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- W4308668832 abstract "Approximate integer programming is the following: For a convex body $K subseteq mathbb{R}^n$, either determine whether $K cap mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity $c$. Approximate integer programming can be solved in time $2^{O(n)}$ while the fastest known methods for exact integer programming run in time $2^{O(n)} cdot n^n$. 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 $x^* in (K cap mathbb{Z}^n)$ can be found in time $2^{O(n)}$, provided that the remainders of each component $x_i^* mod{ell}$ for some arbitrarily fixed $ell geq 5(n+1)$ of $x^*$ 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 $2^{O(n)}n^n$ algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carath'eodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form $Ax = b, 0 leq x leq u, , x in mathbb{Z}^n$ . Such a problem can be reduced to the solution of $prod_i O(log u_i +1)$ approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range $0 leq x_i leq p(n)$ can be solved in time $(log n)^{O(n)}$. For these problems, the best running time so far was $n^n cdot 2^{O(n)}$." @default.
- W4308668832 created "2022-11-14" @default.
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- W4308668832 date "2022-11-07" @default.
- W4308668832 modified "2023-10-17" @default.
- W4308668832 title "From approximate to exact integer programming" @default.
- W4308668832 doi "https://doi.org/10.48550/arxiv.2211.03859" @default.
- W4308668832 hasPublicationYear "2022" @default.
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