Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E)a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x ∈ V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form what is the value of f on x? We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous[4] Ald and Aaronson [1] Aar and solves the main open problem in Aar.
Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E)a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x ∈ V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form what is the value of f on x? We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous[4] Ald and Aaronson [1] Aar and solves the main open problem in Aar.
We study the problem of \emph{local search} on a graph. Given a real-valued black-box function f on the graph's vertices, this is the problem of determining a local minimum of f---a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires Ω(2n/2−o(n)) queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to Ω(2n/2/n2) and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are Ω(N/dlogN) and Ω(4N/dlogN), respectively.
We study the problem of \emph{local search} on a graph. Given a real-valued black-box function f on the graph's vertices, this is the problem of determining a local minimum of f--a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires Ω(2n/2−o(1)) queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to Ω(2n/2/n2) and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are Ω(N1/2/dlogN) and Ω(N1/4/dlogN), respectively.
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.
Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to combinatorial optimization, complexity theory and many other areas in theoretical computer science. In this paper, we study the problem in the randomized and quantum query models and give new lower and upper bound techniques in both models.The lower bound technique works for any graph that contains a product graph as a subgraph. Applying it to the Boolean hypercube (0, 1)n and the constant dimensional grids [n]d, two particular product graphs that recently drew much attention, we get the following tight results: RLS((0, 1)n) = Θ(2n/2n1/2), QLS((0, 1)n) = Θ(2n/3n1/6); RLS([n]d) = Θ(nd/2), ∀ d ≥ 4, QLS([n]d) = Θ(nd/3), ∀ d ≥ 6. Here RLS(G) and QLS(G) are the randomized and quantum query complexities of Local Search on G, respectively. These improve the previous results by Aaronson [2], Ambainis (unpublished) and Santha and Szegedy [20].Our new algorithms work well when the underlying graph expands slowly. As an application to [n]2, a new quantum algorithm using O(☂n(log log n)1.5) queries is given. This improves the previous best known upper bound of O(n2/3) (Aaronson, [2]), and implies that Local Search on grids exhibits different properties in low dimensions.
The problem Local Search, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to combinatorial optimization, complexity theory, and many other areas in theoretical computer science. In this paper, we study the problem in both the randomized and the quantum query models and give new lower and upper bound techniques in both models. The lower bound technique works for any graph that contains a product graph as a subgraph. Applying it to the Boolean hypercube {0,1}n and the constant-dimensional grids [n]d, two particular product graphs that recently drew much attention, we get the following tight results: $\text{{\it RLS\/}}(\{0,1\}^n)=\Theta(2^{n/2}n^{1/2})$, $\text{{\it QLS\/}}(\{0,1\}^n)=\Theta(2^{n/3}n^{1/6})$, $\text{{\it RLS\/}}([n]^d)=\Theta(n^{d/2})$ for d≥4, $\text{{\it QLS\/}}([n]^d)=\Theta(n^{d/3})$ for d≥6. Here $\text{{\it RLS\/}}(G)$ and $\text{{\it QLS\/}}(G)$ are the randomized and quantum query complexities of Local Search on G, respectively. These improve the previous results by Aaronson [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 465–474], Ambainis (unpublished), and Santha and Szegedy [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 494–501]. Our new algorithms work well when the underlying graph expands slowly. As an application to [n]2, a new quantum algorithm using O(n(loglogn)1.5) queries is given. This improves the previously best known upper bound of O(n2/3) (see Aaronson [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 465–474]), and implies that Local Search on grids exhibits different properties in low dimensions.
The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube $\B^n$, the randomized query complexity of Local Search is Θ(2n/2n1/2) and the quantum query complexity is Θ(2n/3n1/6). We also show that for the constant dimensional grid [N1/d]d, the randomized query complexity is Θ(N1/2) for d≥4 and the quantum query complexity is Θ(N1/3) for d≥6. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for [N1/2]2 a new upper bound of O(N1/4(loglogN)3/2) on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.