Place the potatoes in a medium saucepan or pressure cooker (see Note). Add 2 teaspoons salt and enough water to cover the potatoes. Boil the potatoes until they are tender when pierced with the point of a knife but hold their shape, 30 minutes. Drain the potatoes, transfer them to a large mixing bowl, and allow them to cool completely. Add the chopped eggs, mayonnaise, and sweet relish, and fold gently to combine. Add salt and pepper to taste. Refrigerate until ready to serve.
The potatoes may be cooked in a pressure cooker. Sprinkle salt over the potatoes. Follow the manufacturer's instructions and pressure-cook for 5 minutes. Release the pressure immediately and drain and cool the potatoes.