Place sliced cucumber in a bowl and sprinkle with a bit of salt. Let sit until tender, about 5 minutes. Squeeze cucumber to remove excess water. Place drained cucumber in a small bowl.
Place potatoes in a pot and rinse to remove extra starch. Drain. Add fresh water to cover. Add eggs and 1/2 teaspoons salt. Place pot over high heat and bring to a boil. Reduce heat to medium and cook until potatoes are tender, 10 to 15 minutes.
Transfer cooked eggs to a bowl filled with cold water. Drain potatoes well and return them to the empty pot. Season with salt and black pepper and mash potatoes while still hot; leave some chunks for texture. Peel eggs, cut into chunks, and add to the pot with the potatoes. Add cucumber, imitation crab, and corn. Add mayonnaise, vinegar, salt, and black pepper to taste.
Chill potato salad before serving, about 30 minutes.
