Who should pay for it? Fathers? Mothers? The state?

Most answers to such tests are fairly straightforward, but there are complicated cases such as chimeras, abusive and nonconsensual situations, group sex/sex with different partners within tight timeframes which includes the potential for semen transfer, maybe even genetic mutations (if only one gene is off, how do you know it wasn't a random mutation?) which make it necessary that the staff who deliver test results of an unexpected nature need to know the science and have great people skills so they know when to calm people down and investigate further. So no matter how cheap the actual test gets, to be done with appropriate duty of care the process of mass testing is always going to involve individualised attention from trained staff, and that gets spendy.

Should men pay because theirs is the DNA in question? Should women pay because we know the baby is theirs? Should the state pay because it is generally good for society? If a child is found to be the progeny of an unexpected father, should he be hunted down and the costs extracted from him?