We study the integer minimization of a quasiconvex polynomial with
quasiconvex polynomial constraints. We propose a new algorithm that is an
improvement upon the best known algorithm due to Heinz (Journal of Complexity,
2005). This improvement is achieved by applying a new modern Lenstra-type
algorithm, finding optimal ellipsoid roundings, and considering sparse
encodings of polynomials. For the bounded case, our algorithm attains a
time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound
on the number of monomials in each polynomial and r is the binary encoding
length of a bound on the feasible region. In the general case, s l^{O(1)}
d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total
degree of the polynomials and l bounds the maximum binary encoding size of the
input.Comment: 28 pages, 10 figure