Index. 0/1 linear integer minimization, colorability, 245 4ti2, 80, 83

Transcription

1 Index 0/1 linear integer minimization, colorability, 245 4ti2, 80, 83 absolute convergence, 107, 130 affine combination, 3 affine hull, 18 affine hyperplane, 4 algebraic closure, 193, 238, 242, 256 algebraic variety, 239 algorithm ɛ-approximation, 157 approximation, 157 Barvinok s, 129, 134, 146 Buchberger s, 210, 218 Euclidean, 256 geometric Buchberger, 226 Gram Schmidt orthogonalization, 50 Graver proximity, 100 incremental polynomial time, 141 Lenstra s, 53, 79, 144 LLL, 50 Nullstellensatz linear algebra, 242 polynomial total time, 141 polynomial-delay, 141 polynomial-space polynomial-delay, 141 Pottier s, 71 regular triangulation, 121 weak approximation, 158 all-primal Barvinok decomposition, 153 analytic center, 282 analytic function, 137 appoximation scheme weak, 158 approximate continuous convex optimization, 93 approximate continuous convex optimization oracle, 101 approximate shortest vector, 133 approximation algorithm, 157 weak, 158 approximation scheme, 157 arithmetic mean, 163 Artin s theorem, 269 ascending chain, 215 assignment problem, 287 augmentation algorithms, 93 augmentation step, 92 Graver-best, 66, 97, 99 augmentation vector, 63, 96 auxiliary objective function, 95 barvinok, 135 Barvinok s algorithm, 129, 134, 146 for integer linear optimization, 142 homogenized, 153 Barvinok s signed decomposition, 132 all-primal, 153 dual, 152 primal, 152 Barvinok s theorem, 134, 135 basic feasible solution, 19 basic semialgebraic set, 187, 273 basic solution, 19, 151 basic variables, 19 basis, 19, 168 biorthonormal, 133 Gröbner, 208, 217, 286 integral, 176 Markov, 221, 228 of a lattice, 34 Bernoulli l Hôpital rule, 106 binary encoding scheme, 21 binary search, 143 binomial ideal, 218 binomials, 217 biorthonormal basis, 133 Birkhoff polytope, 288 bisection algorithm, 163 bit length, 21 bit-scaling technique, 66,

2 316 Index Blichfeldt s theorem, 41 block-structured integer programs, 77 Boolean operations, 150, 164, 184 Boolean operations lemma, 148 bounded set, 16 branch and bound algorithm, 176 branching on hyperplanes, 54 brick, 81, 90, 92 Brion s theorem, 108, 118, 120, 124 Buchberger s algorithm, 210, 218 geometric version, 226 Buchberger s S-pair criterion, 210 Buchberger s theorem, 210 building block, 81 Carathéodory s theorem, 5, 97, 168 Cauchy integral formula, 285 central curve, 282 central path, 281 certificate of infeasibility, 239 chamber, 151 characteristic polynomial, 280 Cholesky factor, 23 circuit, 97 colorable, 238 combinatorial moment matrix, 279 combinatorial parity conditions, 251 combinatorial system of equations, 239 commutative algebra, 92 comparison oracle, 66, 85, 99 comparison point, 181 complementary slackness, 21, 281 completion algorithm, 92, 226 complex analysis, 106, 137 complex integration, 285 complex number, 238 complex plane, 106 compressed lattice polytope, 279 computational complexity, 21, 58 cone, 5, 274 finitely generated, 5, 12, 13 index descent, 133 pointed, 15 pointed rational, 130 polyhedral, 5 recession, 16, 17 simplicial, 120, 130 tangent, 118 unimodular, 112 conformal (orthant-compatible), 97 conic combination, 3 continuous function, 198 continuous Hirsch conjecture, 282 continuous measure, 89 continuous relaxation, 93, 97 convex combination, 3 convex hull, 3 convex position, 120 convex set, 3 convex subdivision, 122 counting, 105 counting oracle, 143 critical path, 224 cut, 237 De Morgan formula, 118 decision making under uncertainty, 89 decomposition primal Barvinok, 152 signed, 113 decomposition algorithm, 89 decomposition of polyhedra, 117 decomposition tree, 134 decreasing path, 222 degree, 193, 194 Delaunay triangulations, 122 derivative, 198 Descartes rule of signs, 199, 213 determinant of a lattice, 40 diameter of a polytope, 282 Dickson s lemma, 207 dictionary, 227 dictionary order, 203 differential operator, 109, 159 digging algorithm, 145, 146 dimension, 18 directed augmentation, 66, 96 directed graph, 87 directed path, 87 discrete measure, 89 discretization, 165 division algorithm, 194 domain of convergence, 106, 115 dual basis, 133 dual linear programming, 19 duality strong, 20 weak, 20 duality trick, 152

3 Index 317 dynamic programming problem, 86, 99, 285 ɛ-approximation algorithm, 157 elementary square matrices, 6 elimination ideal, 212 elimination of variables, 212 elimination theorem, 212 ellipsoid, 22 ellipsoid method, 22, 25 Euclidean algorithm, 256 evaluation oracle, 93 evaluation problem, 160 exact set, 279 explicit enumeration, 147 exponent vector, 203, 268 exponential integrals, 115, 176 exponential sum, 114, 287 extended Euclidean algorithm, 196, 257 extended group relaxation, 286 extended integer program, 220 extreme point, 18 face, 18 factoring, 176 Farkas lemma, 9, 21, 219, 229, 239, 273 fast Fourier transform, 285 feasibility oracle, 143 finite field, 238 finitely generated cone, 5, 12, 13 finitely generated ideal, 208 fixed dimension, 134 flatness constant, 151 flatness theorem, 54, 151 formal Laurent series, 129 Fourier Motzkin elimination, 6, 7 FPTAS (fully polynomial-time approximation scheme), 158 Fredholm s alternative theorem, 5, 239, 274 fully polynomial-time approximation scheme (FPTAS), 158, 182 for a maximization problem, 158 for maximizing nonnegative polynomials, 159 for a minimization problem, 182, 187 function analytic, 137 continuous, 198 rational, 105, 106 separable convex, 63, 79, 85, 98 separable convex p-piecewise affinelinear, 87 fundamental parallelepiped, 39, 109, 130 fundamental theorem of algebra, 169 gaps, 149, 152 Gaussian elimination, 6, 201, 203 gcd (greatest common divisor), 194, 195, 203 general position, 282 generating function, 105, 108, 129, 287 Boolean operation, 147 evaluation, 135 integer projection, 149 intermediate, 176 mixed-integer, 176 output-sensitive enumeration, 141 positively weighted, 136, 147, 149, 165 projection theorem, 149 rational, 132 specialization, 135 substitution, 135 generating set, 221 generic, 122, 219 geometric Buchberger algorithm, 226 geometric improvement, 67 geometric series, 105, 110, 130 geometry of numbers, 29 global criterion, 179, 181, 185 global mixed-integer polynomial optimization, 157 global nonconvex polynomial optimization, 176 Gomory s group relaxation, 154 Gomory Chvátal closure, 56 Gordan Dickson lemma, 43, 44, 90, 207, 227 Gröbner basis, 208, 217, 286 of a lattice ideal, 222 reduced, 212, 218 reduced minimal, 219 Gröbner complexity, 233 Grötzsch graph, 248 graded lexicographic order, 204 graded reverse lexicographic order, 204 Gram Brianchon theorem, 118 Gram Schmidt orthogonalization algorithm, 50

4 318 Index graph, 221, 237, 286 of a polyhedron, 282 Graver-based dynamic programming, 86, 96, 100 Graver basis, 63, 64, 219, 232 for stochastic IPs (integer programs), 91 length bound, 94 Graver-best augmentation step, 66, 97, 99 Graver complexity, 80, 82, 102 Graver proximity algorithm, 100 Graver test set, 63 greatest common divisor (gcd), 194, 195, 203, 256 of univariate polynomials, 195 grid approximation, 165 grid problem, 165, 171 group relaxation, 285 H-representation, 10 Hadamard product, 147, 184 half-open decomposition, 124, 152, 153 half-open polyhedron, 124 half-open simplicial cone, 130 half-open triangulation, 127 heights, 121 Hermite normal form (HNF), 35, 95, 231 Hilbert basis, 45, 47, 83, 154 element enumeration, 149 Hilbert s 10th problem, 58 Hilbert s 17th problem, 269 Hilbert s basis theorem, 208, 218 Hilbert s Nullstellensatz, 213, 237, 239, 256 Hirsch conjecture, 283 HNF (Hermite normal form), 35, 95, 231 Hochbaum Shanthikumar sproximity, 97 Hölder s inequality, 159 holes, 180 homogeneous polynomial, 243 homogenization, 14, 130, 269 homogenized Barvinok algorithm, 153 hyperplane, 4 hyperplane arrangement, 282 ideal, 183, 201 finitely generated, 208 generated by a set, 202 monomial, 206 radical, 262, 278 toric, 217, 286 vanishing, 202 ideal intersection, 213 ideal membership, 202, 212 ideal point, 181 ILF (integer linear feasibility problem), 54 ILP (integer linear optimization problem), 53 inapproximability, 58 inclusion-exclusion principle, 112, 117, 124 incomputable, 58 indecomposable lattice point, 47 independent set, 252 index of a cone, 111, 130 index of a sublattice, 40 indicator function, 41, 117, 124 infeasibility certificate, 239 infinite geometric series, 106 initial feasible solution, 220 integer hull, 55, 151 integer linear feasibility problem (ILF), 54 integer linear optimization problem (ILP), 53 integer linear program in fixed dimension, 79 integer programming game, 190 integer projection, 136 integral, 176 integral basis, 44 integral generating set, 44 integral polyhedron, 56 intermediate generating function, 176 intersection lemma, 147, 148, 186 irrational decomposition, 152 iterated bisection, 143 Kannan s partitioning theorem, 151 Khinchin s flatness theorem, 54, 151 knapsack, 285 knapsack problem, 284 k-sos ideal, 278 k-th theta body, 278 l 1 -norm, 83, 93, 96, 99, 101 Laplace transform, 115 large scale problem, 89

5 Index 319 largest term, 204 LattE integrale, 135 lattice, 34 sublattice, 40 lattice basis LLL-reduced, 133 lattice-point-free convex bodies, 151 lattice program, 223 lattice width, 54 lattice width direction, 54 Laurent expansion, 106 Laurent polynomial, 130 Laurent series, 106, 284 Lawrence Khovanskii Pukhlikov theorem, 131 layers, 81 leading coefficient, 204 leading monomial, 204, 207 leading term, 194, 204 Lenstra s algorithm, 53, 79, 144 lexicographic order, 141, 181, 203 LF (linear feasibility problem), 22 lift-and-project method, 277 lifted configuration, 121 limit point, 166 lineality space, 15, 17 linear feasibility problem (LF), 22 linear optimization problem (LP), 21 linearization technique, 277 Lipschitz constant, 171 LLL (Lenstra Lenstra Lovász) algorithm, 50 LLL-reduced lattice basis, 133 logarithmic barrier function, 281 Lovász encoding, 252 lower envelope, 121 LP (linear optimization problem), 21 M-ellipsoid coverings, 50 Maclagan s theorem, 91 Markov basis, 221, 228 matrix term order, 204 matrix-cut method, 277 matroid, 282 max-cut problem, 237 maximal decreasing path algorithm, 226 maximal lattice-free convex set, 287 meromorphic function, 115, 130 minimal critical path, 225 Minkowski sum, 12, 14, 47 Minkowski s first theorem, 31, 42 Minkowski Hlawka theorem, 43 mixed-integer generating function, 176 mixed-integer summation technique, 176 modulo nonpointed polyhedra, 120 moment, 89 moment curve, 139 monic polynomial, 261 monomial, 105 monomial ideal, 206 monomial map, 136 monomial order, 203 monomial substitution, 144, 150, 184 Monte-Carlo Markov-chain, 222 Moore Bellman Ford s algorithm, 88 multicommodity network flow problem, 77 multicriterion integer linear programming problem, 179, 181 multiepigraph, 183 multiexponent notation, 129 multigraph, 184 multiplicity, 198 multistage stochastic integer linear programs, 101 multivariate division algorithm, 205 N-fold 4-block decomposable, 77 N-fold 4-block decomposable integer program, 93 N-fold 4-block decomposable matrix, 77 N-fold integer program, 77, 96, 99 N-fold matrix, 79 Nash equilibrium, 190 Newton polytope, 268, 280 Noether s normalization lemma, 256 nonbasic variable, 19 nonnegative modulo an ideal, 278 nonstandard monomial, 214 normal form Smith, 38, 133 normal form algorithm, 71 NP-complete, 245 NP-hard, 58 NulLA (Nullstellensatz linear algebra algorithm), 242 NulLA rank, 243, 253 Nullstellensatz, 213, 237, 239, 256 Nullstellensatz linear algebra algorithm (NulLA), 242

6 320 Index objective function, 108 auxiliary, 95 odd cycle, 237 odd wheel, 247 one-to-one projection, 152, 184 optimality certificate, 63, 65, 96, 223 optimization oracle, 93 oracle, 141, 148 comparison, 66, 85, 99 counting, 143 feasibility, 143 optimization, 93 separation, 22 orbit polytope, 288 oriented chordless 4-cycle, 246 oriented partial 3-cycle, 246 outcome vector, 179 output-polynomial time, 75 output-sensitive complexity analysis, 141 overcounting, 112 Pólya exponent, 269 Pólya s lemma, 269 Pareto optimum, 179, 180, 184 Pareto strategy, 179, 180, 184 partition generating function, 283 permutahedron, 288 phase I, 68, 96, 100, 220 Pick s theorem, 34 piecewise affine linear, 86, 96 pivot, 201 pivot rule, 282 point configuration, 120 pointed cone, 132 pointed polyhedron, 17, 130 pointed rational cone, 130 pointed rational polyhedron, 130 polar, 10 of a cone, 11, 130 of a set, 10 pole, 106, 135 polygon, 31 simple, 32 polyhedral cone, 5 polyhedral norm, 181, 185 polyhedron, 4 pointed, 17, 130 pointed rational, 130 simple, 120, 283 polynomial monic, 261 polynomial map, 217 polynomial system, 237 polynomial-space polynomial-delay enumeration algorithm, 148, 182, 185 polynomial-space polynomial-delay prescribedorder enumeration algorithm, 150, 181 polynomial-time approximation scheme (PTAS), 157 polytopal subdivision, 121 polytope, 4 positive definite form, 269 positive semidefinite (PSD) matrix, 26 positive semidefinite (PSD) polynomial, 269 positively weighted generating function, 136, 147, 149, 165 Positivstellensatz, 274 Pottier s algorithm, 71 preorder, 274 primal Barvinok decomposition, 152 primal-dual interior point algorithm, 281 probability measure, 89 project-and-lift, 73, 230 project-and-lift algorithm, 228 projection, 164, 180, 185 projection theorem, 182, 184 proximity, 93, 97 proximity-scaling technique, 93, 97 PSD (positive semidefinite) matrix, 26 pseudonorm, 187 pseudopolynomial, 97 PTAS (polynomial-time approximation scheme), 157 Putinar s theorem, 277 quadratic assignment problem, 287 quadratic module, 276 quotient ring, 217 quotient rule, 160 radical ideal, 262, 278 Radon s lemma, 5 range of the objective function, 173 rational function, 105, 106 rational generating function, 132 real roots, 198 recession cone, 16, 17

7 Index 321 reduced Gröbner basis, 212, 218 reduced lattice basis, 51 reduced minimal Gröbner basis, 219 reduction path, 223 regular subdivision, 122 regular triangulation, 286 regular triangulation algorithm, 121 remainder of polynomial division, 204, 209 removable singularity, 135 representation, 287 representation theorem for cones, 15 residue, 285 residue technique, 160 residue techniques, 162 resolution of polyhedra, 15 resultant, 256, 257, 259 reverse lexicographic triangulation, 279 Rolle s theorem, 198 root, 194 multiplicity of, 197 root of unity, 238 row reduction, 201 S-polynomial, 209, 218, 224, 227 S-vector, 227 sample, 154, 177 saturated ideal, 233 saturation, 92, 232 scenario, 89 Schmüdgen s theorem, 276 SDP (semidefinite program), 27 selecting a Pareto optimum, 185 semialgebraic set, 274 semidefinite optimization problem (SDP), 27, 239 semidefinite programming, 27, 239, 275 semigroup, 110 separable convex function, 63, 79, 85, 93, 98 separation oracle, 22 separation problem, 26 series expansion, 145 set covering, 287 set packing, 287 shelling, 124 shortest path, 88 shortest path problem, 286 shortest vector, 49 approximate, 133 shortest vector problem (SVP), 49 sign variation, 199 sign-compatible, 44 signed decomposition, 113 simple polygon, 32 simple polyhedron, 120, 283 simplex, 120 simplicial complex, 286 simplicial cone, 120, 130 singularity, 106 removable, 135 slack variable, 17 small-gaps theorem, 151 Smith normal form, 38, 133 SOS (sum of squares), 274 modulo an ideal, 278 spanning set, 221 specialization, 184 square-free, 254 stability number, 252 stable set, 237, 252 stable set polytope, 252, 277 stochastic integer multicommodity flow problem, 78 stochastic integer program with secondorder dominance relations, 78 strong duality, 20 Sturm sequence, 200 subdeterminant, 55, 97 subdivision convex, 122 regular, 122 sublattice, 40 substitution, 136 sum of squares (SOS), 274 modulo an ideal, 278 summation formula, 105 summation method for optimizing polynomials, 159, 161 sums of squares (SOS), 265 superadditivity, 98 support, 97 supported Pareto outcome, 180 supporting hyperplane, 18 Sylvester matrix, 258 symbolic differentiation, 109 symmetric groups, 288 system of linear equations, 5 system of polynomial equations, 193

8 322 Index TH k -exact, 278 tangent cone, 118 Taylor expansion, 137 term, 194 term order, 203 graded lexicographic, 204 graded reverse lexicographic, 204 lexicographic, 141, 181, 203 matrix, 204 test set, 63, 223 theorem Artin s, 269 fundamental of algebra, 169 Gram Brianchon, 118 Hilbert s basis, 208, 218 Minkowski s first, 31, 42 Minkowski Hlawka, 43 Schmüdgen s, 276 small-gaps, 151 Weyl Minkowski, 4, 9, 26 theorem of the alternative, 239 theta body, 277 k-th, 278 theta-rank, 278 three-way transportation problems, 79 tiling, 109 Todd polynomial, 137 toric ideal, 217, 286 toric ring, 217 total curvature, 282 total degree, 204 total dual integrality, 287 total order, 203 totally unimodular, 57 totally unimodular matrix, 77, 79 transportation matrix, 83, 101 transportation polytope, 79 transportation problem, 79 traveling salesman problem, 287 triangulation, 111, 121, 130 of a cone, 120 of a polytope, 120 of a vector configuration, 123 regular, 286 reverse lexicographic, 279 truncated Gröbner basis, 233 truncated Taylor series, 139 Turán graph, 255 two-stage stochastic integer optimization problem, 77 two-stage stochastic integer programming, 88 type, 81, 87 uniform convergence, 107, 130 unimodular, 150, 279, 287 unimodular cone, 112, 133 unimodular matrix, 34, 36, 152 universality theorem, 79 V-representation, 10 valid inequality, 18 valuation, 131 vanishing ideal, 202 variety, 201, 239, 262 zero-dimensional, 214, 239 vector configuration, 123 vector partition function, 283 vertex, 18 volume, 23 von Neumann, 20 Voronoi cell, 50 weak approximation algorithm, 158 weak approximation scheme, 158 weak composition, 95 weak duality, 20 well-ordering, 203 Weyl Minkowski theorem, 4, 9, 26 width, 54 zero divisor, 131 zero-dimensional variety, 214, 239