On approximating the rank of graph divisors

Kristóf Bérczi, Hung P. Hoang, Lilla Tóthmérész

Abstract: 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. The importance of the rank is well illustrated by Baker's 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 Tóthmérész 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(2log1−ε⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">log</mi></mrow><mrow is="true"><mn is="true">1</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ε</mi></mrow></msup><mo is="true">⁡</mo><mi is="true">n</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> for any ε>0<math><mi is="true">ε</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math> unless P=NP<math><mi is="true">P</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">N</mi><mi is="true">P</mi></math>. Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to approximate to within a factor of O(n1/4−ε)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">4</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ε</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> for any ε>0<math><mi is="true">ε</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math>.