{"paper":{"title":"New upper and lower bounds for randomized and quantum Local Search","arxiv_id":"quant-ph/0603034","primary_cat":"quant-ph","submitted_at":"2006-03-06T00:21:25+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"contribution","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.13,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":2,"p":0.85,"id":"s2","own":true,"role":"method","text":"The lower bound technique works for any graph that contains a product graph as a subgraph.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.85,"result":0.0,"background":0.0,"limitation":0.01,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"result","text":"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) = \\Theta(2^{n/2}n^{1/2}), QLS({0,1}^n) = \\Theta(2^{n/3}n^{1/6}), RLS([n]^d) = \\Theta(n^{d/2}) for d \\geq 4, QLS([n]^d) = \\Theta(n^{d/3}) for d \\geq 6.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.78,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.71,"id":"s4","own":false,"role":"background","text":"Here RLS(G) and QLS(G) are the randomized and quantum query complexities of Local Search on G, respectively.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.26,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.04,"background":0.71,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.22}},{"i":5,"p":0.5,"id":"s5","own":false,"role":"contribution","text":"These improve the previous results by Aaronson [STOC'04], Ambainis (unpublished) and Santha and Szegedy [STOC'04].","hedge":{"p":1.0,"label":"asserted"},"own_p":0.46,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.51,"role_probabilities":{"gap":0.08,"method":0.0,"result":0.2,"background":0.04,"limitation":0.0,"future_work":0.0,"contribution":0.5,"interpretation":0.18}},{"i":6,"p":0.49,"id":"s6","own":true,"role":"limitation","text":"Our new algorithms work well when the underlying graph expands slowly.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.82,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.27,"result":0.02,"background":0.0,"limitation":0.49,"future_work":0.0,"contribution":0.21,"interpretation":0.01}},{"i":7,"p":0.67,"id":"s7","own":true,"role":"result","text":"As an application to [n]^2, a new quantum algorithm using O(\\sqrt{n}(\\log\\log n)^{1.5}) queries is given.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.67,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.33,"interpretation":0.0}},{"i":8,"p":0.66,"id":"s8","own":true,"role":"interpretation","text":"This improves the previous best known upper bound of O(n^{2/3}) (Aaronson, [STOC'04]), and implies that Local Search on grids exhibits different properties in low dimensions.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.32,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.66}}]},"compiled":{"main_claim":{"p":0.85,"text":"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) = \\Theta(2^{n/2}n^{1/2}), QLS({0,1}^n) = \\Theta(2^{n/3}n^{1/6}), RLS([n]^d) = \\Theta(n^{d/2}) for d \\geq 4, QLS([n]^d) = \\Theta(n^{d/3}) for d \\geq 6.","location":"abstract s3","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":7187,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.06,"states_limitation":0.66,"human_participants":0.01,"controlled_comparison":0.08,"randomized_assignment":0.03,"states_quantitative_result":0.99,"compares_with_prior_results":0.98,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or 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