Characterize type thresholds for exact and constant-factor online MMS allocation

Characterize the values of the number of known binary additive cost-function types for which exact MMS allocations or, more generally, constant-factor online MMS approximations are possible.

Background

The paper establishes a deterministic 3-competitive online algorithm for known binary additive cost functions when the number of types satisfies k ≤ n, while proving that no deterministic algorithm can guarantee a ratio strictly below 2 even when k is constant. In an appendix, it further shows that exact MMS allocations are achievable for k = 2, whereas a ratio strictly below 3/2 is impossible for k = 7.

These results leave unresolved the precise dependence on the number of known types: in particular, the authors do not determine the threshold values at which exact MMS allocations become impossible or at which constant-factor approximations cease to be achievable.

References

We leave open the question of characterizing the values of $k$ for which exact MMS allocations, or more generally constant-factor approximations, are possible.

— MMS Allocation for Chores with Online Agent Arrivals  (2609.10960 - Li et al., 10 Sep 2026) in Section 6, subsection “Lower Bound for Bounded k”

Furthermore, it remains unknown whether the idea of maintaining a universal residual can be extended to the general additive setting to obtain a constant competitive ratio when $k \le n$.

— MMS Allocation for Chores with Online Agent Arrivals  (2609.10960 - Li et al., 10 Sep 2026) in Section 7, “Conclusion and Open Problems”