A Faster Algorithm for Quasi-convex Integer Polynomial Optimization

We present a faster exponential-time algorithm for integer optimization over quasi-convex polynomials. We study the minimization of a quasiconvex polynomial subject to s quasi-convex polynomial constraints and integrality constraints for all variables. The new algorithm is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). A lower time complexity is reached through applying a stronger ellipsoid rounding method and applying a recent advancement in the shortest vector problem to give a smaller exponential-time complexity of a Lenstra-type algorithm. For the bounded case, our algorithm attains a time-complexity of s(rlMd)O(1) 22n log2(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, slO(1) dO(n) 22n log2(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 inpu...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Robert Hildebrand, Matthias Köppe
• Keywords: Algorithm, CORR 2010, Education, Polynomials, Quasi-convex Polynomial