# On the Complexity of Toric Ideals

## TL;DR

This paper explores the computational complexity of problems related to toric ideals, establishing NP-hardness in general but identifying efficient algorithms for sparse cases using graph parameters like treedepth.

## Contribution

It introduces parameterized algorithms for toric ideal problems based on graph sparsity measures, particularly treedepth, enabling efficient computations in specific cases.

## Key findings

- Normal form problem is fixed-parameter tractable with respect to treedepth.
- Efficient membership testing for reduced Gr"obner bases is developed.
- Algorithms for computing Graver bases are also made efficient under certain graph parameters.

## Abstract

We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases. We show that all these problems are strongly NP-hard in the general case. Nonetheless, we can derive efficient algorithms by taking advantage of the sparsity pattern of the matrix. We describe this sparsity pattern with a graph, and study the parameterized complexity of toric ideals in terms of graph parameters such as treewidth and treedepth. In particular, we show that the normal form problem can be solved in parameter-tractable time in terms of the treedepth. An important application of this result is in multiway ideals arising in algebraic statistics. We also give a parameter-tractable membership test to the reduced Gr\"obner basis. This test leads to an efficient procedure for computing the reduced Gr\"obner basis. Similar results hold for Graver bases computation.