In the matroid intersection problem, we are given two matroids ℳ₁ = (V, ℐ₁) and ℳ₂ = (V, ℐ₂) defined on the same ground set V of n elements, and the objective is to find a common independent set S ∈ ℐ₁ ∩ ℐ₂ 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(n/(ε) + r log r)-independence-query (2/3-ε)-approximation algorithm for any ε > 0. Our idea is very simple: we apply a recent Õ(n √r/ε)-independence-query (1 - ε)-approximation algorithm of Blikstad [ICALP 2021], but terminate it before completion. Moreover, we also present a semi-streaming algorithm for (2/3 -ε)-approximation of matroid intersection in O(1/ε) passes.

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 presented a matroid partition algorithm that uses \(O(np^{3/2}+kn)\) 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 \(\tilde{O}(k^{\prime 1/3}np + kn)\) independence oracle queries, where \(k^{\prime}=\min\{k,p\}\) . This query complexity is \(\tilde{O}(n^{7/3})\) when \(k\leq n\) , which improves upon that of Cunningham’s algorithm. To obtain our algorithm, we present a new approach edge recycling augmentation , which can be attained through new ideas: an efficient utilization of the binary search technique by Nguy \(\tilde{{\hat{\text{e}}}}\) n and Chakrabarty et al. 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.

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. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.

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.