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Approximating the rank of graph divisors is hard

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 rank of a divisor on a graph. Kiss and Tóthmérész reformulated the problem using chip-firing games, and showed that computing the rank of a divisor on a graph is NP-hard. In a recent paper on Approximating the rank of graph divisors, 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.