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(ϕ) · n n+o(n) (where n and ϕ 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). 1. Introduction. Bounded integer programming is a familiar problem with many computer scientists and mathematicians. BIP asks for an integral vector x satisfying the system Ax = b of equations and some constraints 0 ≤ x ≤ u. If there is no upper bound on the variables, it is called Integer Programming (IP). It is called the Bounded Knapsack Problem (BKP) if BIP has unique equation. If there is no upper bound on the variables in BKP, then it is called the Knapsack Problem (KP). These problems are extensively surveyed in the literature, [10]. However, up to now they still need much time to be solved. In 1983, Lenstra [7] first showed that IP is solvable in polynomial time when the dimension is fixed. After this breakthrough result, researchers continue to improve it and thus many substantial improvements were proposed. The most remarkable result is from [5], where Kannan showed that IP is solvable in deterministic time poly(ϕ) · n 2.5n. In his proof of this time bound, lattice problems and approximating subspaces play an important role. Recently, Khoát [6] showed that BIP is solvable in deterministic time poly(ϕ) · n 2n+o(n). Moreover, there are some more interesting results for some other problems. For example, IP in standard form was shown to be solvable in time poly(ϕ) · n n+o(n). He obtained these results by reducing these problems to KP, and then reducing KP to SAP, a lattice problem. The reduction from IP to KP is almost efficient in the sense that it preserves the time bound. However, the reduction from BIP to KP is inefficient. The reason is that the number of variables increases doubly after the reduction. Thus, the time …