# Combinatorial n-fold Integer Programming and Applications

## TL;DR

This paper introduces a single-exponential algorithm for combinatorial n-fold integer programs, enabling faster solutions for various NP-hard problems by allowing variable dimensions and improving over traditional ILP methods.

## Contribution

The authors develop a novel single-exponential algorithm for combinatorial n-fold integer programs that handles variable dimensions and applies augmentation techniques, improving efficiency over Lenstra's algorithm.

## Key findings

- Achieved exponential speedups for problems like Closest String and Swap Bribery.
- Demonstrated the effectiveness of augmentation techniques in combinatorial n-fold IPs.
- Provided new insights into problem structures through the existence of local augmenting steps.

## Abstract

Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra solves ILPs in time that is exponential only in the dimension of the program, and polynomial in the size of the ILP. That algorithm became a ubiquitous tool in the design of fixed-parameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problemby some parameter. However, in many cases using Lenstra's algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension cannot easily express problems involving many different costs. Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a single-exponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both. Unlike Lenstra's algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs, existence of an augmenting step implies existence of a \local" augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.