arXiv subjects

Tatsuya Terao

Publications and source records attributed to Tatsuya Terao.

6 recordsLinked to original sources

arXiv2026

Faster Approximate Linear Matroid Intersection

We consider a fast approximation algorithm for the linear matroid intersection problem. In this problem, we are given two $r \times n$ matrices $M_1$ and $M_2$, and the objective is to find a largest set of columns that are linearly independent in both $M_1$ and $M_2$. We design a $(1 - \varepsilon)$-approximation algorithm with time complexity $\tilde{O}_{\varepsilon}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + r_{*}^ω)$, where $\mathrm{nnz}(M_i)$ denotes the number of nonzero entries in $M_i$ for $i = 1, 2$, $r_{*}$ denotes the maximum size of a common independent set, and $ω< 2.372$ denotes the matrix multiplication exponent. Our approximation algorithm is faster than the exact algorithm by Harvey [FOCS'06 & SICOMP'09] and Cheung--Kwok--Lau [STOC'12 & JACM'13], which runs in $\tilde{O}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + n r_{*}^{ω- 1})$ time. We also develop a fast $(1 - \varepsilon)$-approximation algorithm for the weighted version of the linear matroid intersection problem. In fact, we design a $(1 - \varepsilon)$-approximation algorithm for weighted linear matroid intersection with time complexity $\tilde{O}_{\varepsilon}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + r_{*}^ω)$. Our algorithm improves upon the $(1 - \varepsilon)$-approximation algorithm by Huang--Kakimura--Kamiyama [SODA'16 & Math. Program.'19], which runs in $\tilde{O}_{\varepsilon}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + nr_{*}^{ω- 1})$ time. To obtain these results, we combine Quanrud's adaptive sparsification framework [ICALP'24] with a simple yet effective method for efficiently checking whether a given vector lies in the linear span of a subset of vectors, which is of independent interest.

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arXiv2026

Polynomial Kernels with Reachability for Weighted $d$-Matroid Intersection

This paper studies randomized polynomial kernelization for the weighted $d$-matroid intersection problem. While the problem is known to have a kernel of size $O(d^{(k - 1)d})$ where $k$ is the solution size, the existence of a polynomial kernel is not known, except for the cases when either all the given matroids are partition matroids~(i.e., the $d$-dimensional matching problem) or all the given matroids are linearly representable. The main contribution of this paper is to develop a new kernelization technique for handling general matroids. We first show that the weighted $d$-matroid intersection problem admits a polynomial kernel when one matroid is arbitrary and the other $d-1$ matroids are partition matroids. Interestingly, the obtained kernel has size $\tilde{O}(k^d)$, which matches the optimal bound~(up to logarithmic factors) for the $d$-dimensional matching problem. This approach can be adapted to the case when $d-1$ matroids in the input belong to a more general class of matroids, including graphic, cographic, and transversal matroids. We also show that the problem has a kernel of pseudo-polynomial size when given $d-1$ matroids are laminar. Our technique finds a kernel such that any feasible solution of a given instance can reach a better solution in the kernel, which is sufficiently versatile to allow us to design parameterized streaming algorithms and faster EPTASs.

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arXiv2025

Deterministic $(2/3-\varepsilon)$-Approximation of Matroid Intersection Using Nearly-Linear Independence-Oracle Queries

In the matroid intersection problem, we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ defined on the same ground set $V$ of $n$ elements, and the objective is to find a common independent set $S \in \mathcal{I}_1 \cap \mathcal{I}_2$ of largest possible cardinality, denoted by $r$. In this paper, we consider a deterministic matroid intersection algorithm with only a nearly linear number of independence oracle queries. Our contribution is to present a deterministic $O(\frac{n}{\varepsilon} + r \log r)$-independence-query $(2/3-\varepsilon)$-approximation algorithm for any $\varepsilon > 0$. Our idea is very simple: we apply a recent $\tilde{O}(n \sqrt{r}/\varepsilon)$-independence-query $(1 - \varepsilon)$-approximation algorithm of Blikstad [ICALP 2021], but terminate it before completion. Moreover, we also present a semi-streaming algorithm for $(2/3 -\varepsilon)$-approximation of matroid intersection in $O(1/\varepsilon)$ passes.

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arXiv2024

Parameterized Quantum Query Algorithms for Graph Problems

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for $k$-vertex cover and $k$-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small.

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arXiv2024

Subquadratic Submodular Maximization with a General Matroid Constraint

We consider fast algorithms for monotone submodular maximization with a general matroid constraint. We present a randomized $(1 - 1/e - ε)$-approximation algorithm that requires $\tilde{O}_ε(\sqrt{r} n)$ independence oracle and value oracle queries, where $n$ is the number of elements in the matroid and $r \leq n$ is the rank of the matroid. This improves upon the previously best algorithm by Buchbinder-Feldman-Schwartz [Mathematics of Operations Research 2017] that requires $\tilde{O}_ε(r^2 + \sqrt{r}n)$ queries. Our algorithm is based on continuous relaxation, as with other submodular maximization algorithms in the literature. To achieve subquadratic query complexity, we develop a new rounding algorithm, which is our main technical contribution. The rounding algorithm takes as input a point represented as a convex combination of $t$ bases of a matroid and rounds it to an integral solution. Our rounding algorithm requires $\tilde{O}(r^{3/2} t)$ independence oracle queries, while the previously best rounding algorithm by Chekuri-Vondrák-Zenklusen [FOCS 2010] requires $O(r^2 t)$ independence oracle queries. A key idea in our rounding algorithm is to use a directed cycle of arbitrary length in an auxiliary graph, while the algorithm of Chekuri-Vondrák-Zenklusen focused on directed cycles of length two.

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arXiv2023

Faster Matroid Partition Algorithms

In the matroid partitioning problem, we are given $k$ matroids $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$ defined over a common ground set $V$ of $n$ elements, and we need to find a partitionable set $S \subseteq V$ of largest possible cardinality, denoted by $p$. Here, a set $S \subseteq V$ is called partitionable if there exists a partition $(S_1, \dots , S_k)$ of $S$ with $S_i \in \mathcal{I}_i$ for $i = 1, \ldots, k$. In 1986, Cunningham [SICOMP 1986] presented a matroid partition algorithm that uses $O(n p^{3/2} + k n)$ independence oracle queries, which was the previously known best algorithm. This query complexity is $O(n^{5/2})$ when $k \leq n$. Our main result is to present a matroid partition algorithm that uses $\tilde{O}(k'^{1/3} n p + k n)$ independence oracle queries, where $k' = \min\{k, p\}$. This query complexity is $\tilde{O}(n^{7/3})$ when $k \leq n$, and this improves upon the one of previous Cunningham's algorithm. To obtain this, we present a new approach \emph{edge recycling augmentation}, which can be attained through new ideas: an efficient utilization of the binary search technique by Nguyen [2019] and Chakrabarty-Lee-Sidford-Singla-Wong [FOCS 2019] and a careful analysis of the independence oracle query complexity. Our analysis differs significantly from the one for matroid intersection algorithms, because of the parameter $k$. We also present a matroid partition algorithm that uses $\tilde{O}((n + k) \sqrt{p})$ rank oracle queries.

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