2 Search Results for "Xu, Lei"

Document
DOI: 10.4230/LIPIcs.ESA.2020.33
New Bounds on Augmenting Steps of Block-Structured Integer Programs

Authors: Lin Chen, Martin Koutecký, Lei Xu, and Weidong Shi

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the 𝓁₁- or 𝓁_∞-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of 𝓁_∞-norm at most 𝒪_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and 𝒪_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to 𝒪_FPT(1), perhaps at least in some restricted cases. We prove that the 𝓁_∞-norm of the Graver elements of 4-block n-fold IP is upper bounded by 𝒪_FPT(n^{s_D}), improving significantly over the previous bound 𝒪_FPT(n^{2^{s_D}}). We also provide a matching lower bound of Ω(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the 𝓁_∞-norm of its Graver elements is Ω(n^{s_D}), there exists a different decomposition into lattice elements whose 𝓁_∞-norm is bounded by 𝒪_FPT(1), which allows us to provide improved upper bounds on the 𝓁_∞-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle.
Cite as

Lin Chen, Martin Koutecký, Lei Xu, and Weidong Shi. New Bounds on Augmenting Steps of Block-Structured Integer Programs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{chen_et_al:LIPIcs.ESA.2020.33,
  author =	{Chen, Lin and Kouteck\'{y}, Martin and Xu, Lei and Shi, Weidong},
  title =	{{New Bounds on Augmenting Steps of Block-Structured Integer Programs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{33:1--33:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.33},
  URN =		{urn:nbn:de:0030-drops-128994},
  doi =		{10.4230/LIPIcs.ESA.2020.33},
  annote =	{Keywords: Integer Programming, Graver basis, Fixed parameter tractable}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2017.21
Smart Contract Execution - the (+-)-Biased Ballot Problem

Authors: Lin Chen, Lei Xu, Zhimin Gao, Nolan Shah, Yang Lu, and Weidong Shi

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract
Transaction system build on top of blockchain, especially smart contract, is becoming an important part of world economy. However, there is a lack of formal study on the behavior of users in these systems, which leaves the correctness and security of such system without a solid foundation. Unlike mining, in which the reward for mining a block is fixed, different execution results of a smart contract may lead to significantly different payoffs of users, which gives more incentives for some user to follow a branch that contains a wrong result, even if the branch is shorter. It is thus important to understand the exact probability that a branch is being selected by the system. We formulate this problem as the (+-)-Biased Ballot Problem as follows: there are n voters one by one voting for either of the two candidates A and B. The probability of a user voting for A or B depends on whether the difference between the current votes of A and B is positive or negative. Our model takes into account the behavior of three different kinds of users when a branch occurs in the system -- users having preference over a certain branch based on the history of their transactions, and users being indifferent and simply follow the longest chain. We study two important probabilities that are closely related with a blockchain based system - the probability that A wins at last, and the probability that A receives d votes first. We show how to recursively calculate the two probabilities for any fixed n and d, and also discuss their asymptotic values when n and d are sufficiently large.
Cite as

Lin Chen, Lei Xu, Zhimin Gao, Nolan Shah, Yang Lu, and Weidong Shi. Smart Contract Execution - the (+-)-Biased Ballot Problem. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{chen_et_al:LIPIcs.ISAAC.2017.21,
  author =	{Chen, Lin and Xu, Lei and Gao, Zhimin and Shah, Nolan and Lu, Yang and Shi, Weidong},
  title =	{{Smart Contract Execution - the (+-)-Biased Ballot Problem}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{21:1--21:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.21},
  URN =		{urn:nbn:de:0030-drops-82388},
  doi =		{10.4230/LIPIcs.ISAAC.2017.21},
  annote =	{Keywords: Blockchain, Probability, Random Walk, Smart Contract}
}
  • Refine by Author
  • 2 Chen, Lin
  • 2 Shi, Weidong
  • 2 Xu, Lei
  • 1 Gao, Zhimin
  • 1 Koutecký, Martin
  • Show More...

  • Refine by Classification
  • 1 Mathematics of computing → Combinatorial algorithms
  • 1 Theory of computation → Parameterized complexity and exact algorithms

  • Refine by Keyword
  • 1 Blockchain
  • 1 Fixed parameter tractable
  • 1 Graver basis
  • 1 Integer Programming
  • 1 Probability
  • Show More...

  • Refine by Type
  • 2 document

  • Refine by Publication Year
  • 1 2017
  • 1 2020

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact