Verification of NP-Hardness Reduction Functions for Exact Lattice Problems
نویسندگان
چکیده
Abstract This paper describes the formal verification of NP-hardness reduction functions two key problems relevant in algebraic lattice theory: closest vector problem and shortest problem, both infinity norm. The formalization uncovered a number with existing proofs literature. how these were corrected formalization. work was carried out proof assistant Isabelle.
منابع مشابه
Approximation Hardness for Small Occurrence Instances of NP-Hard Problems
The paper contributes to the systematic study (started by Berman and Karpinski) of explicit approximability lower bounds for small occurrence optimization problems. We present parametrized reductions for some packing and covering problems, including 3-Dimensional Matching, and prove the best known inapproximability results even for highly restricted versions of them. For example, we show that i...
متن کاملLattice Problems in NP ∩ coNP
We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √ n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [7], Goldreich and Goldwasser [14], and Aharonov and Regev [2]. Our technique is based on a simple fact regarding succinct approximati...
متن کاملStrong NP-Hardness for Sparse Optimization with Concave Penalty Functions
We show that finding a global optimal solution for the regularized Lq-minimization problem (q ≥ 1) is strongly NP-hard if the penalty function is concave but not linear in a neighborhood of zero and satisfies a very mild technical condition. This implies that it is impossible to have a fully polynomial-time approximation scheme (FPTAS) for such problems unless P = NP. This result clarifies the ...
متن کاملProof Verification and Hardness of Approximation Problems
The class PCP(f(n), g(n)) consists of all languages L for which there exists a polynomial-time probabilistic oracle machine that uses O(f(n)) random bits, queries O(g(n)) bits of its oracle and behaves as follows: If x ∈ L then there exists an oracle y such that the machine accepts for all random choices but if x 6∈ L then for every oracle y the machine rejects with high probability. Arora and ...
متن کاملHardness of Lattice Problems in `p Norm
We show that for any integer p ≥ 46, the Shortest Vector Problem in `p norm is hard to approximate within factor (4/3)1−45.7/p. We also show a hardness factor of (3/2)1−111.7/p for p ≥ 112. As p grows, these factors approach 4/3 and 3/2 respectively. Both results hold under the assumption NP 6⊆ ZPP. We give a very simple reduction from known hardness results for Hypergraph Independent Set Probl...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Lecture Notes in Computer Science
سال: 2023
ISSN: ['1611-3349', '0302-9743']
DOI: https://doi.org/10.1007/978-3-031-38499-8_21