3 research outputs found
From approximate to exact integer programming
Approximate integer programming is the following: For a convex body , either determine whether is
empty, or find an integer point in the convex body scaled by from its
center of gravity . Approximate integer programming can be solved in time
while the fastest known methods for exact integer programming run in
time . 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 can be
found in time , provided that the remainders of each component for some arbitrarily fixed of 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
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 . Such a problem can be
reduced to the solution of approximate integer
programming problems. This implies, for example that knapsack or subset-sum
problems with polynomial variable range can be solved in
time . For these problems, the best running time so far was
Lattice sparsification and the Approximate Closest Vector Problem
We give a deterministic algorithm for solving the
(1+\eps)-approximate Closest Vector Problem (CVP) on any
-dimensional lattice and in any near-symmetric norm in
2^{O(n)}(1+1/\eps)^n time and 2^n\poly(n) space. Our algorithm
builds on the lattice point enumeration techniques of Micciancio and
Voulgaris (STOC 2010, SICOMP 2013) and Dadush, Peikert and Vempala
(FOCS 2011), and gives an elegant, deterministic alternative to the
"AKS Sieve"-based algorithms for (1+\eps)-CVP (Ajtai, Kumar, and
Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the
existence of a \poly(n)-space and -time algorithm for
exact CVP in the norm, the space complexity of our algorithm
can be reduced to polynomial.
Our main technical contribution is a method for "sparsifying" any
input lattice while approximately maintaining its metric structure. To
this end, we employ the idea of random sublattice restrictions, which
was first employed by Khot (FOCS 2003, J. Comp. Syst. Sci. 2006) for
the purpose of proving hardness for the Shortest Vector Problem (SVP)
under norms.
A preliminary version of this paper appeared in the Proc. 24th Annual
ACM-SIAM Symp. on Discrete Algorithms (SODA'13)
(http://dx.doi.org/10.1137/1.9781611973105.78)