Determine the threshold for constant-competitive and exact online algorithms in the binary setting

Determine the threshold value of the number of known binary additive cost-function types at which constant-competitive online algorithms, and in particular 1-competitive online algorithms, exist.

Background

For known binary additive cost functions, the paper gives a 3-competitive algorithm when k ≤ n and shows that the optimal deterministic ratio is between 2 and 3 in the small-k regime. It also proves exact MMS feasibility for k = 2 and gives a lower bound of 3/2 for k = 7.

The conclusion explicitly identifies an unresolved threshold question: the dependence of the optimal competitive ratio on k is not fully characterized, and the boundary between constant-competitive or exact algorithms and regimes where such guarantees are impossible remains undetermined.

References

For example, our results suggest that the optimal competitive ratio for the binary setting grows with $k$. However, the threshold value of $k$ for which constant-competitive or even $1$-competitive algorithms exist is still unclear.

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