Assume that $G=(V,E)$ is an undirected connected graph and that $H: V \to \mathbb R$ is a function that assign at each vertex $v \in V$ its height $H(v)$. Think of the pair $(G,H)$ as an energy landscape where local minima of $H$ are "valleys" and local maxima are "mountain peaks".

We indicate a path in $G$ from $v$ to $w$ as $\omega: v \to w$ and we denote by $\Phi_{\omega} := \max _{z \in \omega} H(z)$ the maximum height that the path $\omega$ reaches. Given $v,w \in V$, define the energy barrier between $v$ and $w$ by $$\Phi(v,w):= \min_{\omega: v\to w} \Phi_{\omega},$$ where the minimum is taken over all possible paths from $v$ to $w$.

Is there in literature any fast algorithm to find $\Phi(v,w)$? What is the best way to design such algorithm if we need to compute all the energy barriers $\{\Phi(v,w)\}_{v\in V}$ for a fixed target vertex $w$. Is there a clever way to map this problem into one that is well-known, such as shortest path problem on an opportunely modified weighted graph $G'$? Are you aware of any implementation of such optimization problem in software such as Mathematica or Matlab?

Any reference, idea or fruitful comment is welcomed!

(Crossposted on StackExchange since I received no answer).

Assume that $G=(V,E)$ is an undirected connected graph and that $H: V \to \mathbb R$ is a function that assign at each vertex $v \in V$ its height $H(v)$. Think of the pair $(G,H)$ as an energy landscape where local minima of $H$ are "valleys" and local maxima are "mountain peaks".

We indicate a path in $G$ from $v$ to $w$ as $\omega: v \to w$ and we denote by $\Phi_{\omega} := \max _{z \in \omega} H(z)$ the maximum height that the path $\omega$ reaches. Given $v,w \in V$, define the energy barrier between $v$ and $w$ by $$\Phi(v,w):= \min_{\omega: v\to w} \Phi_{\omega},$$ where the minimum is taken over all possible paths from $v$ to $w$.

Is there in literature any fast algorithm to find $\Phi(v,w)$? What is the best way to design such algorithm if we need to compute all the energy barriers $\{\Phi(v,w)\}_{v\in V}$ for a fixed target vertex $w$. Is there a clever way to map this problem into one that is well-known, such as shortest path problem on an opportunely modified weighted graph $G'$? Are you aware of any implementation of such optimization problem in software such as Mathematica or Matlab?

Any reference, idea or fruitful comment is welcomed!

(Crossposted on StackExchange since I received no answer).