Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor theory is the {\it rank} of a divisor on a graph. The importance of the rank is well illustrated by Baker's {\it Specialization lemma}, stating that the dimension of a linear system can only go up under specialization from curves to graphs, leading to a fruitful interaction between divisors on graphs and curves. Due to its decisive role, determining the rank is a central problem in graph divisor theory. Kiss and T\'othm\'eresz reformulated the problem using chip-firing games, and showed that computing the rank of a divisor on a graph is NP-hard via reduction from the Minimum Feedback Arc Set problem. In this paper, we strengthen their result by establishing a connection between chip-firing games and the Minimum Target Set Selection problem. As a corollary, we show that the rank is difficult to approximate to within a factor of $O(2^{\log^{1-\varepsilon}n})$ for any $\varepsilon > 0$ unless $P=NP$. Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to approximate to within a factor of $O(n^{1/4-\varepsilon})$ for any $\varepsilon>0$.

翻译：Baker与Norine开创了图除子的研究，将其作为黎曼面上Riemann-Roch理论的图论类比。图除子理论的核心概念之一是图上除子的{\it 秩}。Baker的{\it 特化引理}充分说明了秩的重要性，该引理指出线性系统的维数在从曲线特化到图的过程中只能增加，这导致了图除子与曲线除子之间的有效互动。由于秩的关键作用，确定秩成为图除子理论的核心问题。Kiss与Tóthmérész利用芯片点火博弈重新表述了该问题，并证明通过最小反馈弧集问题的归约，计算图上除子的秩是NP难的。本文通过建立芯片点火博弈与最小目标集选取问题之间的联系，加强了他们的结论。作为推论，我们证明除非P=NP，否则对于任意$\varepsilon>0$，秩难以在$O(2^{\log^{1-\varepsilon}n})$因子内近似。此外，假设植入稠密子图猜想成立，对于任意$\varepsilon>0$，秩难以在$O(n^{1/4-\varepsilon})$因子内近似。