Year: 2024
Type: article
Abstract: Approximate integer programming is the following: For a given convex body $$K \subseteq {\mathbb {R}}^n$$ K ⊆ R n , either determine whether $$K \cap {\mathbb {Z}}^n$$ K ∩ Z n is empty, or find an integer point in the convex body $$2\cdot (K - c) +c$$ 2 · ( K - c ) + c which is K , scaled by 2 from its center of gravity c . Approximate integer programming can be solved in time $$2^{O(n)}$$ 2 O ( n ) while the fastest known methods for exact integer programming run in time $$2^{O(n)} \cdot n^n$$ 2 O ( n ) · n n . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point $$x^* \in (K \cap {\mathbb {Z}}^n)$$ x ∗ ∈ ( K ∩ Z n ) can be found in time $$2^{O(n)}$$ 2 O ( n ) , provided that the remainders of each component $$x_i^* \mod \ell $$ x i ∗ mod ℓ for some arbitrarily fixed $$\ell \ge 5(n+1)$$ ℓ ≥ 5 ( n + 1 ) of $$x^*$$ x ∗ are given. The algorithm is based on a cutting-plane technique , iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a <
Source: Mathematical Programming
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4395082536