when v and the vertex added just before v are in the two different parts of C.Let w(C)be the weight of C,A v the set of all vertices added before v(excluding v),C v the cut of A vഫ{v} induced by C,and w(C v)the weight of the induced cut.We show that for every active vertex vw͑A v,v͒Յw͑C v͒by induction on the set of active vertices:For the first active vertex,the inequality is satisfied with equality.Let the inequality be true for all active vertices added up to the active vertex v,and let u be the next active vertex that is added.Then we havew͑A u,u͒ϭw͑A v,u͒ϩw͑A uگA v,u͒ϭ:␣Now,w(A v,u)Յw(A v,v)as v was chosen as the vertex most tightly connected with A v.By induction w(A v,v)Յw(C v).All edges between A uگA v and u connect the different parts of C.Thus they contribute to w(C u)but not to w(C v).So␣Յw͑C v͒ϩw͑A uگA v,u͒Յw͑C u͒As t is always an active vertex with respect to C we can conclude that w(A t,t)Յw(C t)which says exactly that the cut-of-the-phase is at most as heavy as C.4.Running TimeAs the running time of the algorithm M INIMUM C UT is essentially equal to the added running time of the͉V͉Ϫ1runs of M INIMUM C UT P HASE,which is called on graphs with decreasing number of vertices and edges,it suffices to show that a single M INIMUM C UT P HASE needs at mostᏻ(͉E͉ϩ͉V͉log͉V͉)time yielding an overall running time ofᏻ(͉VʈE͉ϩ͉V͉2log͉V͉).The key to implementing a phase efficiently is to make it easy to select the next vertex to be added to the set A,the most tightly connected vertex.During execution of a phase,all vertices that are not in A reside in a priority queue based on a key field.The key of a vertex v is the sum of the weights of the edges connecting it to the current A,that is,w(A,v).Whenever a vertex v is added to A we have to perform an update of the queue.v has to be deleted from the queue,and the key of every vertex w not in A,connected to v has to be increased by the weight of the edge v w,if it exists.As this is done exactly once for every edge,overall we have to perform͉V͉E XTRACT M AX and͉E͉I NCREASE K EY ing Fibonacci heaps[Fredman and Tarjun1987],we can perform an E XTRACT M AX operation inᏻ(log͉V͉)amortized time and an I NCREASE K EY operation inᏻ(1)amortized time.Thus,the time we need for this key step that dominates the rest of the phase, isᏻ(͉E͉ϩ͉V͉log͉V͉).5.AnExample F IG .1.A graph G ϭ(V ,E )withedge-weights.F IG .2.The graph after the first M INIMUM C UT P HASE (G ,w ,a ),a ϭ2,and the induced ordering a ,b ,c ,d ,e ,f ,s ,t of the vertices.The first cut-of-the-phase corresponds to the partition {1},{2,3,4,5,6,7,8}of V with weight w ϭ5.F IG .3.The graph after the second M INIMUM C UT P HASE (G ,w ,a ),and the induced ordering a ,b ,c ,d ,e ,s ,t of the vertices.The second cut-of-the-phase corresponds to the partition {8},{1,2,3,4,5,6,7}of V with weight w ϭ5.F IG .4.After the third M INIMUM C UT P HASE (G ,w ,a ).The third cut-of-the-phase corresponds to the partition {7,8},{1,2,3,4,5,6}of V with weight w ϭ7.589A Simple Min-Cut AlgorithmACKNOWLEDGMENT .The authors thank Dorothea Wagner for her helpful re-marks.REFERENCESA HUJA ,R.K.,O RLIN ,J.B.,AND T ARJAN ,R.E.1989.Improved time bounds for the maximum flow problem.SIAM put.18,939–954.A LON ,N.1990.Generating pseudo-random permutations and maximum flow algorithms.Inf.Proc.Lett.35,201–204.C HERIYAN ,J.,H AGERUP ,T.,AND M EHLHORN ,K.1990.Can a maximum flow be computed in o (nm )time?In Proceedings of the 17th International Colloquium on Automata,Languages and Programming .pp.235–248.F ORD ,L.R.,AND F ULKERSON ,D.R.1956.Maximal flow through a network.Can.J.Math.8,399–404.F RANK , A.1994.On the Edge-Connectivity Algorithm of Nagamochi and Ibaraki .Laboratoire Artemis,IMAG,Universite ´J.Fourier,Grenoble,Switzerland.F REDMAN ,M.L.,AND T ARJAN ,R.E.1987.Fibonacci heaps and their uses in improved network optimization algorithms.J.ACM 34,3(July),596–615.G OLDBERG ,A.V.,AND T ARJAN ,R.E.1988.A new approach to the maximum-flow problem.J.ACM 35,4(Oct.),921–940.H AO ,J.,AND O RLIN ,J.B.1992.A faster algorithm for finding the minimum cut in a graph.In Proceedings of the 3rd ACM-SIAM Symposium on Discrete Algorithms (Orlando,Fla.,Jan.27–29).ACM,New York,pp.165–174.K ARGER ,D.,AND S TEIN ,C.1993.An O˜(n 2)algorithm for minimum cuts.In Proceedings of the 25th ACM Symposium on the Theory of Computing (San Diego,Calif.,May 16–18).ACM,New York,pp.757–765.F IG .5.After the fourth and fifth M INIMUM C UT P HASE (G ,w ,a ),respectively.The fourth cut-of-the-phase corresponds to the partition {4,7,8},{1,2,3,5,6}.The fifth cut-of-the-phase corresponds to the partition {3,4,7,8},{1,2,5,6}with weight w ϭ4.F IG .6.After the sixth and seventh M INIMUM C UT P HASE (G ,w ,a ),respectively.The sixth cut-of-the-phase corresponds to the partition {1,5},{2,3,4,6,7,8}with weight w ϭ7.The last cut-of-the-phase corresponds to the partition {2},V گ{2};its weight is w ϭ9.The minimum cut of the graph G is the fifth cut-of-the-phase and the weight is w ϭ4.590M.STOER AND F.WAGNERM ATULA ,D.W.1993.A linear time 2ϩ⑀approximation algorithm for edge connectivity.In Proceedings of the 4th ACM–SIAM Symposium on Discrete Mathematics ACM,New York,pp.500–504.M EHLHORN ,K.,AND N ¨AHER ,S.1995.LEDA:a platform for combinatorial and geometric mun.ACM 38,96–102.N AGAMOCHI ,H.,AND I BARAKI ,T.1992a.Linear time algorithms for finding a sparse k -connected spanning subgraph of a k -connected graph.Algorithmica 7,583–596.N AGAMOCHI ,H.,AND I BARAKI ,puting edge-connectivity in multigraphs and capaci-tated graphs.SIAM J.Disc.Math.5,54–66.N ISHIZEKI ,T.,AND P OLJAK ,S.1989.Highly connected factors with a small number of edges.Preprint.Q UEYRANNE ,M.1995.A combinatorial algorithm for minimizing symmetric submodular functions.In Proceedings of the 6th ACM–SIAM Symposium on Discrete Mathematics ACM,New York,pp.98–101.RECEIVED APRIL 1995;REVISED FEBRUARY 1997;ACCEPTED JUNE 1997Journal of the ACM,Vol.44,No.4,July 1997.591A Simple Min-Cut Algorithm。

一般图上的推广的最小k——cut问题的求解算法

一般图上的推广的最小k——cut问题的求解算法摘要原始割集问题是图论几大经典问题之一，在实际中应用广泛。

割集问题有很多较为复杂的推广问题，如最小multicut问题和最小multiwaycut问题等。

本文主要讨论的是割集问题的另一推广问题最小k--cut问题，并且给出了一个时间复杂度为的近似算法求得该问题的可行解。

关键词推广；最小k--cut问题；近似算法0 引言割集问题在图论和组合优化中占有举足轻重的地位，几十年来一直受到广泛的重视。

原始的割集问题是指给定一个连通赋权图以及在中指定两个点，，问题是找的一个最小权重的边子集，使得与在中不连通。

1 推广的最小k--cut问题定义：给定连通赋权图及正整数和，其中，求的一个边子集，使得中恰有个连通分支，且每个连通分支，的顶点数满足条件，目标是：达到最小。

当且时，即连通分支数为2而每个连通分支的顶点数不受限制时，此问题就为图的原始cut问题它是多项式可解的，调用求最大流的MPM算法再根据最大流最小割集定理就可以求得图的原始cut问题的最优解且时间复杂度。

当时通过最大流算法不一定会找到最优解。

因为最大流算法只保证有两个连通分支，但是每个连通分支的顶点数不一定会小于。

当时此问题是难的，方法是将最小边multiwaycut问题归约到此问题。

任给连通赋权图的最小multiway cut问题的一个实例：，和的顶点子集，其中，称为终端点，最小multiway cut问题就是要求使得中的点在中不连通且达到最小。

构造限制性最小k--cut问题的一个实例，构造的方法：在中的每一个终端点处加入个点，，边的权重为无穷大，这样构造了新图求图的一个权重最小的边子集。

，使得有连通分支且每个连通分支的顶点数都小于。

这样就得到了限制性最小k--cut问题的一个实例。

若有一个多项式时间算法能够解决那么它也能解决实例。

因为通过算法求得的最优解，使得有连通分支且每个连通分支的顶点数都小于，因此中的任意两点是不可能在同一个连通分支的，若与在的同一连通分支，那么此连通分支的顶点数至少为与已知条件矛盾。

Given

COMP9314
Xuemin Lin @DBG.unsw
15
Example
Data Stream: 12, 10, 11, 10, 1, 10, 7, 9, 6, 11, 8, 11, 4, 5, 2
Median
A sliding window Current item
(in ordered set) (N=9)
After “3” arrived: 12, 10, 11, 10, 1, 10, 7, 9, 6, 11, 8, 11, 4, 5, 2, 3
Expired elements
Median (in ordered set)
Current item
COMP9314
Xuemin Lin @DBG.unsw
Iteratively:
⎧⎪rmerged ,k −
=
r− merged ,k −1
+ ri, j−
−
ri
,
j
− −1
∑ ε ⎨
⎪⎩rmerged
,k
+
=
r− merged ,k −1
+
i Ni
Where ri,j is from the j th tuple in the i th local sketch
-Insert tuple before the ith tuple. gnew = 1; Δnew = gi + Δi - 1;
Delete all “superfluous” entries gi := gi + gi-1 -1 General strategy:
-Delete tuples with small capacity and preserve tuples with large capacity.

The Sequential Quadratic Programming Method

The Sequential Quadratic Programming Method
167
2 ewton Methods and Local Optimality
In this and subsequent sections we trace the development of Newton methods from the simplest case of nonlinear equations, through to the general case of nonlinear programming with equations and inequalities.
x∈IR
subject to ci (x) ≥ 0
i = 1 , 2 , . . . , m.
(1.1)
In this formulation, equation constraints must be encoded as two opposed inequality constraints, that is c(x) = 0 is replaced by c(x) ≥ 0 and −c(x) ≥ 0, which is usually not convenient. Thus in practice a more detailed formulation is appropriate, admitting also equations, linear constraints and simple bounds. One way to do this is to add slack variables to the constraints, which
The Sequential Quadratic Programming Method

中科大算法第二章近似算法--黄刘生(调整后适合打印版)

12
NP-完全性理论
Karp的贡献
理查德·卡普（Richard Karp ， 1935- ） 1972 年论文 ”Reducibility among Combinatorial Problems” 发展和加强了由库克提出的“NP完全性”理论。尤其是库克仅证明了命题演算的可满足问题是NP完全的，而卡普则证明了从组合优化中引出的大多数经典问题(背包问题、覆盖问题、匹配问题、分区问题、路径问题、调度问题等)都是NP完全问题。只要证明其中任一个问题是属于P类的，就可解决计算复杂性理论中最大的一个难题，即P=?NP。
SAT∈P当且仅当P=NP
Cook 于1961 年获 Michigan 大学学士学位， 1962 和 1966年分获哈佛大学硕士与博士学位。 1966-1970 ，他在 UC Berkeley 担任助教授； 1970年加盟多伦多大学，现为该校CS 和数学系教授，他的论文开启了NP完备性的研究，令该领域于之后的十年成为计算机科学中最活跃和重要的研究。因其在计算复杂性理论方面的贡献，尤其是在奠定NP完全性理论基础上的突出贡献而荣获1982年度的图灵奖。
9
P、NP及NPC类问题
NP=？P
∵确定型图灵机是非确定型图灵机的特例，∴P⊆NP 是否有NP⊆P？即是否NP=P？
美国麻省的Clay数学研究所于2000年5月24日在巴黎法兰西学院宣布：对七个“千年数学难题”中的每一个均悬赏 100 万美元，而问题NP=？P位列其首:
1.P问题对NP问题 2.霍奇猜想 3. 庞加莱猜想 (2002.11-2003.7 ，俄罗斯数学家佩雷尔曼在 3 篇论文预印本中证明了几何化猜想，2006被授予菲尔兹奖) 4.黎曼假设 5.杨－米尔斯存在性和质量缺口 6.纳维叶－斯托克斯方程的存在性与光滑性 7.贝赫和斯维讷通－戴尔猜想

2014年中科院SCI分区

期刊全称ACM Transactions on Intelligent Systems and TechnologyACM Transactions on Intelligent Systems and TechnologyIEEE Communications Surveys and TutorialsIEEE Communications Surveys and TutorialsIEEE TRANSACTIONS ON FUZZY SYSTEMSIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE International Journal of Neural SystemsIEEE Transactions on Industrial InformaticsMIS QUARTERLYIEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATIONIEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATIONCOMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERINGIEEE WIRELESS COMMUNICATIONSIEEE WIRELESS COMMUNICATIONSIEEE WIRELESS COMMUNICATIONSJournal of Statistical SoftwareIEEE Transactions on Neural Networks and Learning SystemsIEEE Transactions on Neural Networks and Learning SystemsIEEE Transactions on Neural Networks and Learning SystemsMEDICAL IMAGE ANALYSISMEDICAL IMAGE ANALYSISACM COMPUTING SURVEYSIEEE COMMUNICATIONS MAGAZINEINTEGRATED COMPUTER-AIDED ENGINEERINGINTEGRATED COMPUTER-AIDED ENGINEERINGENVIRONMENTAL MODELLING & SOFTWAREINTERNATIONAL JOURNAL OF COMPUTER VISIONIEEE Computational Intelligence MagazineIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONSACM TRANSACTIONS ON GRAPHICSSIAM Journal on Imaging SciencesSIAM Journal on Imaging SciencesIEEE Transactions on Affective ComputingIEEE Transactions on Affective ComputingINFORMATION SCIENCESIEEE Transactions on CyberneticsIEEE Transactions on CyberneticsARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING JOURNAL OF INFORMATION TECHNOLOGYKNOWLEDGE-BASED SYSTEMSIEEE TRANSACTIONS ON IMAGE PROCESSINGJOURNAL OF MACHINE LEARNING RESEARCHIEEE NETWORKIEEE NETWORKIEEE NETWORKIEEE TRANSACTIONS ON INFORMATION THEORY JOURNAL OF LIGHTWAVE TECHNOLOGYCOMPUTERS & EDUCATIONKNOWLEDGE AND INFORMATION SYSTEMSKNOWLEDGE AND INFORMATION SYSTEMSIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS JOURNAL OF THE ACMJOURNAL OF THE ACMJOURNAL OF THE ACMJOURNAL OF THE ACMIEEE TRANSACTIONS ON MOBILE COMPUTINGIEEE TRANSACTIONS ON MOBILE COMPUTINGPATTERN RECOGNITIONAPPLIED SOFT COMPUTINGAPPLIED SOFT COMPUTINGCOMMUNICATIONS OF THE ACMCOMMUNICATIONS OF THE ACMCOMMUNICATIONS OF THE ACMInformation FusionInformation FusionARTIFICIAL INTELLIGENCEACM TRANSACTIONS ON MATHEMATICAL SOFTWAREIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION EVOLUTIONARY COMPUTATIONEVOLUTIONARY COMPUTATIONCOMPUTERS & CHEMICAL ENGINEERINGIEEE TRANSACTIONS ON SOFTWARE ENGINEERINGANNUAL REVIEW OF INFORMATION SCIENCE AND TECHNOLOGYIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGYCHEMOMETRICS AND INTELLIGENT LABORATORY SYSTEMSIEEE Transactions on Systems Man Cybernetics-SystemsFUTURE GENERATION COMPUTER SYSTEMSIEEE TRANSACTIONS ON BROADCASTINGJOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TE NEURAL NETWORKSDATA MINING AND KNOWLEDGE DISCOVERYDATA MINING AND KNOWLEDGE DISCOVERYIEEE Transactions on Human-Machine SystemsIEEE Transactions on Human-Machine SystemsHUMAN-COMPUTER INTERACTIONHUMAN-COMPUTER INTERACTIONIEEE INTERNET COMPUTINGIEEE-ACM TRANSACTIONS ON NETWORKINGIEEE-ACM TRANSACTIONS ON NETWORKINGIEEE-ACM TRANSACTIONS ON NETWORKINGIEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS EXPERT SYSTEMS WITH APPLICATIONSIEEE INTELLIGENT SYSTEMSIEEE MICROIEEE MICRODECISION SUPPORT SYSTEMSDECISION SUPPORT SYSTEMSIEEE Transactions on Services ComputingIEEE Transactions on Services ComputingIEEE Transactions on Autonomous Mental DevelopmentARTIFICIAL LIFEARTIFICIAL LIFEIEEE Journal of Biomedical and Health InformaticsIEEE Journal of Biomedical and Health InformaticsIEEE PERVASIVE COMPUTINGIEEE PERVASIVE COMPUTINGINTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY & DECISION MA INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY & DECISION MA INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY & DECISION MA INFORMATION & MANAGEMENTINTERNATIONAL JOURNAL OF APPROXIMATE REASONINGBIOLOGICAL CYBERNETICSCOMPUTERS & STRUCTURESJOURNAL OF STRATEGIC INFORMATION SYSTEMSAd Hoc NetworksAd Hoc NetworksJOURNAL OF MATHEMATICAL IMAGING AND VISIONJOURNAL OF MATHEMATICAL IMAGING AND VISIONIEEE TRANSACTIONS ON MULTIMEDIAIEEE TRANSACTIONS ON MULTIMEDIAIEEE TRANSACTIONS ON MULTIMEDIAIEEE TRANSACTIONS ON COMMUNICATIONSIEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMSMobile Information SystemsMobile Information SystemsIEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERINGIEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING COMPUTERS & OPERATIONS RESEARCHNEURAL COMPUTATIONIEEE Transactions on Information Forensics and SecurityIEEE MICROWAVE MAGAZINEIMAGE AND VISION COMPUTINGIMAGE AND VISION COMPUTINGIMAGE AND VISION COMPUTINGENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCEIEEE TRANSACTIONS ON RELIABILITYIEEE TRANSACTIONS ON RELIABILITYNEUROCOMPUTINGAUTONOMOUS ROBOTSADVANCED ENGINEERING INFORMATICSINTERNATIONAL JOURNAL OF ELECTRONIC COMMERCEInternational Journal of Web and Grid ServicesInternational Journal of Web and Grid ServicesCOMPUTERS AND ELECTRONICS IN AGRICULTUREIEEE Antennas and Wireless Propagation LettersUSER MODELING AND USER-ADAPTED INTERACTIONIEEE Transactions on HapticsSTRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATIONCOMPUTER GRAPHICS FORUMJournal of Optical Communications and NetworkingJournal of Optical Communications and NetworkingJournal of Optical Communications and NetworkingCOMPUTERS & INDUSTRIAL ENGINEERINGMACHINE LEARNINGACM Transactions on Sensor NetworksACM Transactions on Sensor NetworksEUROPEAN JOURNAL OF INFORMATION SYSTEMSJournal of Computational ScienceJournal of Computational ScienceCOMPUTERS IN INDUSTRYIEEE-ACM Transactions on Computational Biology and BioinformaticsEMPIRICAL SOFTWARE ENGINEERINGVLDB JOURNALVLDB JOURNALJOURNAL OF MANAGEMENT INFORMATION SYSTEMSCOMPUTER SPEECH AND LANGUAGECOMPUTERCOMPUTERJournal of Grid ComputingJournal of Grid ComputingInternational Journal of Bio-Inspired ComputationInternational Journal of Bio-Inspired ComputationPervasive and Mobile ComputingPervasive and Mobile ComputingINDUSTRIAL MANAGEMENT & DATA SYSTEMSINTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMSIEEE Transactions on Computational Intelligence and AI in GamesIEEE Transactions on Computational Intelligence and AI in GamesDATA & KNOWLEDGE ENGINEERINGDATA & KNOWLEDGE ENGINEERINGPARALLEL COMPUTINGCOMPUTING IN SCIENCE & ENGINEERINGMECHATRONICSCOMPUTER METHODS IN BIOMECHANICS AND BIOMEDICAL ENGINEERING IEEE SOFTWARESOFT COMPUTINGSOFT COMPUTINGJOURNAL OF NETWORK AND COMPUTER APPLICATIONSJOURNAL OF NETWORK AND COMPUTER APPLICATIONSJOURNAL OF NETWORK AND COMPUTER APPLICATIONSACM TRANSACTIONS ON SOFTWARE ENGINEERING AND METHODOLOGY TELECOMMUNICATIONS POLICYElectronic Commerce Research and ApplicationsElectronic Commerce Research and ApplicationsINTERNET RESEARCH-ELECTRONIC NETWORKING APPLICATIONS AND POLI INTERNET RESEARCH-ELECTRONIC NETWORKING APPLICATIONS AND POLIACM Transactions on Knowledge Discovery from DataACM Transactions on Knowledge Discovery from DataROBOTICS AND COMPUTER-INTEGRATED MANUFACTURING INFORMATION SYSTEMSWiley Interdisciplinary Reviews-Data Mining and Knowledge DiscoveryWiley Interdisciplinary Reviews-Data Mining and Knowledge DiscoveryCOMPUTER METHODS AND PROGRAMS IN BIOMEDICINECOMPUTER METHODS AND PROGRAMS IN BIOMEDICINEINFORMATION AND SOFTWARE TECHNOLOGYINFORMATION AND SOFTWARE TECHNOLOGYSPEECH COMMUNICATIONJOURNAL OF HEURISTICSJOURNAL OF HEURISTICSARTIFICIAL INTELLIGENCE IN MEDICINECOMPUTER-AIDED DESIGNJOURNAL OF COMPUTING IN CIVIL ENGINEERINGFuzzy Optimization and Decision MakingJournal of the Association for Information SystemsIEEE TRANSACTIONS ON COMPUTERSIEEE Systems JournalIEEE Systems JournalCOMPUTER VISION AND IMAGE UNDERSTANDINGOPTICAL FIBER TECHNOLOGYJournal of Web SemanticsJournal of Web SemanticsJournal of Web SemanticsIEEE Vehicular Technology MagazineINTERNATIONAL JOURNAL OF SYSTEMS SCIENCEACM Transactions on the WebACM Transactions on the WebIEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITYIEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMSIEEE COMPUTER GRAPHICS AND APPLICATIONSINTERNATIONAL JOURNAL OF HUMAN-COMPUTER STUDIESADVANCES IN ENGINEERING SOFTWAREADVANCES IN ENGINEERING SOFTWARECOMPUTERS IN BIOLOGY AND MEDICINECOMPUTER NETWORKS-THE INTERNATIONAL JOURNAL OF COMPUTER AND COMPUTER NETWORKS-THE INTERNATIONAL JOURNAL OF COMPUTER AND COMPUTER NETWORKS-THE INTERNATIONAL JOURNAL OF COMPUTER AND Swarm IntelligenceInternational Journal of Fuzzy SystemsPersonal and Ubiquitous ComputingPersonal and Ubiquitous ComputingJOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION ARTIFICIAL INTELLIGENCE REVIEWINFORMATICANEURAL COMPUTING & APPLICATIONSJOURNAL OF INFORMATION SCIENCECOMPUTATIONAL LINGUISTICSCOMPUTATIONAL LINGUISTICSIEEE COMMUNICATIONS LETTERSIEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEM IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCU IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCU INFORMS JOURNAL ON COMPUTINGINTERNATIONAL JOURNAL OF HIGH PERFORMANCE COMPUTING APPLICAT INTERNATIONAL JOURNAL OF HIGH PERFORMANCE COMPUTING APPLICAT INTERNATIONAL JOURNAL OF HIGH PERFORMANCE COMPUTING APPLICAT MACHINE VISION AND APPLICATIONSMACHINE VISION AND APPLICATIONSJOURNAL OF HYDROINFORMATICSRADIO SCIENCEJOURNAL OF COMPLEXITYAutomated Software EngineeringENGINEERING COMPUTATIONSCOMPUTER COMMUNICATIONSCOMPUTER COMMUNICATIONSACM TRANSACTIONS ON INFORMATION SYSTEMSMOBILE NETWORKS & APPLICATIONSMOBILE NETWORKS & APPLICATIONSMOBILE NETWORKS & APPLICATIONSIEEE Design & TestCOMPUTER STANDARDS & INTERFACESCOMPUTER STANDARDS & INTERFACESGenetic Programming and Evolvable MachinesGenetic Programming and Evolvable MachinesMULTIDIMENSIONAL SYSTEMS AND SIGNAL PROCESSINGPATTERN RECOGNITION LETTERSIEEE Transactions on Dependable and Secure ComputingIEEE Transactions on Dependable and Secure ComputingIEEE Transactions on Dependable and Secure ComputingWORLD WIDE WEB-INTERNET AND WEB INFORMATION SYSTEMSWORLD WIDE WEB-INTERNET AND WEB INFORMATION SYSTEMSONLINE INFORMATION REVIEWROBOTICS AND AUTONOMOUS SYSTEMSJournal of Real-Time Image ProcessingINTERNATIONAL JOURNAL OF UNCERTAINTY FUZZINESS AND KNOWLEDGE ADAPTIVE BEHAVIORJOURNAL OF INTELLIGENT MANUFACTURINGJOURNAL OF COMPUTER AND SYSTEM SCIENCESJOURNAL OF COMPUTER AND SYSTEM SCIENCESNEURAL PROCESSING LETTERSJOURNAL OF SYSTEMS AND SOFTWAREJOURNAL OF SYSTEMS AND SOFTWARESOFTWARE TESTING VERIFICATION & RELIABILITYCOMPUTERS & SECURITYACM Transactions on Applied PerceptionIEEE MULTIMEDIAIEEE MULTIMEDIAIEEE MULTIMEDIAIEEE MULTIMEDIAIEEE TRANSACTIONS ON CONSUMER ELECTRONICSSIMULATION MODELLING PRACTICE AND THEORYSIMULATION MODELLING PRACTICE AND THEORYREQUIREMENTS ENGINEERINGREQUIREMENTS ENGINEERINGComputer Supported Cooperative Work-The Journal of Collaborative Computing DISPLAYSSoftware and Systems ModelingINTERACTING WITH COMPUTERSAUTONOMOUS AGENTS AND MULTI-AGENT SYSTEMSBusiness & Information Systems EngineeringJOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCHINTERNATIONAL JOURNAL OF SATELLITE COMMUNICATIONS AND NETWOR ACM Transactions on Autonomous and Adaptive SystemsACM Transactions on Autonomous and Adaptive SystemsACM Transactions on Autonomous and Adaptive SystemsWIRELESS COMMUNICATIONS & MOBILE COMPUTINGWIRELESS COMMUNICATIONS & MOBILE COMPUTINGINTERNATIONAL JOURNAL OF COMPUTER INTEGRATED MANUFACTURING JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTINGApplied OntologyApplied OntologyApplied OntologyJournal of Ambient Intelligence and Smart EnvironmentsJournal of Ambient Intelligence and Smart EnvironmentsJournal of Ambient Intelligence and Smart EnvironmentsINFORMATION PROCESSING & MANAGEMENTInternational Journal on Semantic Web and Information SystemsInternational Journal on Semantic Web and Information SystemsInternational Journal of Applied Mathematics and Computer Science TELECOMMUNICATION SYSTEMSOPEN SYSTEMS & INFORMATION DYNAMICSJOURNAL OF CRYPTOLOGYSIAM JOURNAL ON COMPUTINGJournal of Software-Evolution and ProcessIET OptoelectronicsCOMPUTATIONAL INTELLIGENCEIET Radar Sonar and NavigationCOMPUTERS & GRAPHICS-UKINTERNATIONAL JOURNAL OF HUMAN-COMPUTER INTERACTIONIEEE Transactions on Learning TechnologiesAPPLIED COMPUTATIONAL ELECTROMAGNETICS SOCIETY JOURNALACM TRANSACTIONS ON PROGRAMMING LANGUAGES AND SYSTEMS COMPUTER AIDED GEOMETRIC DESIGNCOMPUTERS & ELECTRICAL ENGINEERINGCOMPUTERS & ELECTRICAL ENGINEERINGJOURNAL OF DATABASE MANAGEMENTJOURNAL OF DATABASE MANAGEMENTBEHAVIOUR & INFORMATION TECHNOLOGYACM Transactions on Multimedia Computing Communications and ApplicationsACM Transactions on Multimedia Computing Communications and ApplicationsACM Transactions on Multimedia Computing Communications and Applications MULTIMEDIA TOOLS AND APPLICATIONSMULTIMEDIA TOOLS AND APPLICATIONSMULTIMEDIA TOOLS AND APPLICATIONSIEEE Computer Architecture LettersCognitive Systems ResearchSOFTWARE-PRACTICE & EXPERIENCEScientific ProgrammingGRAPHICAL MODELSCYBERNETICS AND SYSTEMSACM SIGCOMM Computer Communication ReviewInformation VisualizationPRESENCE-TELEOPERATORS AND VIRTUAL ENVIRONMENTSPRESENCE-TELEOPERATORS AND VIRTUAL ENVIRONMENTSACM TRANSACTIONS ON DATABASE SYSTEMSACM TRANSACTIONS ON DATABASE SYSTEMSOptical Switching and NetworkingOptical Switching and NetworkingJOURNAL OF FUNCTIONAL PROGRAMMINGACM TRANSACTIONS ON COMPUTER SYSTEMSACM Transactions on Computer-Human InteractionACM Transactions on Computer-Human InteractionComputers and ConcreteETRI JOURNALIEEE SECURITY & PRIVACYIEEE SECURITY & PRIVACYR JournalCOMPUTINGVISUAL COMPUTERACM Transactions on StorageACM Transactions on StorageCONSTRAINTSCONSTRAINTSOPTIMIZATION METHODS & SOFTWAREINFORMATION SYSTEMS FRONTIERSINFORMATION SYSTEMS FRONTIERSEUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS PERFORMANCE EVALUATIONPERFORMANCE EVALUATIONIET Microwaves Antennas & PropagationACM Transactions on Modeling and Computer SimulationJOURNAL OF INTELLIGENT & ROBOTIC SYSTEMSACM Transactions on Embedded Computing SystemsACM Transactions on Embedded Computing SystemsCIN-COMPUTERS INFORMATICS NURSINGCOMPUTER JOURNALCOMPUTER JOURNALCOMPUTER JOURNALCOMPUTER JOURNALENGINEERING WITH COMPUTERSDESIGNS CODES AND CRYPTOGRAPHYNETWORKSNETWORK-COMPUTATION IN NEURAL SYSTEMSInformation Technology and ControlInformation Technology and ControlIETE TECHNICAL REVIEWInternational Journal of Data Warehousing and MiningJOURNAL OF SUPERCOMPUTINGJOURNAL OF SUPERCOMPUTINGNEW GENERATION COMPUTINGNEW GENERATION COMPUTINGWIRELESS NETWORKSWIRELESS NETWORKSPATTERN ANALYSIS AND APPLICATIONSJOURNAL OF INTELLIGENT & FUZZY SYSTEMS CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE ACM Transactions on Internet TechnologyACM Transactions on Internet TechnologyInternational Journal of Ad Hoc and Ubiquitous ComputingInternational Journal of Ad Hoc and Ubiquitous ComputingACM Transactions on Computational LogicACM Transactions on Information and System SecurityCluster Computing-The Journal of Networks Software Tools and ApplicationsCluster Computing-The Journal of Networks Software Tools and Applications INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSEURASIP Journal on Wireless Communications and NetworkingEXPERT SYSTEMSEXPERT SYSTEMSINTERNATIONAL JOURNAL OF RF AND MICROWAVE COMPUTER-AIDED ENG INTERNATIONAL JOURNAL OF GENERAL SYSTEMSINFORMATION RETRIEVALMODELING IDENTIFICATION AND CONTROLCONNECTION SCIENCECONNECTION SCIENCESOFTWARE QUALITY JOURNALSIMULATION-TRANSACTIONS OF THE SOCIETY FOR MODELING AND SIMULA SIMULATION-TRANSACTIONS OF THE SOCIETY FOR MODELING AND SIMULA Digital InvestigationDigital InvestigationDISTRIBUTED AND PARALLEL DATABASESDISTRIBUTED AND PARALLEL DATABASESKNOWLEDGE ENGINEERING REVIEWSIGMOD RECORDSIGMOD RECORDJOURNAL OF INTELLIGENT INFORMATION SYSTEMSJOURNAL OF INTELLIGENT INFORMATION SYSTEMSInternational Journal of Wavelets Multiresolution and Information Processing International Journal on Document Analysis and RecognitionJOURNAL OF COMPUTER INFORMATION SYSTEMSINTERNATIONAL JOURNAL OF COOPERATIVE INFORMATION SYSTEMSAI MAGAZINEIET Computer VisionACM Journal on Emerging Technologies in Computing SystemsDISTRIBUTED COMPUTINGInternational Journal of Antennas and PropagationJOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTINGJOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTINGBELL LABS TECHNICAL JOURNALBELL LABS TECHNICAL JOURNALJournal on Multimodal User InterfacesJournal on Multimodal User InterfacesCryptography and Communications-Discrete-Structures Boolean Functions and Sequences Science China-Information SciencesAPPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTIN APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTIN IBM JOURNAL OF RESEARCH AND DEVELOPMENTIBM JOURNAL OF RESEARCH AND DEVELOPMENTIBM JOURNAL OF RESEARCH AND DEVELOPMENTIBM JOURNAL OF RESEARCH AND DEVELOPMENTREAL-TIME SYSTEMSCOMPUTER MUSIC JOURNALWIRELESS PERSONAL COMMUNICATIONSINFORMATION AND COMPUTATIONJOURNAL OF SYSTEMS ARCHITECTUREJOURNAL OF SYSTEMS ARCHITECTUREACM TRANSACTIONS ON DESIGN AUTOMATION OF ELECTRONIC SYSTEMS ACM TRANSACTIONS ON DESIGN AUTOMATION OF ELECTRONIC SYSTEMS International Journal of Distributed Sensor NetworksInternational Journal of Distributed Sensor NetworksTHEORY AND PRACTICE OF LOGIC PROGRAMMINGTHEORY AND PRACTICE OF LOGIC PROGRAMMINGACM Transactions on Architecture and Code OptimizationACM Transactions on Architecture and Code OptimizationInternational Journal of Information SecurityInternational Journal of Information SecurityInternational Journal of Information SecurityAEU-INTERNATIONAL JOURNAL OF ELECTRONICS AND COMMUNICATIONS Natural ComputingNatural ComputingJOURNAL OF SYMBOLIC COMPUTATIONJournal of Signal Processing Systems for Signal Image and Video Technology MULTIMEDIA SYSTEMSMULTIMEDIA SYSTEMSMATHEMATICAL STRUCTURES IN COMPUTER SCIENCEJOURNAL OF LOGIC AND COMPUTATIONINFORMATION SYSTEMS MANAGEMENTAdvances in Electrical and Computer EngineeringComputer Science and Information SystemsComputer Science and Information SystemsJOURNAL OF AUTOMATED REASONINGAI EDAM-ARTIFICIAL INTELLIGENCE FOR ENGINEERING DESIGN ANALYSIS AI EDAM-ARTIFICIAL INTELLIGENCE FOR ENGINEERING DESIGN ANALYSIS INTERNATIONAL JOURNAL OF PATTERN RECOGNITION AND ARTIFICIAL INT SCIENCE OF COMPUTER PROGRAMMINGComputational and Mathematical Organization TheoryMICROPROCESSORS AND MICROSYSTEMSMICROPROCESSORS AND MICROSYSTEMSAdvances in Mathematics of CommunicationsJOURNAL OF VISUAL LANGUAGES AND COMPUTINGJournal of SimulationJOURNAL OF COMPUTER SCIENCE AND TECHNOLOGYJOURNAL OF COMPUTER SCIENCE AND TECHNOLOGYPHOTONIC NETWORK COMMUNICATIONSPHOTONIC NETWORK COMMUNICATIONSDISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE THEORETICAL COMPUTER SCIENCEALGORITHMICAASLIB PROCEEDINGSKYBERNETIKAANNALES DES TELECOMMUNICATIONS-ANNALS OF TELECOMMUNICATION INTEGRATION-THE VLSI JOURNALInternational Journal of Critical Infrastructure ProtectionACM Transactions on AlgorithmsInternational Journal of Computers Communications & ControlFORMAL ASPECTS OF COMPUTINGJOURNAL OF UNIVERSAL COMPUTER SCIENCEJOURNAL OF UNIVERSAL COMPUTER SCIENCECONCURRENT ENGINEERING-RESEARCH AND APPLICATIONSANALOG INTEGRATED CIRCUITS AND SIGNAL PROCESSINGInternational Journal of Microwave and Wireless TechnologiesJOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING JOURNAL OF VISUALIZATIONIET SoftwareInternational Journal of Unconventional ComputingTurkish Journal of Electrical Engineering and Computer SciencesLanguage Resources and EvaluationIEEE ANNALS OF THE HISTORY OF COMPUTINGINTERNATIONAL JOURNAL OF PARALLEL PROGRAMMINGIT ProfessionalIT ProfessionalIT ProfessionalJOURNAL OF NEW MUSIC RESEARCHINFORMATION PROCESSING LETTERSIntelligent Data AnalysisNeural Network WorldJOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMINGACM SIGPLAN NOTICESAI COMMUNICATIONSJournal of Internet TechnologyJournal of Internet TechnologyIET Information SecurityIET Information SecurityJournal of Applied LogicJournal of Applied LogicFORMAL METHODS IN SYSTEM DESIGNTHEORY OF COMPUTING SYSTEMSNatural Language EngineeringVIRTUAL REALITYVIRTUAL REALITYInternational Journal of Computational Intelligence SystemsInternational Journal of Computational Intelligence SystemsJOURNAL OF COMMUNICATIONS AND NETWORKSJOURNAL OF COMMUNICATIONS AND NETWORKSMINDS AND MACHINESACTA INFORMATICAJournal of Network and Systems ManagementJournal of Network and Systems ManagementAd Hoc & Sensor Wireless NetworksAd Hoc & Sensor Wireless NetworksKSII Transactions on Internet and Information SystemsKSII Transactions on Internet and Information SystemsLogical Methods in Computer ScienceUniversal Access in the Information SocietyCOMPUTER ANIMATION AND VIRTUAL WORLDSInternational Journal of Network ManagementInternational Journal of Network ManagementJOURNAL OF ORGANIZATIONAL COMPUTING AND ELECTRONIC COMMERCE JOURNAL OF ORGANIZATIONAL COMPUTING AND ELECTRONIC COMMERCE FUNDAMENTA INFORMATICAEADVANCES IN COMPUTERSADVANCES IN COMPUTERSAPPLIED ARTIFICIAL INTELLIGENCEJOURNAL OF EXPERIMENTAL & THEORETICAL ARTIFICIAL INTELLIGENCE PROGRAM-ELECTRONIC LIBRARY AND INFORMATION SYSTEMSSecurity and Communication NetworksSecurity and Communication NetworksProblems of Information TransmissionNew Review of Hypermedia and MultimediaINTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE COMPUTER APPLICATIONS IN ENGINEERING EDUCATIONIET Computers and Digital TechniquesIET Computers and Digital TechniquesJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Frontiers of Computer ScienceFrontiers of Computer ScienceFrontiers of Computer SciencePeer-to-Peer Networking and ApplicationsPeer-to-Peer Networking and ApplicationsANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCEChina CommunicationsCOMPUTER LANGUAGES SYSTEMS & STRUCTURESCOMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEM International Journal of Web Services ResearchInternational Journal of Web Services ResearchJournal of Organizational and End User ComputingJournal of Zhejiang University-SCIENCE C-Computers & ElectronicsJournal of Zhejiang University-SCIENCE C-Computers & ElectronicsKYBERNETESJournal of Cellular AutomataIEICE TRANSACTIONS ON COMMUNICATIONSRomanian Journal of Information Science and TechnologyInternational Arab Journal of InformationInternational Arab Journal of InformationCANADIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING-REVU JOURNAL OF CIRCUITS SYSTEMS AND COMPUTERSMalaysian Journal of Computer ScienceMalaysian Journal of Computer ScienceCOMPUTING AND INFORMATICSJOURNAL OF INFORMATION SCIENCE AND ENGINEERINGJournal of Web EngineeringJournal of Web EngineeringInternational Journal on Artificial Intelligence ToolsInternational Journal on Artificial Intelligence ToolsRAIRO-THEORETICAL INFORMATICS AND APPLICATIONSIEEE Latin America TransactionsDESIGN AUTOMATION FOR EMBEDDED SYSTEMSDESIGN AUTOMATION FOR EMBEDDED SYSTEMSIEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICAT IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICAT INTERNATIONAL JOURNAL OF SOFTWARE ENGINEERING AND KNOWLEDGE INTERNATIONAL JOURNAL OF SOFTWARE ENGINEERING AND KNOWLEDGE COMPUTER SYSTEMS SCIENCE AND ENGINEERINGCOMPUTER SYSTEMS SCIENCE AND ENGINEERINGJOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONALIETE JOURNAL OF RESEARCHMICROWAVE JOURNALIEICE TRANSACTIONS ON INFORMATION AND SYSTEMSIEICE TRANSACTIONS ON INFORMATION AND SYSTEMSCRYPTOLOGIAINTELLIGENT AUTOMATION AND SOFT COMPUTINGPROGRAMMING AND COMPUTER SOFTWARETraitement du SignalJournal of the Institute of Telecommunications ProfessionalsMICROWAVES & RFELECTRONICS WORLDCOMPUTER GRAPHICS WORLDJCR期刊简称ISSN所属小类所属小类(中文)小类分区大类复分大类分区ACM T INTEL SYST TEC2157-6904C OMPUTER SCIENCE, ARTI计算机：人工智能1N1 ACM T INTEL SYST TEC2157-6904C OMPUTER SCIENCE, INFO计算机：信息系统1N1 IEEE COMMUN SURV TUT1553-877X T ELECOMMUNICATIONS电信学1N1 IEEE COMMUN SURV TUT1553-877X C OMPUTER SCIENCE, INFO计算机：信息系统1N1 IEEE T FUZZY SYST1063-6706C OMPUTER SCIENCE, ARTI计算机：人工智能1N1 IEEE T PATTERN ANAL0162-8828C OMPUTER SCIENCE, ARTI计算机：人工智能1N1 INT J NEURAL SYST0129-0657C OMPUTER SCIENCE, ARTI计算机：人工智能1N1 IEEE T IND INFORM1551-3203C OMPUTER SCIENCE, INTE计算机：跨学科应1N1 MIS QUART0276-7783C OMPUTER SCIENCE, INFO计算机：信息系统1N1 IEEE T EVOLUT COMPUT1089-778X C OMPUTER SCIENCE, THEO计算机：理论方法1N1 IEEE T EVOLUT COMPUT1089-778X C OMPUTER SCIENCE, ARTI计算机：人工智能1N1 COMPUT-AIDED CIV INF1093-9687C OMPUTER SCIENCE, INTE计算机：跨学科应1N1 IEEE WIREL COMMUN1536-1284T ELECOMMUNICATIONS电信学1N1 IEEE WIREL COMMUN1536-1284C OMPUTER SCIENCE, INFO计算机：信息系统1N1 IEEE WIREL COMMUN1536-1284C OMPUTER SCIENCE, HAR计算机：硬件1N1 J STAT SOFTW1548-7660C OMPUTER SCIENCE, INTE计算机：跨学科应1N1 IEEE T NEUR NET LEAR2162-237X C OMPUTER SCIENCE, THEO计算机：理论方法1N1 IEEE T NEUR NET LEAR2162-237X C OMPUTER SCIENCE, ARTI计算机：人工智能1N1 IEEE T NEUR NET LEAR2162-237X C OMPUTER SCIENCE, HAR计算机：硬件1N1 MED IMAGE ANAL1361-8415C OMPUTER SCIENCE, INTE计算机：跨学科应1N1 MED IMAGE ANAL1361-8415C OMPUTER SCIENCE, ARTI计算机：人工智能2N1 ACM COMPUT SURV0360-0300C OMPUTER SCIENCE, THEO计算机：理论方法1N1 IEEE COMMUN MAG0163-6804T ELECOMMUNICATIONS电信学1N2 INTEGR COMPUT-AID E1069-2509C OMPUTER SCIENCE, INTE计算机：跨学科应2N2 INTEGR COMPUT-AID E1069-2509C OMPUTER SCIENCE, ARTI计算机：人工智能2N2 ENVIRON MODELL SOFTW1364-8152C OMPUTER SCIENCE, INTE计算机：跨学科应2N2 INT J COMPUT VISION0920-5691C OMPUTER SCIENCE, ARTI计算机：人工智能2N2 IEEE COMPUT INTELL M1556-603X C OMPUTER SCIENCE, ARTI计算机：人工智能2N2 IEEE J SEL AREA COMM0733-8716T ELECOMMUNICATIONS电信学1N2 ACM T GRAPHIC0730-0301C OMPUTER SCIENCE, SOFT计算机：软件工程1N2 SIAM J IMAGING SCI1936-4954C OMPUTER SCIENCE, ARTI计算机：人工智能2N2 SIAM J IMAGING SCI1936-4954C OMPUTER SCIENCE, SOFT计算机：软件工程1N2 IEEE T AFFECT COMPUT1949-3045C OMPUTER SCIENCE, CYB计算机：控制论1N2。

lec6

e∈δ (v ) △
xe ≤ 1,
where δ (v ) is the set of edges incident on vertex v . Thus our ﬁrst attempt at a linear description of P is xe ≥ 0 ∀e ∈ E . P1 = (xe ∈ R)e∈E : x(δ (v )) ≤ 1 ∀v ∈ V Since P1 is a convex subset of R|E | and χ(M ) ∈ P1 for each matching M , it follows from the deﬁnition of convex hull that P ⊆ P1 . However, as illustrated by the following example, P P1 in general since P1 can have non-integral extreme points. Consider the triangle (K3 )—its matching polytope is P = conv {(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)}. The point (0.5, 0.5, 0.5) ∈ P1 , i.e. it satisﬁes the constraints above; however it is not in the convex hull of the matching vectors. The above example motivates the following family of additional constraints (introduced by Edmonds). Observe that for any matching M , the subgraph induced by M on any odd cardinality vertex subset U has at most (|U | − 1)/2 edges. Thus, without losing any of the matchings, we can introduce the following additional constraints: x(E (U )) =

Kim-Chuan Toh

Abstract. In each iteration of an interior-point method for semide nite programming,
Key words. Semide nite programming, step-length, Lanczos iteration.
Department of Mathematics National University of Singapore 10 Kent Ridge Crescent, Singapore 119260, Singapore e-mail: mattohkc@.sg
1 Introduction
Suppose X is an n n symmetric positive de nite matrix that is the current primal iterate of an interior point method in semide nite programming (SDP), and X is the search direction to be taken. To de ne the next iterate, we need to estimate the maximum value the step-length can take while satisfying the positive semide niteness condition X + X 0. In a primal-dual interior-point method, the same need to be done for the dual variable Z , but we shall concentrate only on the primal variable X in this note. We refer the reader to 1], 3], 8], 10], 16], among others, for details on interior-point methods for SDP. Let max denotes the maximum allowed step-length. So far as we are aware of, there are two methods used to estimate max in all the current implementations of interior-point methods for SDP; see 2], 5], 6], 7], 8], 14] and 17]. The rst method to estimate max is to check the positive semide niteness condition via Cholesky factorization by backtracking. That is, starting with = 1, the method successively reduce by a xed factor (say, 0:8) until the condition is satis ed. The resulting value of is then a lower bound for max. The second method is to compute max exactly by nding the maximum eigenvalue of an n n symmetric matrix via the QR algorithm 9]. The mechanism is as follows. Suppose X = R R is the Cholesky factorization of X . Let

《人工智能》课程习题

《人工智能》课程习题第一章绪论1-1. 什么是人工智能？试从学科和能力两方面加以说明。

1-2. 在人工智能的发展过程中，有哪些思想和思潮起了重要作用？1-3. 为什么能够用机器（计算机）模仿人的智能？1-4. 现在人工智能有哪些学派？它们的认知观是什么？1-5. 你认为应从哪些层次对认知行为进行研究？1-6. 人工智能的主要研究和应用领域是什么？其中，哪些是新的研究热点？第二章知识表示方法2-1状态空间法、问题归约法、谓词逻辑法和语义网络法的要点是什么？它们有何本质上的联系及异同点?2-2设有3个传教士和3个野人来到河边，打算乘一只船从右岸渡到左岸去。

该船的负载能力为两人。

在任何时候，如果野人人数超过传教士人数，那么野人就会把传教士吃掉。

他们怎样才能用这条船安全地把所有人都渡过河去?再定义描述过河方案的谓词：L-R(x, x1, y, y1,S)：x1个修道士和y1个野人渡船从河的左岸到河的右岸条件：Safety(L,x-x1,y-y1,S’)∧Safety(R,3-x+x1,3-y+y1,S’)∧Boat(L,S)动作：Safety(L,x-x1,y-y1,S’)∧Safety(R,3-x+x1,3-y+y1,S’)∧Boat(R,S’)R-L (x, x1, y, y1,S)：x2个修道士和y2个野人渡船从河的左岸到河的右岸条件：Safety(R,3-x-x2,3-y-y2,S’)∧Safety(L,x+x2,y+y2,S’)∧Boat(R,S)动作：Safety(R,3-x-x2,3-y-y2,S’)∧Safety(L,x+x2,y+y2,S’)∧Boat(L,S’)(2) 过河方案Safety(L,3,3,S0)∧Safety(R,0,0,S0)∧Boat(L,S0)L-R(3, 1, 3, 1,S0) L-R(3, 0, 3, 2,S0)Safety(L,2,2,S1)∧Safety(R,1,1,S1)∧Boat(R,S1)Safety(L,3,1,S1’)∧Safety(R,0,2,S1’)∧Boat(R,S1’)R-L (2, 1, 2, 0,S1) R-L (3,0, 1, 1,S1’)Safety(L,3,2,S2)∧Safety(R,0,1,S2)∧Boat(L,S2)L-R(3, 0, 2, 2,S2)Safety(L,3,0,S3)∧Safety(R,0,3,S3)∧Boat(R,S3)R-L (3, 0, 0, 1,S3)Safety(L,3,1,S4)∧Safety(R,0,2,S1)∧Boat(L,S4)L-R(3, 2, 1, 0,S4)Safety(L,1,1,S5)∧Safety(R,2,2,S5)∧Boat(R,S5)R-L (1, 1, 1, 1,S5)Safety(L,2,2,S6)∧Safety(R,1,1,S6)∧Boat(L,S6)L-R(2, 2, 2, 0,S6)Safety(L,0,2,S7)∧Safety(R,3,1,S7)∧Boat(R,S7)R-L (0, 0, 2, 1,S7)Safety(L,0,3,S8)∧Safety(R,3,0,S8)∧Boat(L,S8)L-R(0, 0, 3, 2,S8)Safety(L,0,1,S9)∧Safety(R,3,2,S9)∧Boat(R,S9)R-L (0, 1, 1, 0,S9)Safety(L,1,1,S10)∧Safety(R,2,2,S10)∧Boat(L,S10)2-3利用图2.3，用状态空间法规划一个最短的旅行路程：此旅程从城市A开始，访问其他城市不多于一次，并返回A。

TGEF(3.0)

. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
TGFF can generate visual graphs in both EPS and VCG formats. The EPS ﬁles should be readable by any postscript viewing program. The VCG format ﬁles require the graph visualization program VCG to view the ﬁles, but provides color and better zoom ability. VCG is a very useful graph visualization program which can be found at: • http://rw4.cs.uni-sb.de/users/sander/html/gsvcg1.html. TGFF is still useful without this program, but I highly recommend using it. If you are unfamiliar with VCG, the basic keys you need to know are: • ‘=’: Zoom in • ‘-’: Zoom out • ‘q’: Quit • Arrows: Move the graph left, right, up and down

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

15-854:ApproximationsAlgorithmsLecturer:AnupamGuptaTopic:Max-cut,HardnessofApproximationsDate:9/14Scribe:HaowenChan

S=V\S,andletE(S,S.Wewillusetheterm“cut”torefervariouslytothepartition(S,S)crossingbetweenSand

S)|.IntheweightedversionofMax-Cut,wearealsogivenaedgeweightfunctionw:E→R,andtheproblemistoﬁndacutwithmaximumweight.TheMax-CutproblemisNP-hard,andisinterestingtocontrastwiththeMin-Cutproblem,whichissolvableinpolynomialtime(e.g.,byreducingtothes-tmin-cutproblem,whichisdualtotheMax-Flowproblem,andissolvablebyalgorithmssuchasonebyEdmondsandKarp[1].

2.1.1ApproxMax-CutusingLocalSearchTHegeneralideainlocalsearchalgorithmsisthesame:Startwithasolution,ﬁndanimprovingsteptomakeabettersolution,andrepeatuntilstuck.Inmoredetail:

1.StartwithsomearbitraryS0⊆V.2.Instepi,letourcurrentcutbedenotedbySi.3.If∃v∈Sis.t.movingvto

Si,i.e.If∃v∈Si).Theorem2.1.1ThelocalsearchalgorithmforMax-Cutterminates.Proof:Eachstepmustincrease|E(Si,Si)|≤|E|=m,thealgorithmterminatesinatmostmsteps.

2edges.Proof:LetthecutatthelocaloptimumbeS.Consideranarbitraryvertexvwithdegreedv.Supposev∈S;anidenticalargumentappliesifv∈S)),whiletheotheredgeshavetheirother

1endpointinStogetabettercut.Hence,numberofedgesadjacenttovthatarecrossingthecut≥dv

2󰀁v∈Vv’sedgescrossingthecut

≥12

=2m

opt(I)Formaximizationproblems,wecandeﬁnetheapproximationratiosimilarly:

ρA=maxinstancesI

opt(I)

c(A(I))≤αforallinstancesI,thenwehaveρA≤α.Theorem2.1.3Thelocalsearchalgorithmdescribedaboveisa2-approximationalgorithm.Proof:ItiseasytoseethatthenumberofedgesmisanupperboundonMax-Cut(G),sincethemaximumcutcan’thavemoreedgesthanexistinthegraph.Sincethecutproducedbythelocalsearchalgorithmalwayshasatleastm

c(A(I))≤UBD(I)m/2=2.(2.1.2)2.1.2Max-CutusingGreedyAlgorithmFixanorderingontheverticesv1,v2,...,vn.StartwithtwoemptybinsS,S)|ismaximized.

Theorem2.1.4Thegreedyalgorithmisa2-approximationalgorithmforMax-Cut.Proof:Consideranyedgeinthegraph.Onevertexmustcomeﬁrstintheordering.Callthelatervertexresponsiblefortheedge.

Letribethenumberofedgesthatviisresponsiblefor.Sinceeveryedgehasexactlyoneresponsiblevertex,󰀁ni=1ri=m.

Claim:Whenviisadded,atleastri

S.Thesetwhichcontainsthemostendpointverticesoftheseriedgesmustcontainatleastri

2edgestothecut.

Numberofedgesinﬁnalcut=󰀁ni=1numberofedgesaddedbyeachresponsiblevertex≥󰀁n

i=1r

2Sinceopt(I)≤m,thealgorithmisa2-approximation.

Suniformlyatrandom(i.e.,thereisaprobability1

2Proof:Letusnumbertheedges1tom.DeﬁneanindicatorvariableXiforeachedgeis.t.Xi=1iftheedgecrossesthecutandXi=0iftheedgedoesn’tcrossthecut.Sinceweassignedtheverticesindependentlyrandomly,probabilitythatbothendpointsofiareinthesameset=probabilitythattheendpointsareindiﬀerentsets=12.Expectedtotalnumber

ofedgescrossingthecut=󰀁mi=1E[Xi]=mHencetherandomalgorithmisa2-approxalgorithmonexpectation.

2.1.4ImprovingtheAlgorithm2.1.4.1HowgoodwastheupperboundUBD(I)=m?Itisusuallyagoodideatothinkabouthowgoodthelowerboundsare,whentryingtolookforbetterapproximationratios.Inthiscase,wecanshowinstanceswhereopt(I)≈m2UBD(I).

ConsiderK2n,thecompletegraphon2nnodes:thishasm=n(2n−1)edges.Foranycutwith|S|=i,|theverticesinOPT=2−1

S.Hencethegreedyalgorithmalsoachievesacutof2n2edgesintheworstcaseforthisinstance.

TheBest-knownApproximationAlgorithmforMax-CutIn1994,GoemansandWilliamson[2]usedaSemideﬁniteprogrammingrelaxationoftheMax-CutproblmetoobtainanperformanceguaranteeofρGW=12.2ProvingInapproximabilityRecallthattoprovesome(decision)problemisNP-hard,wereducesomeNP-hardproblemtotheproblemathand.Forexample,considerthedecisionMax-Cutproblem:ForagivengraphGandgivenparameterk,doesthereexistacutofsize≥kinG?ToprovethatthisproblemisNP-hard,wecouldperformapolynomialtimereductionffromsomeNP-hardproblem(e.g.3SAT).

Intheaboveﬁgure,I≥kdenotesgraphinstanceswithasolutionofvaluekormore,whileI

denotesgraphinstanceswithasolutionofvaluelessthank.I.e.,thesetI≥kwouldcontainall

graphswithamax-cutofkedgesormore,whileIoflessthankedges.

Thereductionfmapsaninstanceof3SATintoagraphGsuchthatthe3SATexpressionissatisﬁableiﬀG∈I≥k.TheexistenceofthismappingshowsthatdecidingmembershipinI≥kisatleastashardasdecidingsatisﬁabilityin3SAT,henceitisNP-hard.

SofarthisapproachcanshowthataproblemisNP-hard.Toshowthattheproblemisinap-proximabletosomefactor,wemustnowshowthatourmappingcanmaptoprobleminstanceswherethemax-cutsatisﬁesacertain“gap-property”:wemustexhibitareductionfthatmapsaninstanceof3SATintoagraphGsuchthatG∈I≥k1iﬀthe3SATinstanceissatisﬁableandG∈Ik2,andthe“gap”referstothegapbetweenthevaluesk1andk2.)