On Bounded Integer Programming

We present an efficient reduction from the Bounded integer programming (BIP) to the Subspace avoiding problem (SAP) in lattice theory. The reduction has some special properties with some interesting consequences. The first is the new upper time bound for BIP, $poly(\varphi)\cdot n^{n+o(n)}$ (where $n$ and $\varphi$ are the dimension and the input size of the problem, respectively). This is the best bound up to now for BIP. The second consequence is the proof that #SAP, for some norms, is #P-hard under semi-reductions. It follows that the counting version of the Generalized closest vector problem is also #P-hard under semi-reductions. Furthermore, we also show that under some reasonable assumptions, BIP is solvable in probabilistic time $2^{O(n)}$.

💡 Research Summary

The paper tackles the long‑standing challenge of improving the algorithmic complexity of Bounded Integer Programming (BIP). After reviewing classical results—Le Nstra’s polynomial‑time algorithm for fixed dimension and Kannan’s poly(ϕ)·n^{2n+o(n)} deterministic bound—the author observes that existing reductions from BIP to the Subspace Avoiding Problem (SAP) via the Knapsack Problem (KP) inflate the dimension to 4n+2, destroying the tightness of the bound. To overcome this, the work introduces a direct reduction from the Bounded Knapsack Problem (BKP) to SAP that only adds n+2 variables, yielding a lattice L₀ of dimension 2n+2.

The construction of L₀ relies on a carefully crafted basis matrix B₀ containing a diagonal scaling matrix \hat U_n, large integer parameters s₀, s₁, λ, and auxiliary vectors. The target subspace S is defined by the linear equation Σ a_i u_i x_i = 0. SAP then asks for the shortest lattice vector in L₀ \ S. Lemma 2.10 proves that any vector violating the two structural constraints (zero entries at positions n+1 and 2n+2, and a coupling condition between paired coordinates) must have a coordinate of magnitude at least Θ(p), where p = min{s₀, s₁, λ/(δ·n·u_max)}. This guarantees that any “short” vector must satisfy the constraints and therefore encodes a feasible BKP solution.

Lemmas 2.11–2.13 establish a tight correspondence between shortest vectors and BKP solutions: under the ℓ_∞ norm the shortest vector has norm ≤ 1, while under the ℓ₁ norm its norm is exactly n. Moreover, each feasible BKP solution yields two SAP solutions (sign‑symmetric), and each SAP solution maps back to a BKP solution after scaling by the upper‑bound vector u. Consequently, Theorems 2.6 and 2.7 assert that the BKP→SAP reduction is Karp‑complete and incurs only an n+2 dimension increase.

Since SAP reduces to the Closest Vector Problem (CVP) in polynomial time, the author leverages known CVP algorithms. For the ℓ₁ norm, CVP can be solved in poly(ψ)·n^{n/2+o(n)} time (see reference