Design and Analysis of Approximation Algorithms

This book is intended to be used as a textbook for graduate students studying theoretical computer science. It can also be used as a reference book for researchers in the area of design and analysis of approximation algorithms. Design and Analysis of Approximation Algorithms is a graduate course in theoretical computer science taught widely in the universities, both in the United States and abroad. There are, however, very few textbooks available for this course. Among those available in the market, most books follow a problem-oriented format; that is, they collected many important combinatorial optimization problems and their approximation algorithms, and organized them based on the types, or applications, of problems, such as geometric-type problems, algebraic-type problems, etc. Such arrangement of materials is perhaps convenient for a researcher to look for the problems and algorithms related to his/her work, but is difficult for a student to capture the ideas underlying the various algorithms. In the new book proposed here, we follow a more structured, technique-oriented presentation. We organize approximation algorithms into different chapters, based on the design techniques for the algorithms, so that the reader can study approximation algorithms of the same nature together. It helps the reader to better understand the design and analysis techniques for approximation algorithms, and also helps the teacher to present the ideas and techniques of approximation algorithms in a more unified way.

[1]  Teofilo F. Gonzalez,et al.  Bounds for partitioning rectilinear polygons , 1985, SCG '85.

[2]  Weili Wu,et al.  Improving Construction for Connected Dominating Set with Steiner Tree in Wireless Sensor Networks , 2006, J. Glob. Optim..

[3]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[4]  Uri Zwick,et al.  Computer assisted proof of optimal approximability results , 2002, SODA '02.

[5]  Sanjeev Mahajan,et al.  Derandomizing Approximation Algorithms Based on Semidefinite Programming , 1999, SIAM J. Comput..

[6]  Fan Chung,et al.  A Lower Bound for the Steiner Tree Problem , 1978 .

[7]  Deying Li,et al.  A polynomial‐time approximation scheme for the minimum‐connected dominating set in ad hoc wireless networks , 2003, Networks.

[8]  B. Korte,et al.  Worst case analysis of greedy type algorithms for independence systems , 1980 .

[9]  Rajeev Motwani,et al.  On syntactic versus computational views of approximability , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[10]  Teofilo F. Gonzalez,et al.  Inproved Bounds for Rectangular and Guillotine Partitions , 1989, J. Symb. Comput..

[11]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .

[12]  Zhi-Quan Luo,et al.  Approximation Algorithms for Quadratic Programming , 1998, J. Comb. Optim..

[13]  Klaus Jansen,et al.  Polynomial-time approximation schemes for geometric graphs , 2001, SODA '01.

[14]  Lusheng Wang,et al.  Improved Approximation Algorithms for Tree Alignment , 1996, J. Algorithms.

[15]  Andrzej Lingas.  Heuristics for minimum edge length rectangular partitions of rectilinear figures , 1983 .

[16]  Arthur L. Liestman,et al.  Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks , 2002, MobiHoc '02.

[17]  Stefan Hougardy,et al.  On approximation algorithms for the terminal Steiner tree problem , 2004, Inf. Process. Lett..

[18]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[19]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[20]  Deying Li,et al.  Construction of strongly connected dominating sets in asymmetric multihop wireless networks , 2009, Theor. Comput. Sci..

[21]  F. Frances Yao,et al.  Approximating shortest superstrings , 1997, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[22]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[23]  Weili Wu,et al.  Non-unique probe selection and group testing , 2007, Theor. Comput. Sci..

[24]  Clifford Stein,et al.  Long Tours and Short Superstrings (Preliminary Version) , 1994, FOCS 1994.

[25]  Uri Zwick,et al.  Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems , 1999, STOC '99.

[26]  Weili Wu,et al.  An Approximation for Minimum Multicast Route in Optical Networks with Nonsplitting Nodes , 2005, J. Comb. Optim..

[27]  Weili Wu,et al.  A Better Theoretical Bound to Approximate Connected Dominating Set in Unit Disk Graph , 2008, WASA.

[28]  Samir Khuller,et al.  Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets , 1998, FSTTCS.

[29]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[30]  Alex Zelikovsky.  An approximation algorithm for weighted itk-polymatroids and the Steiner tree problem in graphs , 1993, IPCO.

[31]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.

[32]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[33]  Reuven Bar-Yehuda,et al.  A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem , 1981, J. Algorithms.

[34]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[35]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[36]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[37]  Jiawei Zhang,et al.  An improved rounding method and semidefinite programming relaxation for graph partition , 2002, Math. Program..

[38]  Tao Jiang,et al.  A More Efficient Approximation Scheme for Tree Alignment , 2000, SIAM J. Comput..

[39]  Bing Lu,et al.  A Polynomial Time Approximation Scheme for the Problem of Interconnecting Highways , 2001, J. Comb. Optim..

[40]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[41]  Hsueh-I Lu,et al.  Space-Efficient Approximation Algorithms for MAXCUT and COLORING Semidefinite Programs , 1998, ISAAC.

[42]  Ryan O'Donnell,et al.  Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? , 2007, SIAM J. Comput..

[43]  Alan M. Frieze,et al.  Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION , 1995, IPCO.

[44]  J. Hyam Rubinstein,et al.  The Steiner ratio conjecture for six points , 1991, J. Comb. Theory A.

[45]  Jacques Stern,et al.  The hardness of approximate optima in lattices, codes, and systems of linear equations , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[46]  Wojciech Rytter,et al.  Parallel and Sequential Approximations of Shortest Superstrings , 1994, SWAT.

[47]  Juraj Hromkovic,et al.  An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality , 2000, STACS.

[48]  U. Zwick.  Analyzing the MAX 2-SAT and MAX DI-CUT approximation algorithms of Feige and Goemans , 2000 .

[49]  Mihalis Yannakakis,et al.  The Complexity of Multiterminal Cuts , 1994, SIAM J. Comput..

[50]  Jonathan S. Turner,et al.  Approximation Algorithms for the Shortest Common Superstring Problem , 1989, Inf. Comput..

[51]  Giorgio Ausiello,et al.  Structure Preserving Reductions among Convex Optimization Problems , 1980, J. Comput. Syst. Sci..

[52]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1998, JACM.

[53]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[54]  Uri Zwick,et al.  A 7/8-approximation algorithm for MAX 3SAT? , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[55]  Uriel Feige,et al.  Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[56]  Sanjeev Khanna,et al.  On approximating rectangle tiling and packing , 1998, SODA '98.

[57]  Dror Rawitz,et al.  Local ratio: A unified framework for approximation algorithms. In Memoriam: Shimon Even 1935-2004 , 2004, CSUR.

[58]  Giri Narasimhan,et al.  Resource-constrained geometric network optimization , 1998, SCG '98.

[59]  Mihir Bellare,et al.  Free bits, PCPs and non-approximability-towards tight results , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[60]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[61]  Erich Prisner,et al.  Two algorithms for the subset interconnection design problem , 1992, Networks.

[62]  Dimitris Bertsimas,et al.  On Dependent Randomized Rounding Algorithms , 1996, IPCO.

[63]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[64]  Yinyu Ye,et al.  A .699-Approximation Algorithm for Max-Bisection , 1999 .

[65]  M. Charikar,et al.  An Optimal Bifactor Approximation Algorithm for the Metric Uncapacitated Facility Location Problem , 2006, SIAM J. Comput..

[66]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[67]  Ding-Zhu Du,et al.  Problem solving in automata, languages, and complexity , 2004, IEEE Circuits and Devices Magazine.

[68]  Ding-Zhu Du,et al.  Connected Dominating Sets in Wireless Networks with Different Transmission Ranges , 2007, IEEE Transactions on Mobile Computing.

[69]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[70]  Dror Rawitz,et al.  On the Equivalence between the Primal-Dual Schema and the Local-Ratio Technique , 2001, RANDOM-APPROX.

[71]  Samir Khuller,et al.  The Budgeted Maximum Coverage Problem , 1999, Inf. Process. Lett..

[72]  Magnús M. Halldórsson,et al.  A Still Better Performance Guarantee for Approximate Graph Coloring , 1993, Information Processing Letters.

[73]  Miroslav Chlebík,et al.  Approximation Hardness of the Steiner Tree Problem on Graphs , 2002, SWAT.

[74]  Weili Wu,et al.  Algorithms for connected set cover problem and fault-tolerant connected set cover problem , 2009, Theor. Comput. Sci..

[75]  Magnús M. Halldórsson,et al.  Approximating the Minimum Maximal Independence Number , 1993, Inf. Process. Lett..

[76]  Donghyun Kim,et al.  New dominating sets in social networks , 2010, J. Glob. Optim..

[77]  Roger Wattenhofer,et al.  Wireless Communication Is in APX , 2009, ICALP.

[78]  Ronald L. Graham,et al.  A NEW BOUND FOR EUCLIDEAN STEINER MINIMAL TREES , 1985 .

[79]  David P. Williamson.  The primal-dual method for approximation algorithms , 2002, Math. Program..

[80]  Ding-Zhu Du,et al.  On heuristics for minimum length rectilinear partitions , 2005, Algorithmica.

[81]  Changyuan Yu,et al.  A 5+epsilon-approximation algorithm for minimum weighted dominating set in unit disk graph , 2009, Theor. Comput. Sci..

[82]  Toshihiro Fujito.  On approximation of the submodular set cover problem , 1999, Oper. Res. Lett..

[83]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[84]  Samir Khuller,et al.  Greedy strikes back: improved facility location algorithms , 1998, SODA '98.

[85]  Yi Zhu,et al.  Efficient Distributed Algorithms for Topology Control Problem with Shortest Path Constraints , 2009, Discret. Math. Algorithms Appl..

[86]  Santosh S. Vempala,et al.  Network Design via Iterative Rounding of Setpair Relaxations , 2022 .

[87]  Laurence A. Wolsey,et al.  An analysis of the greedy algorithm for the submodular set covering problem , 1982, Comb..

[88]  David P. Williamson,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[89]  A. A. Lazarev,et al.  An Approximation Scheme for the 1 , 2012 .

[90]  David P. Williamson,et al.  Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming , 2001, STOC '01.

[91]  Alex Zelikovsky,et al.  Improved Steiner tree approximation in graphs , 2000, SODA '00.

[92]  Eden Chlamtác,et al.  Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[93]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[94]  Mihir Bellare,et al.  On Chromatic Sums and Distributed Resource Allocation , 1998, Inf. Comput..

[95]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[96]  Francesco Maffioli,et al.  Approximation algorithms for maximum cut with limited unbalance , 2007, Theor. Comput. Sci..

[97]  Michael Segal,et al.  A simple improved distributed algorithm for minimum CDS in unit disk graphs , 2005, WiMob'2005), IEEE International Conference on Wireless And Mobile Computing, Networking And Communications, 2005..

[98]  George B. Dantzig,et al.  A PRIMAL--DUAL ALGORITHM , 1956 .

[99]  Toshihiro Fujito,et al.  On approximability of the independent/connected edge dominating set problems , 2000, Inf. Process. Lett..

[100]  Weili Wu,et al.  Approximations for Subset Interconnection Designs , 1998, Theor. Comput. Sci..

[101]  D. Hochbaum.  Approximating covering and packing problems: set cover, vertex cover, independent set, and related problems , 1996 .

[102]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[103]  Weili Wu,et al.  (6+epsilon)-Approximation for Minimum Weight Dominating Set in Unit Disk Graphs , 2008, COCOON.

[104]  Erik Jan van Leeuwen,et al.  Approximating geometric coverage problems , 2008, SODA '08.

[105]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[106]  Ding-Zhu Du,et al.  Matroids and Subset Interconnection Design , 1988, SIAM J. Discret. Math..

[107]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

[108]  Ding-Zhu Du,et al.  On greedy construction of connected dominating sets in wireless networks , 2005, Wirel. Commun. Mob. Comput..

[109]  Tao Jiang,et al.  An approximation scheme for some Steiner tree problems in the plane , 1994, Networks.

[110]  R. Courant,et al.  What Is Mathematics , 1943 .

[111]  Ion I. Mandoiu,et al.  A note on the MST heuristic for bounded edge-length Steiner trees with minimum number of Steiner points , 2000, Inf. Process. Lett..

[112]  Tao Jiang,et al.  Approximation algorithms for tree alignment with a given phylogeny , 1996, Algorithmica.

[113]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1993, STOC.

[114]  Harold N. Gabow,et al.  Iterated rounding algorithms for the smallest k-edge connected spanning subgraph , 2008, SODA '08.

[115]  Toshihiro Fujito,et al.  Submodular Integer Cover and Its Application to Production Planning , 2004, WAOA.

[116]  Kamal Jain,et al.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[117]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[118]  Leonard Pitt,et al.  A bounded approximation for the minimum cost 2-sat problem , 1992, Algorithmica.

[119]  Yinyu Ye,et al.  Linear Programming-Based Algorithms for the Fixed-Hub Single Allocation Problem , 2007 .

[120]  Sudipto Guha,et al.  Approximation algorithms for directed Steiner problems , 1999, SODA '98.

[121]  Panos M. Pardalos,et al.  Analysis of greedy approximations with nonsubmodular potential functions , 2008, SODA '08.

[122]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[123]  Ming Gu,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming in Finite Precision , 1999, SIAM J. Optim..

[124]  Henry Wolkowicz,et al.  Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem , 2002, Discret. Appl. Math..

[125]  Santosh S. Vempala,et al.  A Constant-Factor Approximation Algorithm for the Geometric k-MST Problem in the Plane , 1999, SIAM J. Comput..

[126]  Wolfgang Maass,et al.  Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[127]  F. Hwang.  On Steiner Minimal Trees with Rectilinear Distance , 1976 .

[128]  Mahtab Seddigh,et al.  Dominating Sets and Neighbor Elimination-Based Broadcasting Algorithms in Wireless Networks , 2002, IEEE Trans. Parallel Distributed Syst..

[129]  Martin Skutella,et al.  Convex quadratic and semidefinite programming relaxations in scheduling , 2001, JACM.

[130]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[131]  Piotr Berman,et al.  A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem , 1999, SIAM J. Discret. Math..

[132]  Uri Zwick,et al.  Coloring k-colorable graphs using smaller palettes , 2001, SODA '01.

[133]  Piotr Berman,et al.  Efficient Approximation Algorithms for Tiling and Packing Problems with Rectangles , 2001, J. Algorithms.

[134]  János Komlós,et al.  Probabilistic partitioning algorithms for the rectilinear steiner problem , 1985, Networks.

[135]  David P. Williamson,et al.  Approximating the smallest k‐edge connected spanning subgraph by LP‐rounding , 2005, SODA '05.

[136]  Jiawei Zhang,et al.  An approximation algorithm for scheduling two parallel machines with capacity constraints , 2003, Discrete Applied Mathematics.

[137]  Timothy M. Chan.  Polynomial-time approximation schemes for packing and piercing fat objects , 2003, J. Algorithms.

[138]  Guohui Lin,et al.  Steiner Tree Problem with Minimum Number of Steiner Points and Bounded Edge-Length , 1999, Inf. Process. Lett..

[139]  Robert Krauthgamer,et al.  Polylogarithmic inapproximability , 2003, STOC '03.

[140]  Uri Zwick,et al.  Approximation Algorithms for MAX 4-SAT and Rounding Procedures for Semidefinite Programs , 1999, J. Algorithms.

[141]  Deying Li,et al.  Minimum Power Strongly Connected Dominating Sets in Wireless Networks , 2008, ICWN.

[142]  Tao Jiang,et al.  An Approximation Scheme for Some Steiner Tree Problems in the Plane , 1994, ISAAC.

[143]  Man-Tak Shing,et al.  On optimal routing trees , 1988 .

[144]  Tao Jiang,et al.  Aligning sequences via an evolutionary tree: complexity and approximation , 1994, STOC '94.

[145]  Jing-Chao Chen,et al.  Iterative Rounding for the Closest String Problem , 2007, ArXiv.

[146]  Thomas Erlebach,et al.  Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs , 2006, APPROX-RANDOM.

[147]  Stephen A. Vavasis,et al.  Automatic Domain Partitioning in Three Dimensions , 1991, SIAM J. Sci. Comput..

[148]  Dror Rawitz,et al.  Combinatorial Interpretations of Dual Fitting and Primal Fitting , 2003, WAOA.

[149]  Christoph Ambühl,et al.  An Optimal Bound for the MST Algorithm to Compute Energy Efficient Broadcast Trees in Wireless Networks , 2005, ICALP.

[150]  Donghyun Kim,et al.  Construction of Minimum Connected Dominating Set in 3-Dimensional Wireless Network , 2008, WASA.

[151]  Tao Jiang,et al.  Linear approximation of shortest superstrings , 1991, STOC '91.

[152]  Weili Wu,et al.  A greedy approximation for minimum connected dominating sets , 2004, Theor. Comput. Sci..

[153]  Ding-Zhu Du,et al.  The k-Steiner Ratio in Graphs , 1997, SIAM J. Comput..

[154]  Sanjeev Arora,et al.  A combinatorial, primal-dual approach to semidefinite programs , 2007, STOC '07.

[155]  Timothy M. Chan.  A note on maximum independent sets in rectangle intersection graphs , 2004, Inf. Process. Lett..

[156]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[157]  Franz Rendl,et al.  Semidefinite Programming Relaxations for the Quadratic Assignment Problem , 1998, J. Comb. Optim..

[158]  Christos Levcopoulos.  Fast heuristics for minimum length rectangular partitions of polygons , 1986, SCG '86.

[159]  Bing Lu,et al.  Polynomial Time Approximation Scheme for Symmetric Rectilinear Steiner Arborescence Problem , 2001, J. Glob. Optim..

[160]  D. Sankoff.  Minimal Mutation Trees of Sequences , 1975 .

[161]  Uri Zwick,et al.  MAX CUT in cubic graphs , 2002, SODA '02.

[162]  Evangelos Markakis,et al.  Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP , 2002, JACM.

[163]  Mohammad Mahdian,et al.  Improved Approximation Algorithms for Metric Facility Location Problems , 2002, APPROX.

[164]  Q. Feng,et al.  On better heuristic for Euclidean Steiner minimum trees , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[165]  Uriel Feige.  A threshold of ln n for approximating set cover (preliminary version) , 1996, STOC '96.

[166]  David P. Williamson,et al.  A primal-dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs , 1998, Oper. Res. Lett..

[167]  Philip N. Klein,et al.  A polynomial-time approximation scheme for weighted planar graph TSP , 1998, SODA '98.

[168]  Dimitris Bertsimas,et al.  From valid inequalities to heuristics: a unified view of primal-dual approximation algorithms in covering problems , 1995, SODA '95.

[169]  Weili Wu,et al.  New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit disk graphs , 2011, Theor. Comput. Sci..

[170]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[171]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[172]  Hans Eriksson,et al.  MBONE: the multicast backbone , 1994, CACM.

[173]  Le Thi Hoai An,et al.  A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems , 1998, J. Comb. Optim..

[174]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[175]  Jiawei Zhang,et al.  Improved approximations for max set splitting and max NAE SAT , 2004, Discret. Appl. Math..

[176]  Toshihiro Fujito,et al.  A unified approximation algorithm problems ” , 1998 .

[177]  Michael Langberg,et al.  Approximation Algorithms for Maximization Problems Arising in Graph Partitioning , 2001, J. Algorithms.

[178]  R. Ravi,et al.  When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks , 1995, SIAM J. Comput..

[179]  R. Ravi,et al.  A nearly best-possible approximation algorithm for node-weighted Steiner trees , 1993, IPCO.

[180]  Weili Wu,et al.  Minimum connected dominating sets and maximal independent sets in unit disk graphs , 2006, Theor. Comput. Sci..

[181]  Peter Slavík.  A Tight Analysis of the Greedy Algorithm for Set Cover , 1997, J. Algorithms.

[182]  Etienne de Klerk,et al.  Polynomial Primal-Dual Affine Scaling Algorithms in Semidefinite Programming , 1998, J. Comb. Optim..

[183]  Richard M. Karp,et al.  Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane , 1977, Math. Oper. Res..

[184]  Reuven Bar-Yehuda,et al.  A unified approach to approximating resource allocation and scheduling , 2000, STOC '00.

[185]  Sanjeev Arora,et al.  Nearly Linear Time Approximation Schemes for Euclidean TSP and Other Geometric Problems , 1997, RANDOM.

[186]  E. D. Klerk,et al.  Aspects of semidefinite programming : interior point algorithms and selected applications , 2002 .

[187]  Shih-Cheng Yang,et al.  Approximating the selected-internal Steiner tree , 2007, Theor. Comput. Sci..

[188]  Uri Zwick,et al.  A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems , 2001, Random Struct. Algorithms.

[189]  Loren Schwiebert,et al.  Power efficient topologies for wireless sensor networks , 2001, International Conference on Parallel Processing, 2001..

[190]  Dániel Marx,et al.  Efficient Approximation Schemes for Geometric Problems? , 2005, ESA.

[191]  S. Guha,et al.  Approximation Algorithms for Connected Dominating Sets , 1998, Algorithmica.

[192]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[193]  L. Wolsey.  Heuristic analysis, linear programming and branch and bound , 1980 .

[194]  Esko Ukkonen,et al.  A Greedy Approximation Algorithm for Constructing Shortest Common Superstrings , 1988, Theor. Comput. Sci..

[195]  Panos M. Pardalos,et al.  Topics in Semidefinite and Interior-Point Methods , 1998 .

[196]  R. Graham,et al.  The steiner problem in phylogeny is NP-complete , 1982 .

[197]  Clifford Stein,et al.  Approximating Semidefinite Packing Programs , 2011, SIAM J. Optim..

[198]  W. R. Buckland,et al.  Distributions in Statistics: Continuous Multivariate Distributions , 1973 .

[199]  Sergiy Butenko,et al.  On minimum connected dominating set problem in unit-ball graphs , 2007 .

[200]  Judit Bar-Ilan,et al.  Generalized submodular cover problems and applications , 2001, Theor. Comput. Sci..

[201]  F. Frances Yao,et al.  Two-Phased Approximation Algorithms for Minimum CDS in Wireless Ad Hoc Networks , 2008, 2008 The 28th International Conference on Distributed Computing Systems.

[202]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[203]  Subhash Suri,et al.  Label placement by maximum independent set in rectangles , 1998, CCCG.

[204]  L. Wolsey.  Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems , 1982, Math. Oper. Res..

[205]  Uriel Feige,et al.  Approximating the domatic number , 2000, STOC '00.

[206]  Dror Rawitz,et al.  Using fractional primal-dual to schedule split intervals with demands , 2006, Discret. Optim..

[207]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

[208]  R. Ravi,et al.  When trees collide: an approximation algorithm for the generalized Steiner problem on networks , 1991, STOC '91.

[209]  Uri Zwick,et al.  Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint , 1998, SODA '98.

[210]  David P. Williamson,et al.  A primal-dual approximation algorithm for generalized steiner network problems , 1993, Comb..

[211]  Xiaohua Jia,et al.  A Note on Optical Network with Nonsplitting Nodes , 2005, J. Comb. Optim..

[212]  Lusheng Wang,et al.  An approximation scheme for some Steiner tree problems in the plane , 1994, Networks.

[213]  Reuven Bar-Yehuda,et al.  A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem , 1983, WG.

[214]  Decheng Dai,et al.  A 5 +-approximation algorithm for minimum weighted dominating set in unit disk graph , 2009 .

[215]  Satish Rao,et al.  Approximating geometrical graphs via “spanners” and “banyans” , 1998, STOC '98.

[216]  Samir Khuller,et al.  Approximation Algorithms for Connected Dominating Sets , 1996, Algorithmica.

[217]  Adam William Meyerson,et al.  Approximation algorithms for network design problems , 2002 .

[218]  David P. Williamson,et al.  An iterative rounding 2-approximation algorithm for the element connectivity problem , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[219]  B. Korte,et al.  An Analysis of the Greedy Heuristic for Independence Systems , 1978 .

[220]  Weili Wu,et al.  A PTAS for minimum connected dominating set in 3-dimensional Wireless sensor networks , 2009, J. Glob. Optim..

[221]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

[222]  Jitender S. Deogun,et al.  Routing in sparse splitting optical networks with multicast traffic , 2003, Comput. Networks.

[223]  Donghyun Kim,et al.  Two Constant Approximation Algorithms for Node-Weighted Steiner Tree in Unit Disk Graphs , 2008, COCOA.

[224]  Guohui Lin,et al.  On the terminal Steiner tree problem , 2002, Inf. Process. Lett..

[225]  Alfredo Navarra,et al.  Tighter bounds for the minimum energy broadcasting problem , 2005, Third International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt'05).

[226]  Donghyun Kim,et al.  Constructing Minimum Connected Dominating Sets with Bounded Diameters in Wireless Networks , 2009, IEEE Transactions on Parallel and Distributed Systems.

[227]  Chengxian Xu,et al.  Approximation algorithms for MAX RES CUT with limited unbalanced constraints , 2010 .

[228]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[229]  Jiawei Zhang,et al.  On approximation of max-vertex-cover , 2002, Eur. J. Oper. Res..

[230]  Weili Wu,et al.  A Special Case for Subset Interconnection Designs , 1997, Discret. Appl. Math..

[231]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[232]  Peng-Jun Wan,et al.  Message-optimal connected dominating sets in mobile ad hoc networks , 2002, MobiHoc '02.

[233]  Nisheeth K. Vishnoi,et al.  On the Optimality of a Class of LP-based Algorithms , 2009, Electron. Colloquium Comput. Complex..

[234]  Subhash Khot,et al.  Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[235]  Maxim Sviridenko,et al.  Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee , 2004, J. Comb. Optim..

[236]  R. Ravi,et al.  When cycles collapse: A general approximation technique for constrained two-connectivity problems , 1992, IPCO.

[237]  Noga Alon,et al.  Constructing worst case instances for semidefinite programming based approximation algorithms , 2001, SODA '01.

[238]  Éva Tardos,et al.  Algorithms for a network design problem with crossing supermodular demands , 2004, Networks.

[239]  Deborah Estrin,et al.  An architecture for wide-area multicast routing , 1994, SIGCOMM 1994.

[240]  Uri Zwick,et al.  Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems , 2002, IPCO.

[241]  George O. Wesolowsky,et al.  THE WEBER PROBLEM: HISTORY AND PERSPECTIVES. , 1993 .

[242]  David S. Johnson,et al.  The Complexity of Near-Optimal Graph Coloring , 1976, J. ACM.

[243]  Bing Lu,et al.  Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem , 2000, J. Comb. Optim..

[244]  Joseph S. B. Mitchell,et al.  Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem , 1996, SODA '96.

[245]  Sartaj Sahni,et al.  Approximate Algorithms for the 0/1 Knapsack Problem , 1975, JACM.

[246]  Vaduvur Bharghavan,et al.  Routing in ad-hoc networks using minimum connected dominating sets , 1997, Proceedings of ICC'97 - International Conference on Communications.

[247]  Y. Ye,et al.  Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics , 1998 .

[248]  D. Du,et al.  The Steiner ratio conjecture of Gilbert and Pollak is true. , 1990, Proceedings of the National Academy of Sciences of the United States of America.

[249]  Jiawei Zhang,et al.  Geometric rounding: a dependent randomized rounding scheme , 2011, J. Comb. Optim..

[250]  Ker-I Ko.  Computational complexity of real functions and polynomial time approximations , 1979 .

[251]  Dorit S. Hochbaum,et al.  Various notions of approximations: good, better, best, and more , 1996 .

[252]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .

[253]  Robert G. Bland,et al.  New Finite Pivoting Rules for the Simplex Method , 1977, Math. Oper. Res..

[254]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[255]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[256]  Rajiv Gandhi,et al.  Dependent rounding and its applications to approximation algorithms , 2006, JACM.

[257]  Alex Zelikovsky,et al.  A series of approximation algorithms for the acyclic directed steiner tree problem , 1997, Algorithmica.

[258]  Byrav Ramamurthy,et al.  Minimizing the number of optical amplifiers needed to support a multi-wavelength optical LAN/MAN , 1997, Proceedings of INFOCOM '97.

[259]  Ding-Zhu Du,et al.  Connected Domination in Multihop Ad Hoc Wireless Networks , 2002, JCIS.