Partition → Nonuniform-cost Unanimous Vote Complexity open

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Why this question matters

Nonuniform costs remove a symmetry exploited in sequential testing and may change the complexity of choosing an order.

Uniform-cost ordering results do not settle the expected-cost problem when each test has its own cost.

The reduction to construct

The source gives positive binary integer weights. A valid output is a subset whose sum is half the total, or NO-SOLUTION if no such subset exists.

The target gives independent coins with rational probabilities strictly between zero and one, positive test costs, and a rational bound K. Choose a fixed testing order whose expected cost is at most K, stopping at the first two different outcomes or after all coins.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is NP-hardness for the stated target problem.

An expensive test alone does not encode subset selection; the expected-cost expression must enforce the partition globally.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-16. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.