Paper Info
Title
New upper and lower bounds for randomized and quantum local search
Abstract
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 These improve the previous results by Aaronson [2], Ambainis (unpublished) and Santha and Szegedy [20].Our new algorithms work well when the underlying graph expands slowly.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.
Year
DOI
Venue
2006
10.1145/1132516.1132605
Clinical Orthopaedics and Related Research
Keywords
DocType
Volume
local search problem,local search,quantum query complexity,product graph,new algorithm,lower bound,new quantum algorithm,randomized algorithm,quantum query model,lower bound technique work,quantum local search,particular product graph,underlying graph,quantum algorithm,query complexity decision tree complexity,quantum physics
Conference
abs/quant-ph/0603034
ISBN
Citations 
PageRank 
1-59593-134-1
6
0.53
References 
Authors
23
1
Name
Order
Citations
PageRank
Shengyu Zhang1153.78