{"paper":{"title":"Statistical two-round search for one excellent element","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"When finding one excellent element is feasible, two-round search requires only a logarithmic number of tests in the population size.","cross_cats":["math.IT","stat.AP"],"primary_cat":"cs.IT","authors_text":"Jong Sung Kim, Nagananda K G","submitted_at":"2026-05-15T04:50:05Z","abstract_excerpt":"We formulate and study a statistical version of Katona's two-round search problem of finding at least one excellent element in a set. A population of $n$ elements is considered, where each element is independently excellent with probability $\\lambda/n$, $\\lambda > 0$. A subset test is noiseless: it returns positive exactly when the queried subset contains at least one excellent element. The goal is to minimize the expected number of tests subject to finding one excellent element with probability at least $1-\\alpha$, where $0<\\alpha<1$, under the restriction that testing is performed in two rou"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"When the target success probability is feasible, we prove that the optimal expected number of tests grows logarithmically with the population size. The upper bound is obtained by combining an initial existence test with a second-round separating design; the lower bound follows from an information-counting argument.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The excellence indicators are independent Bernoulli random variables with success probability λ/n, enabling the Poisson regime and the feasibility condition α ≥ e^{-λ}. This modeling choice is introduced in the problem formulation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In the sparse Poisson regime, the optimal expected number of two-round tests to identify one excellent element with probability at least 1-α grows logarithmically with n provided α ≥ e^{-λ}.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"When finding one excellent element is feasible, two-round search requires only a logarithmic number of tests in the population size.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f9dfa26206d774e71a2a8d3d5e201a2698c227c50f6dc88643de461f4429473e"},"source":{"id":"2605.15612","kind":"arxiv","version":1},"verdict":{"id":"9550ff22-31ec-45ba-8899-673779ac3d4a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:04:10.924609Z","strongest_claim":"When the target success probability is feasible, we prove that the optimal expected number of tests grows logarithmically with the population size. The upper bound is obtained by combining an initial existence test with a second-round separating design; the lower bound follows from an information-counting argument.","one_line_summary":"In the sparse Poisson regime, the optimal expected number of two-round tests to identify one excellent element with probability at least 1-α grows logarithmically with n provided α ≥ e^{-λ}.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The excellence indicators are independent Bernoulli random variables with success probability λ/n, enabling the Poisson regime and the feasibility condition α ≥ e^{-λ}. 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