L. Lov' asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM Publications, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Vector problem on the generated instance distribution with probability 1 2, then there is a P time probabilistic algorithm that solves Shortest Vector for every lattice, with probability exponentially close to 1. Algorithms for approximately solving the Shortest Vector problem have many uses (see [Lov86] and [Kan87] for surveys) Our results are based on a combinatorial transformation that has many similarities to the proofs of the main theorems in [DF95a] and [DF95b] We feel that one of the most interesting aspects of our work is a demonstration of the potential utility of parametric methods ....

L. Lov' asz. An Algorithmic Theory of Numbers, Graphs and Convexity. NSF-CBMS Conference Series, SIAM, 1986.

....in a certain lattice. One therefore needs to construct a lattice in which any feasible vector is provably short. For the reader wishing to study this topic in more detail we refer to the articles mentioned in this introduction, to the survey article by Kannan [55] and to the textbooks by Lov asz [67], Schrijver [83] Gr otschel, Lov asz, Schrijver [43] Nemhauser Wolsey [71] and Cohen [14] In these references, and in the article by Lenstra, Lenstra, Lov asz [62] several applications of basis reduction are mentioned, other than integer programming, such as nding a short nonzero ....

L. Lovasz (1986), An Algorithmic Theory of Numbers, Graphs and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics Vol 50. SIAM, Philadelphia.

....referred to as the feasibility set. We assume that at least one set, 1, is bounded. Our objective is an algorithm to compute the feasibility set for any finite intersection of arbitrary closed convex sets. Exact representation of an arbitrary convex set is not possible in finite time and memory [17]. Further, the volume of the feasibility set is likewise elusive: in [9] it is shown that there exists no polynomial time algorithm which would compute a number (r) for each convex set r C IR n such that z( vol(Y) 1.999n z(Y) Therefore we choose to adopt a parametric approximation of . ....

....) 2( 1 2) n ) 1 O)1 vol(k ) C 1 2nVOl(k) vol(k) e 2vol(0) Since contains , we obtain that vol( vol( V. Hence k 2 (n21og(2R) nlogV) This gives a polynomial upper bound on the number of ellipsoids in the sequence. Further, the ellipsoids can be updated in polynomial time [17]. Finally, we show that the center, c, of the limit ellipsoid, belongs to the interior of the feasibility set. Assume c . Without loss of generality, this implies that c C1. By Proposition 3.2, we can find an optimal cut HcP 1 for set ] 1 (H for in OCA) such that the smallest ellipsoid ....

L. Lov&sz, An algorithmic theory of numbers, graphs and convexity. SIAM, 1986. 12 a

....mg. Given real numbers 1 ; m ; 1 ; m , a nonhomogeneous diophantine approximation is a sequence of integers P 1 ; Pm ; Q such that Q 0 and for all j 2 [m] jQ j P j j j . Nonhomogeneous diophantine approximation need not exist. Theorem 1. 2 (Kronecker, see [Cas57, Lov86]) Let 1 ; m ; 1 ; m 2 R. Then exactly one of the following holds. For any 0 there are P 1 ; Pm ; Q such that Q 0 and for all j 2 [m] jQ j P j j j . There are integers a 1 ; am such that P a j j is an integer and P a j j is not an ....

.... approximation P 1 ; P n ; Q of 1 ; n excluding p. We can nd in polynomial time a simultaneous diophantine C n 1 p approximation of 1 ; n excluding p, where C n = 4 p n2 n=2 . 6 We will use Babai s modi cation [Bab86] of Lov asz s lattice algorithm [LLL82, Lov86]. In [Bab86] the following result is proven for 1 = m ; the general case follows from the same proof. Theorem 4.2 ( Bab86] Theorem 7.1) Let 1 ; m ; 1 ; m ; 1 0; m 0 be given rational numbers. Let q 0 be the smallest integer Q for which there ....

L. Lovasz. An Algorithmic Theory of Numbers, Graphs, and Convexity. SIAM, Philadelphia, PA, 1986. 8

....the tested algorithms. 1 Introduction The ellipsoidal approximation of polytopes is an important problem in its own right while it is also a basic subroutine in a number of algorithms for different problems. One example is that Lenstra s algorithm for the integer programming feasibility problem [11, 12] uses the ellipsoidal approximation of polytopes as a subroutine. Consider a full dimensional polytope P 2 n defined by m linear inequalities. For brevity, we will call the problem of finding the maximum volume ellipsoid inscribing P the MaxVE problem. The MaxVE problem has its root in the ....

....problems are made available at the web site: http: www.caam.rice.edu zhang maxvep . Test sets 1 and 2 are obtained from two integer programming feasibility problems through the search trees of an integer programming algorithm the Lenstra algorithm for integer programming feasibility problem [11, 12]. This algorithm searches on a tree of subproblems and applies ellipsoidal approximation on each one of them. The polytopes in Sets 1 and 2 are taken from some branches of the search trees for two integer programming feasibility problems, respectively. The problem sizes in Sets 1 and 2 are ....

L. Lov'asz. An Algorithmic Theory of Numbers, Graphs, and Convexity. SIAM, Philadelphia, 1986

....on a set of n elements, the goal is to compute the number of total orders that are linear extensions of the given partial order. We discuss the relationship between this problem and that of computing the volume of a convex polyhedron. For more details on this subject, see Section 2. 4 of Lov asz [20]. Given a convex set S and an element a of n , a weak separation oracle 1. Asserts that a 2 S, or 2. Asserts that a 62 S and supplies a reason why. In particular for closed convex sets in n , if a 62 S then there exists a hyperplane separating a from S. So if a 62 S, the oracle responds ....

L'aszl'o Lov'asz. An algorithmic theory of numbers, graphs and convexity. In CBMS-NSF Regional Conference Series on Applied Mathematics, 1986.

....uw 2 Z n be linearly independent vectors with w n. A lattice L spanned by hu 1 ; uw i is the set of all integer linear combinations of u 1 ; uw . We say that the lattice is full rank if w = n. We state a few basic results about lattices and lattice basis reduction and refer to [9] for an introduction. Lattice basis reductions are frequently used in the cryptanalysis of public key systems [6] Let L be a lattice spanned by hu 1 ; uw i. We denote by u 1 ; u w the vectors obtained by applying the Gram Schmidt process to the vectors u 1 ; uw . ....

L. Lovasz. An algorithmic theory of numbers, graphs, and convexity. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 50, 1986.

....value of t(S) does not change if we require equality in (iv) since we can increase the y i values without violating any inequality in (v) and without changing T . Using the dual program DLP allows us in some cases to compute L(S) even when S is given implicitly, using the Ellipsoid algorithm of [16, 26] (see Section 5.3) Notation: For a vector y 2 [0; 1] n and a set S U , let y(S) P i2S y i . 3.3 The Capacity of a Quorum System Each time that a distributed protocol generates an access to a quorum S 2 S, it causes one unit of work to be done by the elements of S. During the time that ....

L. Lov'asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM, Philadelphia, 1986.

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [75, 64, 109, 52, 30, 69]) In particular, their relevance to 1 The technique is however polynomial time for fixed dimension, which was enough in [74] 2 cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [99, 23] which were early alternatives to the RSA ....

....We will not discuss Ajtai s worst case average case equivalence [3, 27] which refers to special versions of SVP and SBP (see [24, 25, 11] such as SVP when the lattice gap 2 = 1 is at least polynomial in the dimension. 2. 4 Algorithmic results The main algorithmic results are surveyed in [75, 64, 109, 52, 30, 69, 24, 97]. No polynomial time algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based ....

L. Lov'asz. An Algorithmic Theory of Numbers, Graphs and Convexity, volume 50. SIAM, 1986. CBMS-NSF Regional Conference Series in Applied Mathematics.

....list of constraints and maintain polynomial time convergence is a characteristic that is the key to its applications in combinatorial optimization. For an elegant treatment of the many deep theoretical consequences of the Ellipsoid Algorithm, the reader is directed to the monograph by Lov asz [50] and the book by Grotschel, Lov asz and Schrijver [34] Computational experience with the Ellipsoid Algorithm, however, showed a disappointing gap between the theoretical promise and practical efficiency of this method in the solution of linear programming problems. Dense matrix computations as ....

L.Lov'asz, An Algorithmic Theory of Numbers, Graphs and Convexity, SIAM Press, 1986.

....in a certain lattice. One therefore needs to construct a lattice in which any feasible vector is provably short. For the reader wishing to study this topic in more detail we refer to the articles mentioned in this introduction, to the survey article by Kannan [55] and to the textbooks by Lov asz [67], Schrijver [83] Gr otschel, Lov asz, Schrijver [43] Nemhauser Wolsey [71] and Cohen [14] In these references, and in the article by Lenstra, Lenstra, Lov asz [62] several applications of basis reduction are mentioned, other than integer programming, such as nding a short nonzero ....

L. Lovasz (1986), An Algorithmic Theory of Numbers, Graphs and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics Vol 50. SIAM, Philadelphia.

....and number theory, but reviewing these other topics is outside the scope of our chapter. For the reader wishing to study lattices and integer programming in more detail we refer to the articles mentioned in this introduction, to the survey article by Kannan [24] and to the textbooks by Lov asz [34], Schrijver [44] Gr otschel, Lov asz, Schrijver [18] Nemhauser Wolsey [37] and Cohen [8] In these references, and in the article by Lenstra, Lenstra, Lov asz [31] several applications of basis reduction, other than integer programming, are mentioned, such as nding a short nonzero ....

L. Lovasz (1986). An Algorithmic Theory of Numbers, Graphs and Convexity. CBMSNSF Regional Conference Series in Applied Mathematics Vol 50. SIAM, Philadelphia.

....and c 2 N such that jx (a ib)2 c j 2 k The running time for k 5 is k log k w(x) log w(x) Poly(deg[Q[x] Q] Also, w(x) Bitsize(y) Moreover, if x is real, then b is always zero in M(y(x) k) The running time above is better than the O(k 2 ) in the proof of Theorem (1.4. 7) a) in [Lov86] or in Algorithm 2 Step 3 in [Loo83] This theorem gives also an implicit bound for the bit size of y(x) Given y(x) this can be used to decide if an algebraic number is 0, for instance. Theorem C makes sense in the Turing model of computation. However, it also makes sense in the weighted ....

....sequences of 0 s and 1 s. It is possible to compute mixed expressions, like f(x) where f is an integer polynomial and x 2 Q ra . Theorem D. The restriction y : Q ra Z 1 of the function y of Theorem D can be computed in polynomial time over (Q ra ; w) Unlike Theorem (1.4. 7) b) in [Lov86], the machine in Theorem D does not know a priori the weight of the input. Theorem D implies that it is possible to decide in polynomial time whether x 2 Q. The proof relies heavily on the Lenstra Lenstra Lovasz algorithm [LLL82] A non uniform model of computation over number elds was ....

Laszlo Lovasz. An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Reg. Conference Series in Appl. Math. 50. SIAM, Philadelphia, 1986.

.... se[29] 100000001504746621987668885504.000000 se[30] 1000000015047466219876688855040.000000 se[31] 9999999848243207295109594873856.000000 se[32] 100000003318135351409612647563264.000000 se[33] 1000000071866979741764260066230272.000000 se[34] 10000000409184787596297531937521664.000000 se[35]=100000004091847875962975319375216640.000000 se[36] 1000000040918478759629753193752166400.000000 se[37] 10000000567641112624826207124609564672.000000 se[38] 100000006944061726476491472742798852096.000000 se[39] Inf #include stdio.h #define LONGSEQ 39 #define SEED 1 #define GROWING 10 float ....

....se[26] 0.0000000022204458272057 se[27] 0.0000000055511146790366 se[28] 0.0000000138777869196360 se[29] 0.0000000346944659668225 se[30] 0.0000000867361649170562 se[31] 0.0000002168404193980678 se[32] 0.0000005421010200734599 se[33] 0.0000013552526070270687 se[34] 0. 0000033881315175676718 se[35]= 0.0000084703287939191796 se[36] 0.0000211758215300505981 se[37] 0.0000529395547346211970 se[38] 0.0001323488831985741854 se[39] 0.0003308721934445202351 se[40] 0.0008271804545074701309 se[41] 0.0020679510198533535004 se[42] 0.0051698777824640274048 se[43] 0.0129246944561600685120 se[44] ....

[Article contains additional citation context not shown here]

L. Lov'asz. An algorithmic theory of numbers, graphs, and convexity. In SIAM Publications. Philadelphia, 1986.

....NY 14260. Research supported in part by NSF grants CCR 9319393 and CCR 9634665. Email: apn cs.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view [12, 20, 11, 13]. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14] The subject was first conceived as a bridge between geometry and Diophantine approximation and the theory of quadratic forms. The field Geometry of Numbers was christened by Minkowski when ....

L. Lov'asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM, Philadelphia, 1986.

....factor, even probabilistically. The celebrated Lov asz basis reduction algorithm [LLL82] finds a short vector within a factor of 2 n=2 in polynomial time. One major open problem in this field has been whether SVP is NP hard for the natural l 2 norm. This was conjectured e.g. by Lov asz [Lov86]. In a tour de force, Ajtai settled this conjecture very recently [Ajt97] SVP is NP hard for l 2 norm under randomized reductions. Moreover Ajtai showed that to approximate the shortest vector of an n dimensional lattice within a factor of i 1 1 2 n k j (for a sufficiently large constant ....

L. Lov'asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM, Philadelphia, 1986.

....the problem that for any norm k k, min k b Gamma Ax k (3) x 2 R n integral; where b 2 R m , and A is an m Thetan matrix with integer elements. This problem, called the closest vector problem in integer programming, has been proven to be NP complete even for simple norms such as l 2 and l 1 [11, 24, 25]. Another example is related to the solution of a class of more general problems: mixed integer nonlinear programming problems. A mixed integer nonlinear program min g(x; y) 4) y 2 R m x 2 R n integral can be formulated, under appropriate assumptions, as a nonlinear integer program min ....

....the lifting process with a procedure that can compute integer lattice free convex bodies. If such an oracle exists, even better supporting planes can be obtained, and the algorithm will be more effective. Finding such an oracle itself is an important research issue in integer programming [24, 25]. Acknowledgments The author is grateful to Professors John E. Dennis and Robert E. Bixby for many discussions relating to this work, and for their support, encouragement, and guidance during his Ph.D. study. ....

L. Lov'asz [1986]. An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Regional Conference Series in Applied Mathematics 50. SIAM, Philadelphia.

....are far from a total break. 4.1 Coppersmith s Theorem The most powerful attacks on low public exponent RSA are based on a theorem due to Coppersmith [7] Coppersmith s theorem has many applications, only some of which will be covered here. The proof uses the LLL lattice basis reduction algorithm [17] as explained below. Theorem 3 (Coppersmith) Let N be an integer and f 2 Z[x] be a monic polynomial of degree d. Set X = N 1 d Gammaffl for some ffl 0. Then, given hN; fi Marvin can efficiently find all integers jx 0 j X satisfying f(x 0 ) 0 mod N . The running time is dominated by ....

....h(x) satisfying the required bound. Once h(x) is found, Lemma 4 implies that it has x 0 as a root over the integers. Consequently x 0 can be easily found. It remains to show how to find h(x) efficiently. To do so, we must first state a few basic facts about lattices in Z w . We refer to [17] for a concise introduction to the topic. Let u 1 ; uw 2 Z w be linearly independent vectors. A (full rank) lattice L spanned by hu 1 ; uw i is the set of all integer linear combinations of u 1 ; uw . The determinant of L is defined as the determinant of the w Theta w ....

L. Lov'asz. An Algorithmic Theory of Number, Graphs and Convexity. SIAM Publications, 1986.

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L. Lov' asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM Publications, 1986.

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Lovasz L. (1986): An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania.

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L. Lovasz, An Algorithmic Theory of Numbers, Graphs and Convexity, CBMSNSF Regional Conference Series in Applied Mathematics, 50, SIAM Publications, Philadelphia, 1986 25

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L. Lovasz, \An algorithmic theory of Numbers, Graphs and Convexity", SIAM (CBMS-NSF Regional conference series in applied mathematics).

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L. Lov'asz. An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Regional Conference Series on Applied Mathematics, SIAM, 1986. 11

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L. Lov' asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM Publications, 1986.

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L. Lov' asz. An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM Publications, 1986.

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