Deterministic Algorithm and Faster Algorithm for Submodular Maximization subject to a Matroid Constraint
(2408.03583v2)
Published 7 Aug 2024 in cs.DS and cs.DM
Abstract: We study the problem of maximizing a monotone submodular function subject to a matroid constraint, and present for it a deterministic non-oblivious local search algorithm that has an approximation guarantee of $1 - 1/e - \varepsilon$ (for any $\varepsilon > 0$) and query complexity of $\tilde{O}\varepsilon(nr)$, where $n$ is the size of the ground set and $r$ is the rank of the matroid. Our algorithm vastly improves over the previous state-of-the-art $0.5008$-approximation deterministic algorithm, and in fact, shows that there is no separation between the approximation guarantees that can be obtained by deterministic and randomized algorithms for the problem considered. The query complexity of our algorithm can be improved to $\tilde{O}\varepsilon(n + r\sqrt{n})$ using randomization, which is nearly-linear for $r = O(\sqrt{n})$, and is always at least as good as the previous state-of-the-art algorithms.
The paper introduces a deterministic non-oblivious local search algorithm achieving a near-optimal 1-1/e-ε approximation, bridging the gap between deterministic and randomized approaches.
It presents a faster randomized algorithm with query complexity of O(n + r√n), significantly outperforming previous state-of-the-art methods.
The work leverages combinatorial structures and the matroid exchange property to offer practical, scalable solutions for large-scale submodular maximization problems.
Deterministic Algorithm and Faster Algorithm for Submodular Maximization subject to a Matroid Constraint
The paper by Buchbinder and Feldman presents significant progress in the field of submodular function maximization under a matroid constraint. The authors propose a deterministic non-oblivious local search algorithm that achieves an approximation guarantee of 1−e1−ϵ for any ϵ>0. The paper not only provides deterministic results but also introduces a faster randomized algorithm for the same problem, showcasing notable improvements in query complexity compared to previous state-of-the-art algorithms.
Background
Submodular function maximization is a well-studied problem within combinatorial optimization, notable for its extensive applications in machine learning, economics, and network analysis. A function f:2N→R≥0 is submodular if it exhibits a diminishing returns property. When combined with a matroid constraint (N,I), the problem becomes finding a subset S∈I that maximizes f.
The greedy algorithm, known for its simplicity and efficiency, provides an approximation ratio of 1−e1 for cardinality constraints but performs suboptimally for general matroid constraints, offering only a 21 approximation. The continuous greedy algorithm, developed by Călinescu et al., improved this to 1−e1 but at the cost of high runtime and reliance on the multilinear extension—a technique requiring substantial randomization.
Contributions
The primary contributions of this paper include:
Deterministic Non-Oblivious Local Search Algorithm:
Approximation Guarantee: The proposed algorithm achieves an approximation ratio of 1−e1−ϵ, nullifying the gap between deterministic and randomized algorithms in terms of approximation quality.
Query Complexity: The deterministic algorithm utilizes O(nr) value and independence oracle queries, where n is the size of the ground set and r is the rank of the matroid.
Randomized Algorithm:
Improvement in Query Complexity: The randomized variant improves the query complexity to O(n+rn), particularly beneficial when r=O(n). This enhancement positions the algorithm favorably when compared to existing solutions.
Technical Approach
The authors introduce a non-oblivious local search algorithm that eschews the continuous greedy approach's dependency on the multilinear extension. Instead, it maximizes an auxiliary function g defined over disjoint sets of the ground set elements. This function effectively balances contributions from various parts of the solution, ensuring robustness against small perturbations.
To further refine the runtime, a fast variant of the local search algorithm is proposed, incorporating efficient oracle query methods and strategic element updates. Key strengths include:
Algorithm Design: By initializing with an approximate solution and iteratively improving it through localized adjustments, the algorithm leverages both combinatorial structure and submodularity to ensure efficient convergence.
Matroid Exchange Property: The algorithm employs matroid exchange properties to maintain feasibility, adding or removing elements based on marginal gains and ensuring the overall solution remains within the constraint set.
Implications and Future Work
The implications of this research are multifaceted:
Practical Implementations: The reduction in query complexity and the provision of a deterministic solution make the algorithm practical for large-scale applications where runtime and determinism are critical.
Theory and Beyond: The work opens up new avenues for exploring deterministic approaches in submodular maximization, inspiring further research into polynomial-time approximation schemes (PTAS) and fully polynomial-time approximation schemes (FPTAS) for related problems.
Broader Classes of Constraints: Future developments could investigate extending these techniques to handle more complex constraints beyond matroids, such as intersection of multiple matroids or knapsack constraints.
Conclusion
The deterministic and randomized algorithms introduced in this paper represent a substantial advancement in submodular function maximization under matroid constraints. By bridging the approximation guarantee gap between deterministic and randomized methods and significantly improving the query complexity, this work sets a new benchmark in the field. The practical and theoretical implications of these results hold promise for future research, particularly in developing more efficient and broadly applicable optimization algorithms.