matroid intersection algorithms

This work addresses the longstanding gap between weighted and unweighted matroid intersection approximation algorithms, where weighted variants have significantly lagged in both theory and practice. We propose the first general reduction framework that efficiently transforms any α-approximation algorithm for unweighted matroid intersection into an α(1−ε)-approximation algorithm for the weighted case, incurring only an O(log W) overhead in running time, where W denotes the maximum ratio between weights. By leveraging weight scaling and binary search, our method achieves this near-lossless preservation of approximation guarantees while enabling compatibility with constrained computational models such as streaming and one-way communication. This result bridges the performance gap between weighted and unweighted settings across multiple computational paradigms.

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 paper studies the matroid intersection problem under the Minimum Rank Oracle (MRO) model. For the unweighted case, we reconstruct the exchange graph and rederive Edmonds’ min-max theorem solely via MRO queries, yielding the first polynomial-time algorithm. For the weighted case, we prove NP-hardness in general, but identify a tractable special case under circuit-disjointness and design an FPT algorithm parameterized by maximum circuit size. Furthermore, we develop a lexicographic optimization framework achieving a 1/2-approximation for maximum-weight common independent sets. Our key contributions are: (i) the first structural characterization of matroid intersection using only minimum-rank queries—bypassing reliance on standard rank or independence oracles; (ii) a novel paradigm for modeling exchange properties under MRO; and (iii) systematic results on parameterized tractability and approximation guarantees.

This paper studies edge coloring on the intersection of a general matroid and multiple partition matroids—a problem proven to be NP-hard. We propose the first polynomial-time $O(1)$-approximation algorithm for the setting where one matroid is general and the others are partition matroids. Our method embeds the general matroid into a partition matroid structure via combinatorial reduction, then designs an efficient algorithm combining matroid basis exchange with greedy coloring, ensuring feasibility under the independence constraints of the matroid intersection. The algorithm uses at most $1 + sum_i (chi(M_i) - 1)$ colors—significantly improving upon prior algorithms restricted to intersections of solely partition matroids. This is the first constant-factor approximation algorithm for matroid intersection coloring that accommodates a general matroid, thereby extending both the theoretical scope and practical applicability of matroid coloring.

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 work addresses the challenge of efficiently maintaining a directed minimum spanning tree (DMST) under dynamic updates such as edge weight modifications, insertions, or deletions. The authors propose a dynamic algorithm grounded in weighted matroid intersection, which constructs and incrementally maintains an auxiliary graph to iteratively refine the current solution toward the updated DMST, guaranteeing monotonic improvement in solution quality. The key innovation lies in an efficient mechanism for updating the auxiliary graph coupled with a carefully designed iterative optimization strategy that balances theoretical convergence guarantees with practical computational efficiency. Experimental results demonstrate that the proposed method significantly outperforms existing baselines across various dynamic scenarios, offering both superior effectiveness and runtime performance.

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.

This work addresses the multi-choice matroid secretary problem ((J,κ)-MSP), where elements arrive online in random order and the goal is to select a maximum-weight independent set under both multiple-selection and global cardinality constraints. The paper innovatively decouples online storage from the final solution, introducing a multi-track algorithmic framework and modeling the selection process via a union of matroids, combined with a single-threshold routing strategy. It establishes the first systematic theoretical foundation for this problem: when capacity constraints are absent, it proves the existence of an exact optimal algorithm for transversal matroids, whose competitive ratio matches the success probability of the classical J-choice secretary problem; it identifies rank-one instances as worst-case scenarios; and it derives performance guarantees—exponentially converging to 1 for transversal matroids, and approximation ratios of \(1 - O(e^{-J/(k e)})\) for k-column-sparse matroids and \(1 - O(e^{-J/e})\) for laminar matroids—while quantifying the rank loss induced by global capacity and its coupling with parameters \(b\), \(J\), and \(\kappa\).

This work resolves the long-standing open problem of the parallel complexity of computing a matroid basis. By integrating matroid theory, refined randomization, and combinatorial optimization techniques, we devise a near-optimal parallel algorithm that efficiently computes a matroid basis in $O(n^{1/3} \log^{1/3} n)$ rounds. This round complexity exceeds the classic Karp–Upfal–Wigderson lower bound by only a $\log^{2/3} n$ factor, thereby achieving the first parallel algorithm that approximates the optimal bound within a polylogarithmic factor. Our result substantially advances the theoretical frontier for this fundamental problem in parallel computation, pushing it significantly closer to the known lower bound since 1985.

This work addresses the online graph matroid secretary problem under the constraint that the algorithm has access only to an independence oracle for elements that have arrived so far and no prior knowledge of the underlying graph structure. The authors propose a polynomial-time online algorithm that achieves a constant competitive ratio without requiring any additional structural information about the graph. By integrating structural properties of graph matroids, weight optimization under the random arrival model, and an adaptive online decision strategy, the algorithm guarantees a competitive ratio of at least $1/36$ relative to the maximum-weight independent set. This result constitutes the first constant-competitive solution for unknown graph matroids in the secretary setting, overcoming the reliance on extra prior information inherent in previous approaches.