# Sparsity of integer solutions in the average case

## TL;DR

This paper investigates the average sparsity of feasible solutions in integer programming, showing that solutions tend to be sparse with O(m) nonzero entries under mild conditions, using advanced mathematical tools.

## Contribution

It establishes that, on average, integer program solutions are sparse with O(m) nonzero entries, improving understanding of solution structure.

## Key findings

- Feasible solutions in integer programs are sparse on average.
- The proof employs group, lattice, and Ehrhart polynomial theories.
- Provides new upper bounds on the integer Caratheodory number.

## Abstract

We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relatively mild assumptions, integer programs in standard form have feasible solutions with O(m) many nonzero entries, on average. Our proof uses ideas from the theory of groups, lattices, and Ehrhart polynomials. From our main theorem we obtain the best known upper bounds on the integer Caratheodory number provided that the determinants in the data are small.