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 free-order secretary problem in bipartite graphs under two-sided independence systems—where both agents and items are subject to combinatorial constraints (e.g., matroids, independence systems)—and elements are revealed and matched online in an adaptive, order-unaware manner. We introduce the *k-increasing system*, a novel combinatorial structure that generalizes the classical core lemma to both edge-arrival and agent-arrival models. Our main contribution is the first constant-competitive, order-unaware algorithm for multi-item selection. Theoretical analysis establishes an Ω(1/k²) competitive ratio under the intersection of k-matroids; moreover, constant competitive ratios are achieved for k-increasing systems, partition matroids, and k-matchings—significantly broadening the scope and performance guarantees of free-order online bipartite matching.

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 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 studies the Santa Claus problem with matroid constraints: allocate indivisible items (gifts) to children to maximize the minimum utility among children, where child $i$’s value for item $j$ is either $0$ or $p_j$. It introduces matroid constraints—modeling general resource allocation restrictions—for the first time in this context. Methodologically, it simplifies and generalizes Haxell’s augmenting tree technique, yielding a unified and concise approximation framework that integrates matroid theory, hypergraph matching, linear programming relaxation, and rounding. The algorithm achieves a $(4+varepsilon)$-approximation ratio, substantially improving upon the previous best $12.33$-approximation. Moreover, it serves as a black-box improvement for the LP relaxation bound of the classical Santa Claus problem. This work establishes a new paradigm for fair allocation under combinatorial constraints, bridging fairness, discrete optimization, and structural constraint modeling.

This work addresses the problem of maximizing a monotone submodular function subject to a matroid independence constraint. The authors introduce, for the first time, a stochastic Poisson process into this domain and propose a novel algorithm that achieves efficient optimization through only a small number of single-element exchanges, without requiring discretization or rounding. The method features a simple structure, circumventing the need for complex rounding procedures inherent in traditional approaches, while attaining the tight $(1-1/e)$ approximation guarantee. As applications, the framework effectively solves submodular welfare maximization as well as general and separable assignment problems, yielding significant improvements in computational efficiency.

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.

This work addresses the problem of maximizing a monotone submodular function subject to the intersection of $k$ matroids, as well as the weighted $k$-set packing problem. The authors propose a novel ordered local search algorithm that abandons the conventional weight bucketing approach and instead greedily selects elements in decreasing order of marginal value, augmented with threshold-driven feasible exchange operations. By tightly integrating marginal-value ordering with local search, the method achieves an approximation ratio of $k/2 + o(k)$ for matroid $k$-parity matching and $(\ln(4)k)/3 + o(k)$ for weighted $k$-set packing, both improving upon the best previously known results.