Short lists with short programs in short time

Given a machine \(U\), a \(c\)-short program for \(x\) is a string \(p\) such that \(U(p)=x\) and the length of \(p\) is bounded by \(c\) + (the length of a shortest program for \(x\)). We show that for any standard Turing machine, it is possible to compute in polynomial time on input \(x\) a list of polynomial size guaranteed to contain a O\((\log |x|)\)-short program for \(x\). We also show that there exists a computable function that maps every \(x\) to a list of size \(|x|^2\) containing a O\((1)\)-short program for \(x\). This is essentially optimal because we prove that for each such function there is a \(c\) and infinitely many \(x\) for which the list has size at least \(c|x|^2\). Finally we show that for some standard machines, computable functions generating lists with \(0\)-short programs, must have infinitely often list sizes proportional to \(2^{|x|}\).