Complexity of integer quasiconvex polynomial optimization

Abstract

We study a particular case of integer polynomial optimization: Minimize a polynomial F ^ on the set of integer points described by an inequality system F 1 ⩽ 0 , … , F s ⩽ 0 , where F ^ , F 1 , … , F s are quasiconvex polynomials in n variables with integer coefficients. We design an algorithm solving this problem that belongs to the time-complexity class O ( s ) · l O ( 1 ) · d O ( n ) · 2 O ( n 3 ) , where d ⩾ 2 is an upper bound for the total degree of the polynomials involved and l denotes the maximum binary length of all coefficients. The algorithm is polynomial for a fixed number n of variables and represents a direct generalization of Lenstra's algorithm [Math. Oper. Res. 8 (1983) 538–548] in integer linear optimization. In the considered case, our complexity-result improves the algorithm given by Khachiyan and Porkolab [Discrete Comput. Geom. 23 (2000) 207–224] for integer optimization on convex semialgebraic sets.