You (Almost) Can't Beat Brute Force for 3-Matroid Intersection

TL;DR

This paper proves that for the $ ext{l}$-matroid intersection problem, brute force algorithms are essentially optimal, and it provides new algorithms and lower bounds that deepen understanding of the problem's computational complexity.

Contribution

The paper establishes tight bounds on the runtime of algorithms for $ ext{l}$-matroid intersection, showing brute force cannot be significantly improved and introducing a generalized local search technique.

Findings

Brute force cannot be significantly improved for $ ext{l}$-matroid intersection.

An algorithm faster than brute force by a polylogarithmic factor is presented.

A parameterized lower bound based on the matroid rank is established.

Abstract

The $\ell$-matroid intersection ($\ell$-MI) problem asks if $\ell$ given matroids share a common basis. Already for $\ell = 3$, notable canonical NP-complete special cases are $3$-Dimensional Matching and Hamiltonian Path on directed graphs. However, while these problems admit exponential-time algorithms that improve the simple brute force, the fastest known algorithm for $3$-MI is exactly brute force with runtime $2^{n}/poly(n)$, where $n$ is the number of elements. Our first result shows that in fact, brute force cannot be significantly improved, by ruling out an algorithm for $\ell$-MI with runtime $o\left(2^{n-5 \cdot n^{\frac{1}{\ell-1}} \cdot \log (n)}\right)$, for any fixed $\ell\geq 3$. We further obtain: (i) an algorithm that solves $\ell$-MI faster than brute force in time $2^{n-\Omega\left(\log^2 (n)\right)} $ (ii) a parameterized running time lower bound of $2^{(\ell-2) \cdot k \cdot \log k} \cdot poly(n)$ for $\ell$-MI, where the parameter $k$ is the rank of the matroids. We obtain these two results by generalizing the Monotone Local Search technique of Fomin et al. (J. ACM'19). Broadly speaking, our generalization converts any parameterized algorithm for a subset problem into an exponential-time algorithm which is faster than brute-force.