Fine-Grained Equivalence for Problems Related to Integer Linear Programming

TL;DR

This paper proves the equivalence of several combinatorial problems to Integer Linear Programming with binary variables, establishing that improvements in algorithms for one problem would translate to all, and introduces a more efficient algorithm based on variable symmetry.

Contribution

It establishes fine-grained reductions showing the equivalence of multiple problems to ILP, and improves the algorithmic running time by exploiting variable symmetry.

Findings

Problems are equivalent under fine-grained reductions.

Improved algorithm runs in ${n'}^{O(m)} + O(nm)$ time, surpassing previous bounds.

Algorithmic improvements depend on problem symmetry and variable distinctness.

Abstract

Integer Linear Programming with $n$ binary variables and $m$ many $0/1$-constraints can be solved in time $2^{\tilde O(m^2)} \text{poly}(n)$ and it is open whether the dependence on $m$ is optimal. Several seemingly unrelated problems, which include variants of Closest String, Discrepancy Minimization, Set Cover, and Set Packing, can be modelled as Integer Linear Programming with $0/1$ constraints to obtain algorithms with the same running time for a natural parameter $m$ in each of the problems. Our main result establishes through fine-grained reductions that these problems are equivalent, meaning that a $2^{O(m^{2-\varepsilon})} \text{poly}(n)$ algorithm with $\varepsilon > 0$ for one of them implies such an algorithm for all of them. In the setting above, one can alternatively obtain an $n^{O(m)}$ time algorithm for Integer Linear Programming using a straightforward dynamic programming approach, which can be more efficient if $n$ is relatively small (e.g., subexponential in $m$). We show that this can be improved to ${n'}^{O(m)} + O(nm)$, where $n'$ is the number of distinct (i.e., non-symmetric) variables. This dominates both of the aforementioned running times.