On approximating the rank of graph divisors

Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue Riemann-Roch theory for Riemann surfaces. One key concepts divisor is rank on graph. The importance well illustrated by Baker's Specialization lemma, stating that dimension linear system can only go up under specialization from curves to graphs, leading fruitful interaction between graphs curves. Due its decisive role, determining central problem in theory. Kiss Tóthmérész reformulated using chip-firing games, showed computing NP-hard via reduction Minimum Feedback Arc Set problem. In this paper, we strengthen their result establishing connection games Target Selection As corollary, show difficult approximate within factor O(2log1−ε⁡n) any ε>0 unless P=NP. Furthermore, assuming Planted Dense Subgraph Conjecture, O(n1/4−ε) ε>0.