On approximating the rank of graph divisors

TL;DR

This paper explores the computational complexity of approximating the rank of graph divisors, establishing strong hardness results and connecting chip-firing games to the Minimum Target Set Selection problem.

Contribution

It strengthens existing NP-hardness results by linking chip-firing games to the Minimum Target Set problem and derives new inapproximability bounds under standard complexity assumptions.

Findings

Computing the rank of a divisor on a graph is NP-hard.

The rank is hard to approximate within a factor of $O(2^{ ext{log}^{1- ext{epsilon}} n})$ unless P=NP.

Under the Planted Dense Subgraph Conjecture, the rank is hard to approximate within a factor of $O(n^{1/4- ext{epsilon}})$.

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 {\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 $\varepsilon>0$.