A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called kings of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Ω(n^{4/3}) and the best-known upper bound is O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show tight bounds (up to logarithmic factors) of Θ̃(n) and Θ̃(√n) in the randomized and quantum query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.
A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called kings of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Ω(n^{4/3}) and the best-known upper bound is O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show tight bounds (up to logarithmic factors) of Θ̃(n) and Θ̃(√n) in the randomized and quantum query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.
A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among N alternatives. We study the quantum query complexity of computing tournament solutions when the outcome of each comparison has to be learned by a query. Classically, the picture is uniform: the Copeland set, the uncovered set and the top cycle all require Θ(N2) queries, and so does finding a single vertex of maximum out-degree. We show that the quantum query complexity of the top cycle can be computed with Θ~(N) queries, and a Condorcet winner can be found with Θ~(N) queries. Finding a strong king requires Θ(N3/2) queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known Θ(N3/2) bound for finding a vertex of maximum degree, established by Gilani, Mande, Patro, and Raja. Our lower bounds reduce counting problems to tournament problems using Ambainis's adversary method. For the top cycle, we find a Hamiltonian path and locate the prefix forming the top cycle by divide and conquer.
We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2) bound for finding a maximum-out-degree vertex in an n-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an n-vertex undirected graph requires Θ(n3/2) quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum (k,n−k)-Cut, which asks for the largest cut with one side of size k. For every constant k≥1, we give an O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)-Cut involving k-Graph Collision, which asks whether the marked vertices of a known graph contain a k-clique. For k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for k-Graph Collision, the reduction yields a lower bound matching our upper bound above.
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph H is present in an input graph G, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length k. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly k, or at most k. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes.
When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity O(n3/2−αk), where αk∈Θ(c−k) and c=3+17/2≈1.33, beating the previous best upper bound O(n3/2) on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an O(n) query algorithm.
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.
The edge connectivity of a simple graph is the least number of edges whose removal disconnects the graph. We study the quantum query complexity of computing the edge connectivity of a simple graph with n vertices and m edges in both the adjacency matrix and adjacency array models. For classical randomized algorithms, Ω(n2) and Ω(m) lower bounds are known for these models, respectively, showing that the trivial algorithm is optimal in terms of query complexity for both cases. In contrast, we show upper bounds of O~(n3/2) and O~(mn) on the quantum query complexity of computing edge connectivity in the adjacency matrix and adjacency array models, respectively. Our upper bound is tight up to logarithmic factors for the adjacency matrix model, while for the adjacency array model the best quantum query lower bound we know is Ω(n). Our upper bounds follow by combining a lemma on the structure of near-minimum cuts due to Rubinstein, Schramm and Weinberg (ITCS 2018) with a quantum sparsification algorithm by Apers and de Wolf (FOCS 2020).
For a weighted graph, instead of edge connectivity the relevant notion is the weight of a minimum cut, i.e. the minimum total weight of a set of edges whose removal disconnects the graph. For weighted graphs we show that in the worst case Ω(n2) queries can be needed to compute the weight of a minimum cut by a quantum algorithm in both the adjacency matrix and adjacency array models, thus no quantum speedup is possible in this case.
We are presented with a graph, G, on n vertices with m edges whose edge set is unknown. Our goal is to learn the edges of G with as few queries to an oracle as possible. When we submit a set S of vertices to the oracle, it tells us whether or not S induces at least one edge in G. This so-called OR-query model has been well studied, with Angluin and Chen giving an upper bound on the number of queries needed of O(mlogn) for a general graph G with m edges. When we allow ourselves to make *quantum* queries (we may query subsets in superposition), then we can achieve speedups over the best possible classical algorithms. In the case where G has maximum degree d and is O(1)-colorable, Montanaro and Shao presented an algorithm that learns the edges of G in at most O~(d2m3/4) quantum queries. This gives an upper bound of O~(m3/4) quantum queries when G is a matching or a Hamiltonian cycle, which is far away from the lower bound of Ω(m) queries given by Ambainis and Montanaro. We improve on the work of Montanaro and Shao in the case where G has bounded degree. In particular, we present a randomized algorithm that, with high probability, learns cycles and matchings in O~(m) quantum queries, matching the theoretical lower bound up to logarithmic factors.