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

The ℓ-matroid intersection (ℓ-MI) problem asks if ℓ given matroids share a common basis. Already for ℓ = 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 significantly (e.g., Eiben-Koana-Wahlström (SODA’24)), the fastest known algorithm for 3-MI on general matroids is exactly brute force with runtime 2ⁿ/poly(n), where n is the number of elements. Our main result shows that, in fact, brute force cannot be significantly improved, by ruling out an algorithm for ℓ-MI with runtime (Formula presented), for any fixed ℓ ≥ 3. For 3-MI, this gives a lower bound of o(2^{n-5‧√n‧log(n)}). Our negative result raises the following natural questions: (i) Is there an algorithm for 3-MI with runtime strictly better than brute force? (ii) Can we separate the parameterized complexity of 3-MI from the important special case on linear matroids (parameterized by the rank of the matroids k)? In particular, can a lower bound match the existing c^k2 · poly(n) algorithm of Huang-Ward (SIDMA’23) for general ℓ-MI parameterized by the rank? We make progress towards obtaining affirmative answers to the above questions. In particular, we present (i) an algorithm which solves ℓ-MI faster than brute force in time 2^{n−Ω(log2(n))} for any ℓ ≥ 3, and (ii) a parameterized running time lower bound of 2^{(ℓ−2)·k·logk} · poly(n) for ℓ-MI, for any ℓ ≥ 3. We obtain these results by generalizing the Monotone Local Search technique of Fomin-Gaspers-Lokshtanov-Saurabh (J. ACM’19). Broadly speaking, given a subset problem, our generalization transforms any algorithm parameterized by solution size, with runtime of the form f(k) · poly(n), into an exponential-time algorithm with runtime depending on f. This implies that any f(k) · poly(n) time parameterized algorithm for a subset problem yields a 2^{n−ω(log n)} time algorithm beating brute force, which may be of independent interest.