Smoothed complexity of standard policy- and value-iteration rules

Determine whether standard policy-iteration rules, including Howard, lexicographic, and RandomFacet, have polynomial smoothed complexity on deterministic two-player games, and whether the value-iteration algorithms discussed by Cadilhac et al. have polynomial smoothed complexity.

Background

The paper establishes smoothed polynomial complexity for a particular policy-iteration rule and for three value-iteration algorithms on deterministic discounted and mean-payoff games over arbitrary graphs. The authors note that their policy rule is not among the standard rules commonly studied, leaving unresolved whether analogous guarantees hold for those standard rules and for other combinatorial algorithms.

The same paragraph also identifies the versions of value iteration discussed in the cited literature as an unresolved class whose smoothed complexity remains to be determined.

References

Therefore, we can raise the same questions about policy iteration: do these standard rules also have polynomial smoothed complexity on deterministic two-player games? How about other ``combinatorial'' algorithms? For value iteration, one can ask if the different versions of value iteration discussed in have smoothed polynomial complexity.

— Smoothed analysis of policy iteration and value iteration for deterministic discounted and mean-payoff games played on any graph  (2610.22149 - Loff et al., 15 Oct 2026) in Section Conclusion and future work, paragraph 1

In the smoothed analysis of the simplex method, an optimal sub-linear dependence on $\frac{1}{\rho}$ has recently been established . One can ask if such sub-linear dependence can also be achieved in the setting of deterministic games.

— Smoothed analysis of policy iteration and value iteration for deterministic discounted and mean-payoff games played on any graph  (2610.22149 - Loff et al., 15 Oct 2026) in Section Conclusion and future work, paragraph 4