Consider a set $X = \{x_1, \dots, x_n\}$ of positive real numbers (or natural numbers, if you like) to be a set of resistors. For any subset $S \subset X$, we can build resistive circuits and measure the resistance between any two points in a circuit. Is it possible to determine (or even just verify) the maximum number in $X$ (i.e. $\arg\max_i x_i$) with fewer than $n$ measurements? (It is trivial to get $n$, just measure each resistor individually.)
(A resistive circuit is a connected graph $G = (V, E)$, where each edge has a resistance $r(i, j)$, here we ask that $r$ is a bijection between $E$ and some subset $S \subset X$. A measurement between $s, t \in V$ involves inputting 1 unit of current at $s$, which uniquely determines potentials $p(i)$ for every $i \in V$ by physical laws, and the effective resistance $\tilde{r}(s, t)$ between $s$ and $t$ is $|p(s) - p(t)|$ by Ohm's law. So in other words, we are allowed to read $< n$ numbers from the (finite) set $\{\tilde{r}(s, t) : S \subset X, (G, r)$ is a resistive circuit with $r$ a bijection between $E$ and $S$, $s, t \in V\}$. For more details, please see any book about network flows, in particular electrical flows.)
For an easier variant, restrict the resistive circuit to be a series circuit. Then there is only one way to measure each subset $S \subset X$, in particular, by taking the sum of the elements, so we are reading $< n$ numbers from the set $\{\sum_{x_i \in S} x_i : S \subset X\}$.
Is anything known about this problem? Lower bounds, randomized solutions, approximation algorithms, interactive proofs, etc. are all welcome, if relevant.
