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SIAM Journal on Computing

Tight Bounds for Randomized and Quantum Local Search

Shengyu Zhang

Published Jan 1, 2009• 7 citations
Publisher

Abstract

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 $d\geq4$, $\text{{\it QLS\/}}([n]^d)=\Theta(n^{d/3})$ for $d\geq6$. 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Aaronson [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 465–474], Ambainis (unpublished), and Santha and Szegedy [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 494–501].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 $O(\sqrt{n}(\log\log n)^{1.5})$ queries is given. This improves the previously best known upper bound of $O(n^{2/3})$ (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.

DOI
10.1137/06066775x

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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\{0,1\}^n{0,1}n and the constant-dimensional grids [n]d[n]^d[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)\text{{\it RLS\/}}(\{0,1\}^n)=\Theta(2^{n/2}n^{1/2})RLS\/({0,1}n)=Θ(2n/2n1/2), QLS\/({0,1}n)=Θ(2n/3n1/6)\text{{\it QLS\/}}(\{0,1\}^n)=\Theta(2^{n/3}n^{1/6})QLS\/({0,1}n)=Θ(2n/3n1/6), RLS\/([n]d)=Θ(nd/2)\text{{\it RLS\/}}([n]^d)=\Theta(n^{d/2})RLS\/([n]d)=Θ(nd/2) for d≥4d\geq4d≥4, QLS\/([n]d)=Θ(nd/3)\text{{\it QLS\/}}([n]^d)=\Theta(n^{d/3})QLS\/([n]d)=Θ(nd/3) for d≥6d\geq6d≥6. Here RLS\/(G)\text{{\it RLS\/}}(G)RLS\/(G) and QLS\/(G)\text{{\it QLS\/}}(G)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[n]^2[n]2, a new quantum algorithm using O(n(log⁡log⁡n)1.5)O(\sqrt{n}(\log\log n)^{1.5})O(n​(loglogn)1.5) queries is given. This improves the previously best known upper bound of O(n2/3)O(n^{2/3})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.
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 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)Θ(2^{n/2}n^{1/2})Θ(2n/2n1/2) and the quantum query complexity is Θ(2n/3n1/6)Θ(2^{n/3}n^{1/6})Θ(2n/3n1/6). We also show that for the constant dimensional grid [N1/d]d[N^{1/d}]^d[N1/d]d, the randomized query complexity is Θ(N1/2)Θ(N^{1/2})Θ(N1/2) for d≥4d \geq 4d≥4 and the quantum query complexity is Θ(N1/3)Θ(N^{1/3})Θ(N1/3) for d≥6d \geq 6d≥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[N^{1/2}]^2[N1/2]2 a new upper bound of O(N1/4(log⁡log⁡N)3/2)O(N^{1/4}(\log\log N)^{3/2})O(N1/4(loglogN)3/2) on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.
We study the problem of \emph{local search} on a graph. Given a real-valued black-box function fff on the graph's vertices, this is the problem of determining a local minimum of fff---a vertex vvv for which f(v)f(v)f(v) is no more than fff evaluated at any of vvv'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))\Omega(2^{n/2 - o(n)} )Ω(2n/2−o(n)) queries to determine a local minima on the nnn-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)\Omega(2^{n/2}/n^2)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 GGG of NNN vertices and diameter ddd, the randomized and quantum query complexities for local search on GGG are Ω(N/dlog⁡N)\Omega\left({\sqrt{N}}/{d\log N}\right)Ω(N​/dlogN) and Ω(N4/dlog⁡N)\Omega\left({\sqrt[4]{N}}/{\sqrt{d\log N}}\right)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))Ω(2^{n/2 - o(1)})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)Ω(2^{n/2}/n^2)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/dlog⁡N)Ω(N^{1/2}/d\log N)Ω(N1/2/dlogN) and Ω(N1/4/dlog⁡N)Ω(N^{1/4}/\sqrt{d\log N})Ω(N1/4/dlogN​), respectively.
We consider the quantum query complexity of local search as a function of graph geometry. Given a graph G=(V,E)G = (V,E)G=(V,E) with nnn vertices and black box access to a function f:V→Rf : V \to \mathbb{R}f:V→R, the goal is find a vertex vvv that is a local minimum, i.e. with f(v)≤f(u)f(v) \leq f(u)f(v)≤f(u) for all (u,v)∈E(u,v) \in E(u,v)∈E, using as few oracle queries as possible. We show that the quantum query complexity of local search on GGG is Ω(n34g)Ω\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)Ω(g​n43​​), where ggg is the vertex congestion of the graph. For a βββ-expander with maximum degree ΔΔΔ, this implies a lower bound of $ Ω\bigl(\frac{\sqrtβ \; n^{\frac{1}{4}}}{\sqrtΔ \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Brânzei, Choo, and Recker (2024). As a corollary, on constant degree expanders, we derive a lower bound of Ω(n14log⁡n)Ω\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)Ω(logn​n41​​). This improves upon the best prior quantum lower bound of $Ω\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr) $ by Santha and Szegedy (2004). In contrast to the classical setting, a gap remains in the quantum case between our lower bound and the best-known upper bound of O(n13)O\bigl( n^{\frac{1}{3}} \bigr)O(n31​) for such graphs.
We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification (BMPV\mathsf{BMPV}BMPV), answering a longstanding open question in quantum query complexity. For n×nn\times nn×n matrices, our upper bound is O~(n17/12)\widetilde O(n^{17/12})O(n17/12), improving on the standard O(n3/2)O(n^{3/2})O(n3/2) bound obtained using Grover search by Buhrman and Špalek [SODA 2006]. We complement this result by showing an Ω(n5/4)Ω(n^{5/4})Ω(n5/4) lower bound, which improves over the previous best known lower bound of Ω~(n19/18)\widetildeΩ(n^{19/18})Ω(n19/18) by Childs, Kimmel, and Kothari [ESA 2012]. Our approach centers on a connection with Orthogonal Vectors (OV\mathsf{OV}OV), which asks whether an indexed list of nnn Boolean vectors of dimension nnn contains two vectors with disjoint supports. In particular, we prove equivalences between OV\mathsf{OV}OV and BMPV\mathsf{BMPV}BMPV and establish the above bounds for OV\mathsf{OV}OV. We also prove a tight Θ~(n3/2)\widetilde Θ(n^{3/2})Θ(n3/2) bound for a variant of BMPV\mathsf{BMPV}BMPV that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.