Given multiple sets, how do I find the smallest subset of each set that is unique to all other subsets?

Question

Background. I have numbers 1 to 20 (white digits on black background) that can appear on screen, and I wish to identify these numbers. As they cannot be simply copy-pasted, I will be comparing the position of the white pixels of the number on-screen with a list of positions of white pixels of all 20 numbers. However, each number can have a large number of pixels, and comparing all of those pixels may not be necessary to identify that number. Hence, I'd like to make the least number of comparisons as possible.

Algorithmic question: I have multiple sets with elements that are unique within each set, but may not be unique across all sets. How can I find the smallest possible subset of each set such that each subset is unique?

Example 1: Let A = {1, 2}, and B = {3, 4}. The smallest subset of A and B would be {1} and {3} (or {2} and {4}), as those subsets are unique to each original set, and are as small as possible.

Example 2: Let A = {1, 2, 3, 4}, B = {1, 2, 3, 5}, C = {1, 2, 4, 5}. Possible smallest subsets are {3, 4}, {3, 5}, {4, 5}. If any element were removed from any of those subsets, then that subset can also belong to a different set. E.g. removing 4 from the first subset will leave {3}, which makes it ambiguous if the {3} identifies the first or the second set.

Answer 1

It's a solution with O(n^3) time complexity, and O(n) memory complexity. (If I am not wrong)

function isElement(elem, s) {
  return s.includes(elem)
}

function isId(id, sets) {
  let setsWithSuchElementsNumber = 0
  for (const s of sets) {
    if (id.every((e) => isElement(e, s))) {
      setsWithSuchElementsNumber++
    }
  }

  return setsWithSuchElementsNumber === 1
}

function getSetId(s, sets) {
  const count = {}
  const elements = []
  for (const elem of s) {
    if (count[elem] == null) {
      elements.push(elem)
    }
    count[elem] = 0
  }

  for (const otherSet of sets) {
    for (const e of elements) {
      if (isElement(e, otherSet)) {
        count[e]++
      }
    }
  }

  elements.sort((first, second) => {
    return Math.sign(count[first] - count[second])
  })

  for (let idSize = 1; idSize <= elements.length; idSize++) {
    const possibleId = elements.slice(0, idSize)

    if (isId(possibleId, sets)) {
      return possibleId
    }
  }

  return null
}

const getSetIds = (sets) => {
  return sets.map((s) => getSetId(s, sets))
}

const res = getSetIds([
  [1, 2, 3, 4],
  [1, 2, 3, 5],
  [1, 2, 4, 5],
])

console.log(res.join(' '))