A Polynomial-Time Algorithm for the Planar Quantified Integer Programming Problem

This paper is concerned with the design and analysis of a polynomial-time algorithm for an open problem in Planar Quantified Integer Programming (PQIP), whose polynomial-time solution is previously unknown. Since the other three classes of PQIP are known to be in PTIME, we also accomplish the proof that PQIP is in PTIME. Among its practical implications, this problem is a model for an important kind of scheduling. A challenge to solve this problem is that using quantifier elimination is not a valid approach. This algorithm exploits the fact that a PQIP can be horizontally partitioned into slices and that the feasibility of each slice can be checked efficiently. We present two different solutions to implement the subroutine CheckSlice: The first one uses IP(3), i.e., Integer Programming with at most three non-zero variables per constraint; the other is based on counting lattice points in a convex polygon. We compare the features of these two solutions.