Phase transition in a sequential assignment problem on graphs

Abstract

We study the following sequential assignment problem on a finite graph $G=(V,E)$. Each edge $e\in E$ starts with an integer value $n_{e}\ge0$, and we write $n=\sum_{e\in E}n_{e}$. At time $t$, $1\le t\le n$, a uniformly random vertex $v\in V$ is generated, and one of the edges $f$ incident with $v$ must be selected. The value of $f$ is then decreased by $1$. There is a unit final reward if the configuration $(0,\ldots,0)$ is reached. Our main result is that there is a phase transition: as $n\to\infty$, the expected reward under the optimal policy approaches a constant $c_{G}>0$ when $(n_{e}/n:e\in E)$ converges to a point in the interior of a certain convex set $\mathcal{R}_{G}$, and goes to $0$ exponentially when $(n_{e}/n:e\in E)$ is bounded away from $\mathcal{R}_{G}$. We also obtain estimates in the near-critical region, that is when $(n_{e}/n:e\in E)$ lies close to $\partial\mathcal{R}_{G}$. We supply quantitative error bounds in our arguments.