## Approximation AlgorithmsAlthough this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872-1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed con jecture that P -=/= NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientific inquiry in computer science and mathematics. This book presents the theory of ap proximation algorithms as it stands today. It is reasonable to expect the picture to change with time. This book is divided into three parts. In Part I we cover combinato rial algorithms for a number of important problems, using a wide variety of algorithm design techniques. The latter may give Part I a non-cohesive appearance. However, this is to be expected - nature is very rich, and we cannot expect a few tricks to help solve the diverse collection of NP-hard problems. Indeed, in this part, we have purposely refrained from tightly cat egorizing algorithmic techniques so as not to trivialize matters. Instead, we have attempted to capture, as accurately as possible, the individual character of each problem, and point out connections between problems and algorithms for solving them. |

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### Contents

I | 1 |

II | 2 |

III | 3 |

V | 5 |

VI | 7 |

VII | 10 |

VIII | 15 |

IX | 16 |

CXX | 172 |

CXXI | 174 |

CXXII | 175 |

CXXIII | 176 |

CXXIV | 178 |

CXXV | 179 |

CXXVII | 180 |

CXXVIII | 182 |

X | 17 |

XI | 19 |

XII | 22 |

XIII | 26 |

XIV | 27 |

XVI | 28 |

XVII | 30 |

XVIII | 31 |

XIX | 32 |

XX | 33 |

XXI | 37 |

XXII | 38 |

XXIV | 40 |

XXV | 44 |

XXVI | 46 |

XXVII | 47 |

XXIX | 50 |

XXX | 52 |

XXXI | 53 |

XXXII | 54 |

XXXIV | 57 |

XXXV | 60 |

XXXVII | 61 |

XXXIX | 64 |

XL | 66 |

XLII | 67 |

XLIII | 68 |

XLIV | 69 |

XLVI | 71 |

XLVII | 72 |

XLIX | 73 |

L | 74 |

LII | 77 |

LIII | 78 |

LIV | 79 |

LVI | 80 |

LVII | 81 |

LIX | 83 |

LXI | 84 |

LXIII | 87 |

LXIV | 89 |

LXVI | 93 |

LXVIII | 97 |

LXIX | 100 |

LXX | 101 |

LXXI | 103 |

LXXII | 107 |

LXXIII | 108 |

LXXV | 111 |

LXXVI | 112 |

LXXVIII | 116 |

LXXIX | 117 |

LXXX | 118 |

LXXXII | 119 |

LXXXIII | 121 |

LXXXIV | 122 |

LXXXV | 123 |

LXXXVI | 124 |

LXXXVIII | 126 |

LXXXIX | 128 |

XC | 129 |

XCI | 130 |

XCII | 131 |

XCIV | 133 |

XCV | 135 |

XCVI | 136 |

XCVII | 138 |

XCVIII | 139 |

C | 140 |

CI | 141 |

CII | 142 |

CIII | 143 |

CIV | 144 |

CV | 145 |

CVI | 148 |

CVII | 151 |

CVIII | 153 |

CIX | 154 |

CXI | 156 |

CXII | 159 |

CXIII | 162 |

CXIV | 166 |

CXV | 167 |

CXVII | 169 |

CXVIII | 170 |

CXIX | 171 |

CXXX | 184 |

CXXXI | 185 |

CXXXII | 186 |

CXXXIII | 189 |

CXXXIV | 190 |

CXXXV | 191 |

CXXXVIII | 192 |

CXXXIX | 193 |

CXL | 194 |

CXLI | 196 |

CXLII | 197 |

CXLIV | 198 |

CXLV | 203 |

CXLVI | 206 |

CXLVII | 211 |

CXLVIII | 212 |

CL | 216 |

CLI | 218 |

CLII | 220 |

CLIII | 223 |

CLIV | 230 |

CLV | 231 |

CLVI | 232 |

CLVII | 233 |

CLVIII | 234 |

CLIX | 235 |

CLX | 237 |

CLXII | 238 |

CLXIII | 241 |

CLXIV | 242 |

CLXVI | 243 |

CLXVII | 246 |

CLXVIII | 248 |

CLXX | 249 |

CLXXII | 250 |

CLXXIII | 253 |

CLXXIV | 255 |

CLXXVI | 257 |

CLXXVII | 258 |

CLXXVIII | 260 |

CLXXIX | 263 |

CLXXX | 265 |

CLXXXI | 268 |

CLXXXII | 273 |

CLXXXIII | 274 |

CLXXXIV | 276 |

CLXXXV | 278 |

CLXXXVI | 280 |

CLXXXVII | 284 |

CLXXXVIII | 288 |

CLXXXIX | 292 |

CXC | 294 |

CXCI | 295 |

CXCII | 297 |

CXCIII | 298 |

CXCIV | 300 |

CXCV | 302 |

CXCVI | 306 |

CXCVIII | 309 |

CXCIX | 311 |

CC | 313 |

CCI | 316 |

CCII | 318 |

CCIII | 322 |

CCV | 324 |

CCVI | 325 |

CCVII | 326 |

CCVIII | 329 |

CCIX | 332 |

CCX | 334 |

CCXII | 336 |

CCXIII | 338 |

CCXIV | 343 |

CCXV | 344 |

CCXVI | 345 |

CCXVII | 346 |

CCXVIII | 348 |

CCXX | 349 |

CCXXI | 352 |

CCXXII | 353 |

CCXXIV | 354 |

CCXXV | 355 |

357 | |

373 | |

377 | |

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### Common terms and phrases

achieves an approximation approximation algorithm approximation factor approximation guarantee assume Boolean variables bound on OPT Chapter clauses Clearly compute Consider the following constraints corresponding counting the number cut in G cycle defined denote distance labels dual program endpoints Exercise extreme point solution factor algorithm factor approximation algorithm feasible solution feedback vertex set find a minimum FPRAS fractional Give given graph G greedy algorithm Hence Hint instance integer program integrality gap iteration Lemma length linear program lower bound LP-duality LP-relaxation makespan MAX-3SAT maximize maximum minimize multicommodity flow multicut node nonnegative NP-complete NP-hard O(logn objective function value obtain optimal solution output partition path PCP theorem picked polynomial time algorithm primal primal-dual schema probability Proof reduction relaxation s-t cut satisfies semidefinite programming set cover problem shortest vector Show subset Theorem tight example triangle inequality truth assignment undirected graph vector program vertex cover problem

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