apply matroid theory

This work addresses optimal decision-making under functional prerequisite constraints by introducing the Matroidal Prerequisite System (MPS)—a novel model jointly defined by a poset and a matroid, where feasible sequences correspond to matroid-independent sets and dependencies can be satisfied via functional substitution. The study establishes, for the first time, an isomorphism between MPS and strongly polyhedral greedy systems, enabling the design of both deterministic and randomized approximation algorithms. For additive objectives, the algorithms achieve approximation ratios of Δ and (1 + λ_max), respectively. For submodular objectives, they yield a deterministic (2 + λ_max)-approximation and a randomized Δ²·(1 − 1/e − δ)⁻¹-approximation. Furthermore, under the Gap-ETH assumption, the paper proves that no algorithm can attain a min{Δ, λ_max}^{o(1)}-approximation.

This paper studies the submodular partition problem under matroid constraints: minimizing the sum of a submodular function over disjoint subsets, subject to the constraint that the partition separates some basis of a given matroid. This unified framework captures classical problems including multiway cut, hypergraph $k$-cut, and matrix multicut. We introduce and systematically analyze this general model for the first time. For symmetric, monotone, and general submodular functions, we achieve the current best approximation ratios: $(2 - 2/k)$—matching the theoretical lower bound for symmetric functions—and the first nontrivial efficient approximations for monotone and general cases. Our approach integrates submodular optimization, matroid theory, and combinatorial approximation algorithms, combining Lagrangian relaxation with greedy construction techniques. The results advance the unified modeling and tight approximation boundary analysis of submodular partitioning.

This study addresses the adaptive evaluation of random Boolean functions over partition matroids: given a ground set whose elements have unknown active states, the goal is to determine whether there exists a basis consisting entirely of active elements, using the minimum expected number of queries. To this end, the work proposes a novel approach that integrates adaptive randomized strategies, optimization under expected budget constraints, and interleaved solving across multiple instances, yielding the first polynomial-time constant-factor approximation algorithm for this problem. This result overcomes the limitation of prior methods, which lacked provable approximation guarantees, and establishes an effective algorithmic framework for stochastic query problems with expected-cost constraints.

This paper studies monotone submodular maximization under matroid constraints. Addressing a long-standing bottleneck in the approximation ratio of deterministic algorithms—previously capped at 0.5008—it introduces the first deterministic non-blind local search algorithm achieving an approximation ratio of $1 - 1/e - varepsilon$. This bridges the theoretical gap between deterministic and randomized algorithms. The method fully exploits matroid structure to attain nearly linear query complexity $ ilde{O}_varepsilon(nr)$. By incorporating lightweight randomization, the complexity improves to $ ilde{O}_varepsilon(n + rsqrt{n})$. Notably, this is the first deterministic framework—retaining full determinism in its core design—to achieve the $1 - 1/e - varepsilon$ guarantee, significantly surpassing all prior deterministic approaches. The result advances the state-of-the-art both in approximation quality and computational efficiency for constrained submodular optimization.

This paper addresses the discrete maximization of non-monotone submodular functions under cardinality and matroid constraints, breaking the long-standing $1/e approx 0.367$ approximation barrier for combinatorial algorithms. We propose the **guided randomized greedy framework**, integrating fast local search while avoiding costly continuous extensions. We further design **deterministic and nearly-linear-time variants** that preserve the approximation guarantees. Under cardinality constraints, our algorithm achieves a $0.385$ approximation ratio—improving upon the previous best $0.367$; under matroid constraints, it attains $0.305$, surpassing $0.281$. The deterministic variant achieves $0.377$ with nearly-linear time complexity. To our knowledge, this is the first purely combinatorial algorithm—requiring no continuous optimization—that strictly exceeds the $1/e$ barrier, significantly enhancing scalability and practical applicability.

This work addresses the linear matroid intersection problem—finding a maximum-cardinality subset of columns that is linearly independent in both of two given matrices. The paper proposes the first $(1-\varepsilon)$-approximation algorithm for this problem that integrates an adaptive sparsification framework with an efficient subroutine for detecting vector span membership, marking the first application of adaptive sparsification to linear matroid intersection. This approach substantially improves the time complexity for both unweighted and weighted variants, achieving $\tilde{O}_\varepsilon(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + r_*^\omega)$, which surpasses the previous bound of $\tilde{O}_\varepsilon(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + n r_*^{\omega-1})$. The designed span-detection subroutine is also of independent interest.

This work addresses a fundamental limitation in classical matroid algorithms, which assume independence queries can be answered in constant time, disregarding the actual cost dependence on the size of the queried set. The paper introduces a size-sensitive query model where the cost of an independence query scales linearly with the cardinality of the queried set, and investigates algorithmic complexity for three core tasks under this model: computing a matroid basis, approximating the rank, and partitioning into independent sets. The authors establish the first unconditional query complexity lower bound in this setting, showing that Ω(n²) queries are necessary for general matroids. Moreover, for matroids whose largest circuit has size at most c, they design an explicit algorithm achieving an expected query cost of O(n^{2−1/c} log n), thereby circumventing the general lower bound and yielding nearly tight upper and lower bounds.

This study addresses the problem of ranking and rank aggregation under matroid and flag-matroid prefix constraints, measured by Kendall tau distance. It unifies and generalizes existing notions such as k-fairness and block fairness, and for the first time handles more general constraints involving hierarchies and quotas. For the single-input setting, the authors propose a polynomial-time algorithm based on the Bruhat order and a greedy strategy to efficiently compute the nearest feasible ranking satisfying flag-matroid constraints. In the multi-input aggregation setting, they prove that the problem remains NP-hard even under partition matroids. By integrating matroid theory with structural analysis of the symmetric group, this work significantly extends the theoretical foundations of fair ranking.