Optimal coalition structure

The problem is the following: if there is a set S = {x1, ..., x_n} and a function f: set -> number, which takes a set as an input and returns a number as an output, what is the best possible coalition structure (coalition structure a set of subsets of a S.That is, find the subsets,such that the sum of f(s_i) for every subset s_i in S is maximal). The sets in the coalition should not overlap and their union should be S. A template is this:

def optimal_coalition(coalitions):
    """
    :param coalitions: a dictionary of the form {coalition: value}, where coalition is a set, and value is a number
    :return:
    """


optimal_coalition({set(1): 30, set(2): 40, set(1, 2): 71}) # Should return set(set(1, 2))

This is from a paper I found: enter image description here

Answer

I transliterated the pseudocode. No doubt you can make it better — I was hewing very closely to show the connection.

I did fix a bug (Val(C') + Val(C C') > v(C) should be Val(C') + Val(C C') > Val(C), or else we may overwrite the best partition with one merely better than all of C) and two typos (C / C' should be C C'; and CS* is a set, not a tree).

import itertools


def every_possible_split(c):
    for i in range(1, len(c) // 2 + 1):
        yield from map(frozenset, itertools.combinations(tuple(c), i))


def optimal_coalition(v):
    a = frozenset(x for c in v for x in c)
    val = {}
    part = {}
    for i in range(1, len(a) + 1):
        for c in map(frozenset, itertools.combinations(tuple(a), i)):
            val[c] = v.get(c, 0)
            part[c] = {c}
            for c_prime in every_possible_split(c):
                if val[c_prime] + val[c - c_prime] > val[c]:
                    val[c] = val[c_prime] + val[c - c_prime]
                    part[c] = {c_prime, c - c_prime}
    cs_star = {a}
    while True:
        for c in cs_star:
            if part[c] != {c}:
                cs_star.remove(c)
                cs_star.update(part[c])
                break
        else:
            break
    return cs_star


print(
    optimal_coalition({frozenset({1}): 30, frozenset({2}): 40, frozenset({1, 2}): 69})
)