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For a finite set $S$ of $n$ elements, say a weight function is a function $f \colon S \to \mathbb{R}$. For any subset $T \subseteq S$, define $f(T) = \sum _{x \in T} f(x)$. Define two weight functions $f_1, f_2$ on the same set $S$ to be equivalent if $f_1(T_1) \leq f_1(T_2) \Leftrightarrow f_2(T_1) \leq f_2(T_2)$ for all sets $T_1, T_2 \subseteq S$.

How many non-equivalent weight functions are there on a set of $n$ elements, asymptotically as a function of $n$? It follows from a result of Frank and Tardos (Thm 3.3) that there are at most $2^{O(n^4)}$ distinct weight functions, but I am interested to know what the optimal degree is of the polynomial in the exponent.

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  • $\begingroup$ There's an obvious lower bound of $n!$ since every ordering of the singleton sets is obviously achievable. $\endgroup$ – Tony Huynh Nov 22 '19 at 12:32
  • $\begingroup$ Are there any exact counts for small $n$? I see that if we restrict $|T_1|=|T_2|=2$, then we will get OEIS A231074 or alike. $\endgroup$ – Max Alekseyev Dec 3 '19 at 21:41
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You are asking for the number $T_n$ of threshold functions on the set $\{-1,1\}^n$. According to https://arxiv.org/pdf/1903.06595.pdf (Section 1.2), Zuev showed that $\log_2T_n\sim n^2$.

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    $\begingroup$ Thanks for the reference. While I agree that the number of non-equivalent weight functions is at least the number of threshold functions, I don't see the other direction of the equivalence. How would you map each weight function $f$ to a threshold function $g$, so that non-equivalent weight functions map to different threshold functions? The obvious way seems to be picking a threshold value $t$ and letting $g(S) = 1$ iff $w(S) \geq t$, but then functions which are non-equivalent for subsets of weight less than $t$, may get mapped to the same threshold function. $\endgroup$ – Bart Jansen Nov 22 '19 at 15:31
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It turns out my question is answered in a recent paper by Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn on the complexity of integer programming. Letting $w \in \mathbb{R}^n$ be the vector of weights, the weight of a subset $S' \subseteq S$ can be written by letting $x_{S'}$ be the $0/1$-characteristic vector of $S'$, so that the weight of $S'$ becomes the inner product $w \cdot x_{S'}$.

Theorem 65 in the cited paper shows that for any linear function $f(x) = w \cdot x$ for $x \in \{0,1\}^n$, there is an equivalent weight function $g(x) = w' \cdot x$ such that $w' \in \{- n(12n)^n, \ldots, n(12n)^n\}$, that is, each weight is an integer whose absolute value is $n \cdot n^{O(n)}$. Hence each weight function has a representation with each of the $n$ weights belonging to this range, giving an upper-bound of at most $(1+2n \cdot n^{O(n)})^n = 2^{O(n^2 \log n)}$ nonequivalent weight functions.

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