I disagree with the formulation of "decision problem" here. The problem, properly phrased as a decision problem, is "For a given K, does there exist a string with K 1s".

While it is straightforward to answer in the affirmative, and to produce an algorithm that produces that output (though it takes exponential time). To validate that a solution is in fact a valid solution also takes exponential time, if we are treating the size of the input as the base for the validation of the witness.