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

2024-12-03

arXiv.org

AI Summary

This paper investigates the computational complexity of the 3-Matroid Intersection (3-MI) problem, aiming to break the long-standing 2ⁿ/poly(n) brute-force enumeration barrier. Methodologically, it introduces a randomized reduction under the Exponential Time Hypothesis (ETH) to establish the first lower bound: no algorithm solves 3-MI in o(2ⁿ⁻⁵√n log n) time. It further develops a generalized Monotone Local Search framework, yielding the first super-brute-force algorithm running in 2ⁿ⁻Ω(log²n) time. For the weighted variant (EMI), it proves that no randomized polynomial-time algorithm exists unless RP = NP. Finally, it establishes a tight parameterized lower bound of (ℓ−2)k log k for k-element solutions. Collectively, these results provide the strongest known characterization of the intrinsic exponential hardness of 3-MI, sharply delineating the tractability frontier for matroid intersection problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $oleft(2^{n-5 cdot n^{frac{1}{ell-1}} cdot log (n)} ight)$, for any fixed $ellgeq 3$. The complexity gap between $3$-MI and the polynomially solvable $2$-matroid intersection calls for a better understanding of the complexity of intermediate problems. One such prominent problem is exact matroid intersection (EMI). Given two matroids whose elements are either red or blue and a number $k$, decide if there is a common basis which contains exactly $k$ red elements. We show that EMI does not admit a randomized polynomial time algorithm. This bound implies that the parameterized algorithm of Eisenbrand et al. (FOCS'24) for exact weight matroid cannot be generalized to matroid intersection. We further obtain: (i) an algorithm that solves $ell$-MI faster than brute force in time $2^{n-Omegaleft(log^2 (n) ight)} $ (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.

Problem

• Investigating the computational complexity of 3-matroid intersection problems
• Establishing lower bounds for brute-force algorithm performance
• Developing faster algorithms through generalized monotone local search

Innovation

• Generalized Monotone Local Search technique
• Converts parameterized algorithms to exponential-time
• Faster than brute-force for subset problems