Hardness of Approximating the Rank of Graph Divisors

Conceitos essenciais

Computing the rank of a graph divisor is difficult to approximate within reasonable bounds, even in simple graphs.

Resumo

The paper investigates the complexity of computing the rank of a graph divisor, which is a key concept in the graph-theoretic analogue of the Riemann-Roch theory.
The main contributions are:

Establishing a connection between computing the rank of a divisor and the Minimum Target Set Selection (Min-TSS) problem, a central problem in combinatorial optimization that is notoriously hard to approximate.

Showing that the rank of a divisor is difficult to approximate to within a factor of O(2^(log^(1-ε) n)) for any ε > 0 unless P = NP.

Assuming the Planted Dense Subgraph Conjecture, the rank is also difficult to approximate to within a factor of O(n^(1/4-ε)) for any ε > 0.

The authors first show that the Min-TSS problem reduces to computing the distance of a divisor on an auxiliary graph from a recurrent state. They then show that computing the distance of a divisor from a recurrent state reduces to computing the distance of a divisor on a slightly modified graph from a non-halting state. As the computation of the rank is equivalent to computing the distance from a non-halting state for an appropriately modified divisor, the lower bounds on the approximability of the rank follow from known hardness results for the Min-TSS problem.

Estatísticas

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Citações

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Principais Insights Extraídos De

On approximating the rank of graph divisors

by Kris... às arxiv.org 04-12-2024

https://arxiv.org/pdf/2206.09662.pdf

Perguntas Mais Profundas

What are some potential applications of the hardness results for the rank of graph divisors

The hardness results for the rank of graph divisors have implications in various fields such as network analysis, computational biology, and optimization. In network analysis, understanding the rank of graph divisors is crucial for studying information flow, influence propagation, and network resilience. The hardness results can help in designing more robust network structures and strategies for information dissemination. In computational biology, the rank of graph divisors can be used to model genetic interactions, regulatory networks, and disease spread. The hardness results can guide the development of more accurate computational models and algorithms in biological research. In optimization, the rank of graph divisors plays a role in resource allocation, task scheduling, and system efficiency. The hardness results can inform the design of optimization algorithms that are resilient to complex network structures and constraints.

Can the hardness results be extended to other variants or generalizations of the rank computation problem

The hardness results for the rank computation problem can potentially be extended to other variants or generalizations of the problem. For example, if we consider weighted graphs where edges have different capacities or weights, the complexity of computing the rank of graph divisors may increase due to the additional constraints. Similarly, in directed graphs where the direction of edges matters, the rank computation problem could become more challenging. Additionally, if we introduce probabilistic or stochastic elements into the graph divisor model, the hardness results may extend to these probabilistic settings as well. Overall, the hardness results provide insights into the computational complexity of various extensions and variants of the rank computation problem.

Are there any special classes of graphs or divisors for which the rank can be computed efficiently

There are special classes of graphs and divisors for which the rank can be computed efficiently. For instance, in simple graphs with a small number of vertices and edges, the rank computation problem can be solved in polynomial time using specific algorithms tailored to the graph structure. Additionally, for graphs with certain properties such as being bipartite, planar, or tree-like, the rank computation problem may have efficient solutions due to the inherent structure of these graphs. Moreover, for divisors with specific characteristics such as being effective or winnable, the rank computation can be simplified, leading to faster algorithms for computing the rank. Overall, the efficiency of computing the rank of graph divisors depends on the specific properties and structures of the graphs and divisors involved.