Vectors in a box

Buchin, Kevin; Matoušek, Jiřı́; Moser, Robin A.; Pálvölgyi, Dömötör

doi:10.1007/s10107-011-0474-y

Cited by 6 publications

(8 citation statements)

References 9 publications

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“…Suppose that y is a cycle of z * −x * . By i) and iii) of Lemma 5, z * −y is also an optimal solution of the integer program (10). But z * −y −x * 1 < z * −x * 1 contradicting the minimality of z * −x * 1 .…”

Section: Proximity In the 1 -Normmentioning

confidence: 89%

“…The number of inequalities in their linear program is bounded by an exponential in τ(m). Buchin et al [10] have shown that m m/2−o(m) τ(m) m m+o(m) which then yields an algorithm for integer programming that is pseudopolynomial for fixed m but doubly exponential in m. Their upper bound on τ(m) is proved via the Steinitz lemma. We take a different path in applying the Steinitz lemma.…”

Section: The Steinitz Lemmamentioning

confidence: 99%

“…Lemma 6. Let x * be an optimal solution of the linear programming relaxation of (10) and let z * be an optimal integer solution of (10) such that z * − x * 1 is minimal. There does not exist a cycle of z * − x * .…”

Section: Proximity In the 1 -Normmentioning

confidence: 99%

See 2 more Smart Citations

Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma

Eisenbrand

Weismantel

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider integer programming problems in standard form max{c T x : Ax = b, x 0, x ∈ Z n } where A ∈ Z m×n , b ∈ Z m and c ∈ Z n . We show that such an integer program can be solved in time (m·∆) O(m) · b 2 ∞ , where ∆ is an upper bound on each absolute value of an entry in A. This improves upon the longstanding best bound of Papadimitriou (1981) , where in addition, the absolute values of the entries of b also need to be bounded by ∆. Our result relies on a lemma of Steinitz that states that a set of vectors in R m that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by m.We also use the Steinitz lemma to show that the 1 -distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by m · (2 m · ∆ + 1) m . Here ∆ is again an upper bound on the absolute values of the entries of A. The novel strength of our bound is that it is independent of n.We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.

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Section: Proximity In the 1 -Normmentioning

confidence: 89%

Section: The Steinitz Lemmamentioning

confidence: 99%

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Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma

Eisenbrand

Weismantel

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The question of bounding a single partial sum bears similarities to another variation of the Steinitz Lemma in which one permutes a (not necessarily zero-sum) sequence such that some partial sum lies in U [5,12].…”

Section: Corollary 1 Let (U Imentioning

confidence: 99%

“…For Theorem 4 we are interested in bounding ( x, y) ∞ with as little dependence on n as possible. Therefore, when we apply the colorful Steinitz Lemma we remove the term nd and focus instead on 40d 5 .…”

Section: ⊓ ⊔mentioning

confidence: 99%

A Colorful Steinitz Lemma with Applications to Block Integer Programs

Oertel¹,

Paat²,

Weismantel³

2022

Preprint

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The Steinitz constant in dimension d is the smallest value c(d) such that for any norm on R d and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by c(d). Grinberg and Sevastyanov prove that c(d) ≤ d and that the bound of d is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences.Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.1 Introduction.Let • : R d → R be an arbitrary norm with corresponding unit ballGrinberg and Sevastyanov prove the following result on zero-sum sequences. We refer to

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