On any diet those are problems. You're generally limiting your food and calories so you don't always hit the RDA. In fact, I'd be willing to bet most "standard" diets are deficient, or would be if people weren't eating things with vitamins added to them. I take a few supplements on nutrients I'm low on.

I have been constipated on just about EVERY diet I've been on, low carb, low fat, low calorie. The problem is you just have a lot less bulk to pass and it tends to hang out and get hard.

The solution is to get added fiber in the diet. Which is easy to do.