On the complexity of quasiconvex integer minimization problem

TL;DR

This paper investigates the computational complexity of integer minimization for quasiconvex and conic functions, revealing polynomial algorithms for conic functions and proving hardness results for quasiconvex functions.

Contribution

It establishes the non-existence of polynomial algorithms for quasiconvex functions and provides polynomial algorithms for conic functions, along with complexity bounds and reductions.

Findings

No polynomial algorithm exists for quasiconvex functions using comparison oracle.

Polynomial algorithms exist for conic functions in the same setting.

Exponential lower bounds on oracle complexity are shown for conic functions.

Abstract

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered in the paper, assuming that an optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $\log R$ to optimize quasiconvex functions in the ball of integer radius $R$ using only the comparison oracle. On the other hand, if an optimized function is conic, then we show that there is a polynomial on $\log R$ algorithm. We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.