On approximating the rank of graph divisors

TL;DR

The paper addresses the hardness of approximating the rank of a graph divisor, a central concept in Baker–Norine's graph divisor theory, by linking chip-firing dynamics to a chain of reductions from Minimum Target Set Selection. It establishes that the rank cannot be approximated within $O(2^{\log^{1-\varepsilon} n})$ unless $P=NP$, and, under the Planted Dense Subgraph Conjecture, within $O(n^{1/4-\varepsilon})$. The core approach rewrites rank as a distance to non-halting configurations, then reduces Min-TSS to a recurrent-state distance (Dist-Rec) and subsequently to a non-halting distance (Dist-Nonhalt), thereby transferring hardness results. These results also apply to tropical curves, and the authors discuss extensions to simple graphs via subdividing edges, highlighting the broader impact on combinatorial optimization and discrete geometry.

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 Tóthméresz 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 $\varepsilon>0$.

Figures (1)

• Figure 1: An illustration of the reduction of Min-TSS to Dist-Rec.

Theorems & Definitions (25)

• Proposition 1: Baker and Norine
• Remark 2
• Proposition 3: G. Tardos
• Proposition 4
• proof
• Proposition 5: Feige and Kogan
• Proposition 6: Charikar, Naamad, and Wirth
• Claim 7
• proof
• Claim 8