## 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 |

CXXII | 172 |

CXXIII | 174 |

CXXIV | 175 |

CXXV | 176 |

CXXVI | 178 |

CXXVII | 179 |

CXXIX | 180 |

CXXX | 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 |

CVIII | 148 |

CIX | 151 |

CX | 153 |

CXI | 154 |

CXIII | 156 |

CXIV | 159 |

CXV | 162 |

CXVI | 166 |

CXVII | 167 |

CXIX | 169 |

CXX | 170 |

CXXI | 171 |

CXXXII | 184 |

CXXXIII | 185 |

CXXXIV | 186 |

CXXXV | 189 |

CXXXVI | 190 |

CXXXVII | 191 |

CXL | 192 |

CXLI | 193 |

CXLII | 194 |

CXLIII | 196 |

CXLIV | 197 |

CXLVI | 198 |

CXLVII | 203 |

CXLVIII | 206 |

CXLIX | 211 |

CL | 212 |

CLII | 216 |

CLIII | 218 |

CLIV | 220 |

CLV | 223 |

CLVI | 230 |

CLVII | 231 |

CLVIII | 232 |

CLIX | 233 |

CLX | 234 |

CLXI | 235 |

CLXII | 237 |

CLXIV | 238 |

CLXV | 241 |

CLXVI | 242 |

CLXVIII | 243 |

CLXIX | 246 |

CLXX | 248 |

CLXXII | 249 |

CLXXIV | 250 |

CLXXV | 253 |

CLXXVI | 255 |

CLXXVIII | 257 |

CLXXIX | 258 |

CLXXX | 260 |

CLXXXI | 263 |

CLXXXII | 265 |

CLXXXIII | 268 |

CLXXXIV | 273 |

CLXXXV | 274 |

CLXXXVI | 276 |

CLXXXVII | 278 |

CLXXXVIII | 280 |

CLXXXIX | 284 |

CXC | 288 |

CXCI | 292 |

CXCII | 294 |

CXCIII | 295 |

CXCIV | 297 |

CXCV | 298 |

CXCVI | 300 |

CXCVII | 302 |

CXCVIII | 306 |

CC | 309 |

CCI | 311 |

CCII | 313 |

CCIII | 316 |

CCIV | 318 |

CCV | 322 |

CCVII | 324 |

CCVIII | 325 |

CCIX | 326 |

CCX | 329 |

CCXI | 332 |

CCXII | 334 |

CCXIV | 336 |

CCXV | 338 |

CCXVI | 343 |

CCXVII | 344 |

CCXVIII | 345 |

CCXIX | 346 |

CCXX | 348 |

CCXXII | 349 |

CCXXIII | 352 |

CCXXIV | 353 |

CCXXVI | 354 |

CCXXVII | 355 |

357 | |

373 | |

377 | |

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### Common terms and phrases

achieves approximation algorithm assume basis bits called capacity Chapter clauses Clearly components compute connected Consider constant constraints construct contains corresponding cost cycle defined denote directed distance dual edge elements endpoints equality established example Exercise expected facility fact factor feasible solution flow fractional function Give given graph graph G Hence independent inequality instance integral iteration least Lemma length linear lower bound LP-relaxation matching MAX-3SAT maximize maximum metric minimize minimum multicut NP-hard O(log objective Observe obtain optimal solution output packing pair path picked polynomial positive primal probability problem Proof prove random reduction relaxation removal result rounding running satisfies semidefinite program set cover shortest Show simple specified string Suppose Theorem tight truth assignment variables vector verifier vertex cover vertices weight

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