Balancing selection can maintain polymorphisms in natural populations for extended periods of evolutionary time. However, in this paper, Dannemann et al. 2016 identify three archaic haplotypes in the modern human genome that are due to adaptive introgression. Why can we exclude that haplotypes III, IV and VII (the introgressed haplotypes) have remained in the human gene pool since the split between modern humans and Neanderthals because of balancing selection?
Balancing selection and positive selection are not mutually exclusive. Neither are balancing selection and negative selection. Sickle cell is a classic example where there is negative selection against the homozygote for the sickling mutation, and positive selection for the heterozygote. The result is that there both alleles are maintained in the population under balancing selection.