Improving Lagarias-Odlyzko Algorithm For Average-Case Subset Sum: Modular Arithmetic Approach

• URL: http://arxiv.org/abs/2408.16108v1
• Date: Wed, 28 Aug 2024 19:32:14 GMT
• Title: Improving Lagarias-Odlyzko Algorithm For Average-Case Subset Sum: Modular Arithmetic Approach
• Authors: Antoine Joux, Karol Węgrzycki,
• Abstract summary: We introduce a modular arithmetic approach to the Subset Sum problem. We show that density guarantees can be improved by analysing the lengths of the LLL reduced basis vectors.
• Score: 3.0079490585515343
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: Lagarias and Odlyzko (J.~ACM~1985) proposed a polynomial time algorithm for solving ``\emph{almost all}'' instances of the Subset Sum problem with $n$ integers of size $\Omega(\Gamma_{\text{LO}})$, where $\log_2(\Gamma_{\text{LO}}) > n^2 \log_2(\gamma)$ and $\gamma$ is a parameter of the lattice basis reduction ($\gamma > \sqrt{4/3}$ for LLL). The algorithm of Lagarias and Odlyzko is a cornerstone result in cryptography. However, the theoretical guarantee on the density of feasible instances has remained unimproved for almost 40 years. In this paper, we propose an algorithm to solve ``almost all'' instances of Subset Sum with integers of size $\Omega(\sqrt{\Gamma_{\text{LO}}})$ after a single call to the lattice reduction. Additionally, our argument allows us to solve the Subset Sum problem for multiple targets while the previous approach could only answer one target per call to lattice basis reduction. We introduce a modular arithmetic approach to the Subset Sum problem. The idea is to use the lattice reduction to solve a linear system modulo a suitably large prime. We show that density guarantees can be improved, by analysing the lengths of the LLL reduced basis vectors, of both the primal and the dual lattices simultaneously.

Related papers

• Efficient Frameworks for Generalized Low-Rank Matrix Bandit Problems [61.85150061213987]
  We study the generalized low-rank matrix bandit problem, proposed in citelu2021low under the Generalized Linear Model (GLM) framework. To overcome the computational infeasibility and theoretical restrain of existing algorithms, we first propose the G-ESTT framework. We show that G-ESTT can achieve the $tildeO(sqrt(d_1+d_2)3/2Mr3/2T)$ bound of regret while G-ESTS can achineve the $tildeO
  arXiv  Detail & Related papers  (2024-01-14T14:14:19Z)
• Fast algorithms for k-submodular maximization subject to a matroid constraint [10.270420338235237]
  We apply a Threshold-Decreasing Algorithm to maximize $k$-submodular functions under a matroid constraint. We give a $(frac12 - epsilon)$-approximation algorithm for $k$-submodular function.
  arXiv  Detail & Related papers  (2023-07-26T07:08:03Z)
• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• Linear Query Approximation Algorithms for Non-monotone Submodular Maximization under Knapsack Constraint [16.02833173359407]
  This work introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular over a ground set of size $n$ subject to a knapsack constraint. $mathsfDLA$ is a deterministic algorithm that provides an approximation factor of $6+epsilon$ while $mathsfRLA$ is a randomized algorithm with an approximation factor of $4+epsilon$.
  arXiv  Detail & Related papers  (2023-05-17T15:27:33Z)
• Private Isotonic Regression [54.32252900997422]
  We consider the problem of isotonic regression over a partially ordered set (poset) $mathcalX$ and for any Lipschitz loss function. We obtain a pure-DP algorithm that has an expected excess empirical risk of roughly $mathrmwidth(mathcalX) cdot log|mathcalX| / n$, where $mathrmwidth(mathcalX)$ is the width of the poset. We show that the bounds above are essentially the best that can be
  arXiv  Detail & Related papers  (2022-10-27T05:08:07Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization [47.27237492375659]
  We study the bilinearly coupled minimax problem: $min_x max_y f(x) + ytop A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions. No known first-order algorithms have hitherto achieved the lower complexity bound of $Omega(sqrtfracL_xmu_x + frac|A|sqrtmu_x,mu_y) log(frac1vareps
  arXiv  Detail & Related papers  (2022-01-19T05:56:19Z)
• Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint [75.85952446237599]
  We show that a modified greedy algorithm can achieve an approximation factor of $0.305$. We derive a data-dependent upper bound on the optimum. It can also be used to significantly improve the efficiency of such algorithms as branch and bound.
  arXiv  Detail & Related papers  (2020-08-12T15:40:21Z)
• Linear-Time Algorithms for Adaptive Submodular Maximization [17.19443570570189]
  First, we develop a well-studied adaptive submodular problem subject to a cardinality constraint. Second, we introduce the concept of fully adaptive submodularity. Our algorithm achieves a $frac1-1/e-epsilon4-2/e-2epsilon$ approximation ratio using only $O(nlogfrac1epsilon)$ number of function evaluations.
  arXiv  Detail & Related papers  (2020-07-08T15:54:28Z)
• Second-order Conditional Gradient Sliding [79.66739383117232]
  We present the emphSecond-Order Conditional Gradient Sliding (SOCGS) algorithm. The SOCGS algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. It is useful when the feasible region can only be accessed efficiently through a linear optimization oracle.
  arXiv  Detail & Related papers  (2020-02-20T17:52:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.