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Given an positive integer $n$, we define an order of subset sums (or simply, an order, when there is no ambiguity) to be a sequence of all subsets of $\{1,\ldots,n\}$. For example, when $n=2$, the sequence $\emptyset,\{1\},\{2\},\{1,2\}$ is an order.

We call an order $S_1,\ldots,S_{2^n}$ valid if there exist real numbers $0<x_1<\cdots<x_n$ such that $\sum_{i\in S_1}x_i<\cdots<\sum_{i\in S_{2^n}}x_i$. For example, when $n=2$, the sequence $\emptyset,\{1\},\{2\},\{1,2\}$ is a valid order, but the sequence $\emptyset,\{1\},\{1,2\},\{2\}$ is not because we cannot make $x_1+x_2<x_2$.

The question is, given $n$, how to enumerate all possible valid orders. To output an order $S_1,\ldots,S_{2^n}$, it is sufficient to output an algorithm that generates this order, i.e., an algorithm with no input (the parameter $n$ is built-in) that outputs $S_1,\ldots,S_{2^n}$ in order, as long as the algorithm runs in $O(n2^n)$ time. Even so, this problem still cannot be solved in time polynomial in $n$, because there may be exponentially many valid orders, thus an algorithm running in exponential time is welcome.

A trivial algorithm would be to iterate over all possible orders, and check for each one if it is valid. But I cannot even find an (efficient) way to check if an order is valid.

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  • $\begingroup$ As far as I understand, you are asking about Boolean Lattice (please see: en.wikipedia.org/wiki/Boolean_algebra_(structure)), where you introduce a total order using an additional vector $X=(x_1,x_2,...,x_n)$. Your "valid order" is used to be called a chain. It's unclear why you define this total order using existence of $X$ only - it will depend on values in $X$. Also for totally ordered set there will be only one chain (provided that there are no pairwise equal elements). $\endgroup$
    – HEKTO
    Feb 23 '20 at 23:10
  • $\begingroup$ @HEKTO Yes, I'm essentially seeking for a method to enumerate all possible chains as the values of $x_1,\ldots,x_n$ vary. However, I don't understand how this problem is related to the Boolean algebra since this problem talks about orders of subsets. $\endgroup$
    – xskxzr
    Feb 24 '20 at 1:51
  • $\begingroup$ If you want to know what happens with chains when the vector $X$ varies - then you should clarify your question. The set of subsets of a finite set is Boolean algebra and Boolean lattice (= $n$-dimensional hypercube), where each node is a Boolean $n$-tuple $\endgroup$
    – HEKTO
    Feb 24 '20 at 4:04
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    $\begingroup$ @HEKTO I now understand the relation between subsets and Boolean algebra, but how can this help solve the problem? I don't know how to clarify since I don't know the definition of "chain" and I have a sense that your "chain" is not equivalent to my "valid order". Maybe it is more proper to say that I'm essentially seeking for a method to enumerate all possible total orders as the values of $x_1,\ldots,x_n$ vary. I think the question statement is clear though it does not mention any concept about boolean lattice. $\endgroup$
    – xskxzr
    Feb 24 '20 at 5:47
  • $\begingroup$ Chain is a totally ordered subset of a poset. Maximum chain is a chain, which is not contained in any other chain. Totally ordered set consists of a single maximum chain. $\endgroup$
    – HEKTO
    Feb 24 '20 at 15:14
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Here is one algorithm that runs in doubly exponential time -- so probably useless. In other words, this answer is probably not useful.

You can check whether an order is valid in $2^{O(n)}$ time (i.e., $\text{poly}(2^n)$ time) by reducing to linear programming: you obtain $2^n+n-2$ linear inequalities in $n$ variables, and you can test for feasibility in time polynomial in the number of inequalities and variables. Then, enumerating over all possible sums and checking if each is valid would yield a $(2^n)! 2^{O(n)} = 2^{n 2^n + O(2^n)}$ time algorithm. This will be ridiculously slow in practice.


To improve this, you could use an iterative pruning / branch-and-bound style algorithm. Given a prefix $S_1,\dots,S_k$ of an order, we can test whether it is valid using linear programming as above. If it is not, we don't consider any order that starts with this prefix. In other words, do breadth-first search: first choose $S_1$, then choose $S_2$, then choose $S_3$, and so on, at each step checking whether the prefix obtained so far is valid.

This may still be extremely slow, as there is the possibility that a prefix $S_1,\dots,S_k$ is valid but there is no way to complete it to a full order $S_1,\dots,S_{2^n}$ that is valid, so this might do a lot of wasted work.

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