Related papers: FPT-algorithms for The Shortest Lattice Vector and…
In this paper, we present FPT-algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The…
We study integer linear programs (ILP) of the form $\min\{c^\top x\ \vert\ Ax=b,l\le x\le u,x\in\mathbb Z^n\}$ and analyze their parameterized complexity with respect to their distance to the generalized matching problem, following the…
Given a parametric lattice with a basis given by polynomials in Z[t], we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in t: that is, they are given by formulas that are piecewise…
A particular instance of the Shortest Vector Problem (SVP) appears in the context of Compute-and-Forward. Despite the NP-hardness of the SVP, we will show that this certain instance can be solved in complexity order $O(n\psi\log(n\psi))$…
A lattice is the integer span of some linearly independent vectors. Lattice problems have many significant applications in coding theory and cryptographic systems for their conjectured hardness. The Shortest Vector Problem (SVP), which is…
The Shortest Lattice Vector (SLV) problem is in general hard to solve, except for special cases (such as root lattices and lattices for which an obtuse superbase is known). In this paper, we present a new class of SLV problems that can be…
The Minimum Vertex Cover problem, a classical NP-complete problem, presents significant challenges for exact solution on large graphs. Fixed-Parameter Tractability (FPT) offers a powerful paradigm to address such problems by exploiting a…
In the area of parameterized complexity, to cope with NP-Hard problems, we introduce a parameter k besides the input size n, and we aim to design algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some function f(k) and…
In the present paper we study a non-modular variant of the Short Integer Solution problem over the integers. Given a random matrix $A \in \mathbb{Z}^{n\times m}$ with entries $a_{ij}$ such that $0\le a_{ij}< Q,$ for some $Q>0,$ the goal is…
Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable.…
We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…
We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $\min \{f(\mathbf{x}) \mid A\mathbf{x} = \mathbf{b}, \, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \,…
We consider integer programs whose constraint matrix has a special block structure: $\min\{f(x):H_{com}x=b, l\le x\le u,x\in\mathbb{Z}^{t_B+nt_A}\}$, where the objective function $f$ is separable convex and the constraint matrix $H_ {com}$…
In this paper, we study a class of approximation problems, appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials), whose parameters are subject…
Any ideal in a number field can be factored into a product of prime ideals. In this paper we study the prime ideal shortest vector problem (SVP) in the ring $ \Z[x]/(x^{2^n} + 1) $, a popular choice in the design of ideal lattice based…
Lattice-based cryptography has recently emerged as a prime candidate for efficient and secure post-quantum cryptography. The two main hard problems underlying its security are the shortest vector problem (SVP) and the closest vector problem…
We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entrywise $\ell_p$-approximation error, for any $p \geq 1$; the case $p = 2$ is the classical SVD problem. We obtain the first provably good…
The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the…
It has been shown that the parallel Lattice Linear Predicate (LLP) algorithm solves many combinatorial optimization problems such as the shortest path problem, the stable marriage problem and the market clearing price problem. In this…
The Shortest Paths Problem (SPP) is no longer unresolved. Just for a large scalar of instance on this problem, even we cannot know if an algorithm achieves the computing. Those cutting-edge methods are still in the low performance. If we go…