Originally Answered: low carb diet or vegetable/fruit diet work the fastest?
Did you want to lose weight for a few months, or do you prefer to keep weight off permanently? Gina Kolata showed that the most popular diets are the least effective over the long term for the vast majority of people. That included diets called low carb (like Atkins), Weight Watchers, and physician supervised calorie restricted diets.
The fastest way to lose weight is a physician supervised fast where the only thing consumed is water. I do this diet every six months or so for about 5 days at a time. This cleanses the body and allows your digestive system to rest and allow other systems more time to focus on clean up. You will lose water weight and muscle weight, but these masses come back once you resume your normal diet. You can expect to lose as much as 5 pounds per week of fat weight each week. Sadly, when you resume your normal diet, your weight will come back if you eat the Standard American diet.
The best combination is eat a "whole plant based diet" as described by Dr. Caldwell Esselstyn, Jr. This is a permanent dietary change and it is endorsed by most reputable and independent nutrition researchers, including Professor T. Colin Campbell, Dr. Joel Fuhrman, Dr. John McDougall, and Dr. Neal Barnard.
I lost 90 pounds permanently by eating a balanced diet including leafy greens, whole grains, legumes, vegetables, and fruits. The only supplements I take are a teaspoon of ground flax seeds for Omega-3s, vitamin D, and vitamin B12. I avoid animal based products because of the high levels of cholesterol, animal proteins, and fats. My vascular system has improved significantly. My health has never been better and I feel fantastic and mentally alert.
I'd use a water fast as described by "Fasting and Eating for Health" by Dr. Joel Fuhrman. You can expect some "allergy like" reactions the first two or three days when fasting. Make sure to consult with a physician to make sure you don't have medical conditions or take medications which would contraindicate a fast. A water fast lasting more than three days must be supervised by a medical professional with training in the area of fasting. You can find retreats offering medically supervised fasts held periodically throughout the US and around the world.
I lost an average of about 2 pounds a week of fat tissue by just eating a "whole plant based diet" as described by the authors I listed before. Of course the bathroom scale will show water weight loss and muscle weight loss as well.
You might take up a light resistance training program because muscle is more calorie dense than fat tissue.
I lost 90 pounds by just using diet. I maintain that weight loss by eating this same healthy diet. It's optimal for health for most people. Some people may need to eat animal proteins because of genetic problems or specific diseases, but most people would improve their health by going pure vegetarian.