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Applying Aldous’ algorithm to speed up μ-d.s.r.–based PLS^Σ_i^P algorithms

Develop techniques to leverage Aldous’ minimization algorithm to achieve algorithmic speedups for problems that have been placed in PLS with Σ_i^P oracle access via μ-downward self-reductions.

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Background

Aldous’ algorithm yields subexponential-time randomized algorithms for certain PLS-complete problems like Sink-of-DAG. Prior works have transferred such speedups to related problems via fine-grained reductions.

The μ-d.s.r framework can incur blowups that obscure direct application of Aldous’ approach. Finding principled ways to apply Aldous’ algorithm in this setting could produce new, faster algorithms for a range of total search problems.

References

Below we list a few open questions that arise from our work. Aldous' algorithm gives us a randomized procedure for $#1{Sink-of-DAG}$ for an input $S: { 0, 1}n \rightarrow { 0, 1}n, V: { 0, 1}n \rightarrow { 0, 1}n$ that runs in expected time $\poly(n) 2{n/2}$. and gave $\poly(n) 2{n/2}$ time randomized algorithms for #1{P-LCP} and #1{S-Arrival} by showing there exist fine grained reductions from those problems to #1{Sink-of-DAG}, and then applying Aldous' algorithm. Our proofs using $\mu$-d.s.r on the other hand incur a large blowup in instance size (except that of #1{King}). Is there any way to apply Aldous' algorithm to achieve a speedup for problems shown to be in $\cc{PLS{\Sigma_iP}$ via $\mu$-d.s.r?

— Downward self-reducibility in the total function polynomial hierarchy (2507.19108 - Gajulapalli et al., 25 Jul 2025) in Discussion and Open questions, Item 5