(0,1)-CVP: Sub-2^n Algorithmic Advances

• (0,1)-CVP is a lattice problem defined by finding the closest subset sum of basis vectors with binary (0,1) coefficients.
• The breakthrough algorithm reduces complexity from 2^n to approximately (1.7299)^n using split-and-list partitioning and fast triangle detection.
• Reductions to MAX-SAT and minimum-weight clique establish fine-grained equivalences that deepen our understanding of lattice-based cryptographic hardness.

The $(0,1)$-Closest Vector Problem ($(0,1)$-CVP) is a specialized instance of the Closest Vector Problem (CVP) in lattice theory, central to both theoretical computer science and cryptography. In this variant, one seeks the closest lattice point to a target vector where the lattice point must be a subset sum of the basis vectors—that is, coefficients are restricted to $0$ or $1$. This restriction parallels classic problems in combinatorial optimization and connects deeply to fine-grained complexity conjectures through new algorithmic frameworks and reductions.

1. Formal Definition and Problem Structure

Let $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$ denote a full-rank basis of an integer lattice $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$. Given a target vector $\mathbf t \in \mathbb Z^m$ and distance $d > 0$, the decision problem $(0,1)$-$\mathrm{CVP}_2$ is specified as:

Given $(0,1)$0, $(0,1)$1, and $(0,1)$2:

• Return YES if $(0,1)$3 such that $(0,1)$4,
• Otherwise, return NO (i.e., for all $(0,1)$5, $(0,1)$6).

Formally,

$(0,1)$7

This formulation constrains the solution space to the $(0,1)$8 subset sums over the basis vectors, mapping directly to fundamental combinatorial enumeration.

2. Algorithmic Advances: Beating the $(0,1)$9 Barrier

The previously best-known exact algorithm for $0$0-CVP required explicit enumeration over all $0$1 possibilities. The principal contribution is an algorithm achieving a running time of $0$2 for exact $0$3-$0$4 given all entries in $0$5 and $0$6 are bounded by $0$7 (Abboud et al., 7 Jan 2025). This surpasses the naïve exhaustive search limit.

Algorithmic Framework

• Split-and-List: Partition the set of indices $0$8 into $0$9 disjoint blocks of size $1$0. For each block, enumerate all subset sums, resulting in $1$1 lists of $1$2 vectors.
• Pairwise Quadratic Decomposition: Leverage the Euclidean norm, expressing

$1$3

and encode the problem as a minimum-weight $1$4-clique problem in a multipartite graph whose vertices are the enumerated partial sums.

• Triangle (3-Clique) Method with Fast Matrix Multiplication: For $1$5, the multipartite graph contains $1$6 vertices. Fast weighted triangle detection, using state-of-the-art matrix multiplication ($1$7), solves the minimum-weight triangle problem in $1$8 time, yielding a total complexity $1$9.

High-Level Pseudocode for the Algorithm

$(0,1)$5 Enumeration takes $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$0 time; triangle detection runs in $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$1.

3. Reductions to MAX-SAT and Minimum-Weight $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$2-Clique

Equivalence to Weighted Max-$$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$3-SAT

For even $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$4, there is a polynomial-time Karp reduction $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$5-$$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$6 Weighted Max-$$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$7-SAT on $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$8 variables. Each variable encodes inclusion of a particular basis vector. By constructing weighted $$B = (\mathbf b_1,\dots,\mathbf b_n) \in \mathbb Z^{m \times n}$$9-SAT clauses whose satisfaction records quadratic or higher-order combinations, the optimum truth assignment coincides (up to a shift) with the minimum lattice distance.

This reduction is tight for all even $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$0, showing a fine-grained equivalence (up to polynomial factors) between these problems.

Reduction to Minimum-Weight $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$1-Clique

The quadratic structure of the Euclidean norm allows reduction of $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$2-$$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$3 to minimum-weight $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$4-Clique detection. Assign each block’s subset sum to a vertex part, with the edge weights as described above. The optimal subset sum then corresponds exactly to the minimum total clique weight. For $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$5, the fastest algorithm for minimum-weight triangle detection fully translates to the best $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$6-$$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$7 algorithm to date.

4. Complexity Analysis and Fine-Grained Implications

Breaking the $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$8 barrier is achieved through applying triangle-detection with fast matrix multiplication algorithms, giving runtime $$\mathcal L(B) = \{\sum_{i=1}^n z_i\,\mathbf b_i\,|\,z_i \in \mathbb Z\}$$9, where $\mathbf t \in \mathbb Z^m$0 (Abboud et al., 7 Jan 2025). This is a significant super-constant improvement in the exponent over brute force enumeration.

Implications for Complexity Conjectures

• SETH Barriers: Known SETH-based lower bounds for CVP in other $\mathbf t \in \mathbb Z^m$1 norms (where $\mathbf t \in \mathbb Z^m$2 is not even) preclude $\mathbf t \in \mathbb Z^m$3 time algorithms. However, barriers for even $\mathbf t \in \mathbb Z^m$4 (in particular, Euclidean norm $\mathbf t \in \mathbb Z^m$5) were not established, and this result shows explicit progress, demonstrating the Euclidean case is algorithmically easier in this restricted setting.
• Minimum-Weight $\mathbf t \in \mathbb Z^m$6-Clique and APSP: It is widely conjectured that minimum-weight $\mathbf t \in \mathbb Z^m$7-clique remains $\mathbf t \in \mathbb Z^m$8-hard. Any breakthrough in $\mathbf t \in \mathbb Z^m$9-clique detection (e.g., faster than $d > 0$0) would imply faster $d > 0$1-$d > 0$2 solutions, thus providing cryptographic support to the $d > 0$3-clique and, by implication, APSP hardness conjectures.

5. Broader Connections and Cryptographic Consequences

A fine-grained equivalence exists between $d > 0$4-$d > 0$5 (for even $d > 0$6) and Weighted Max-$d > 0$7-SAT, meaning improvements for one translate precisely to the other (up to polynomial factors). This establishes direct links between lattice-based cryptographic hardness assumptions (in the worst case) and central fine-grained complexity conjectures—notably SETH (Strong Exponential Time Hypothesis), MAX-SAT, minimum clique, and APSP.

Additionally, prior fine-grained hardness reductions for general $d > 0$8 all make essential use of the $d > 0$9 case. Consequently, the new algorithms show that, in the Euclidean setting, such restricted reductions cannot exclude sub-$(0,1)$0 time; any unconditional lower bound must leverage the structure of general coefficients outside $(0,1)$1.

6. Summary and Impact

The discovery of a sub-$(0,1)$2 exact algorithm for $(0,1)$3-$(0,1)$4 establishes a new frontier for both the algorithmic understanding of lattice problems and their connections to classical hypotheses in fine-grained complexity. The equivalences and reductions presented inform ongoing debates regarding the fine-grained hardness of cornerstone combinatorial and cryptographic problems, illustrating how algorithmic advances in specialized lattice settings feed directly into broader complexity-theoretic landscapes (Abboud et al., 7 Jan 2025).