Peel the potatoes, boil them in salted water, let them cool and cut into slices. Grate the carrots, core the apple and cut into cubes. Halve the pomegranates and use a spoon to scrape the seeds out of the skin, collecting the juice.
Mix the carrots, potatoes, apple and pomegranate seeds together.
Dressing:
Mix the oil, balsamic vinegar and pomegranate juice together. Season with salt, pepper and sugar and pour over the salad.
Roast the walnuts and serve the salad sprinkled with the kernels and mint.