Approximability Threshold Conjecture for Optimisation Polyamorous Scheduling

Prove that the approximability threshold a* for Optimisation Polyamorous Scheduling satisfies a* ≥ 4; equivalently, show that unless P = NP, no polynomial-time algorithm achieves an approximation ratio strictly less than 4 for Optimisation Polyamorous Scheduling.

Background

The authors define a* as the smallest approximation ratio achievable by polynomial-time algorithms for Optimisation Polyamorous Scheduling (OPS). They present hardness-of-approximation results based on reductions, and an O(log n)-approximation algorithm, leaving a significant gap between lower and upper bounds.

Although Theorem 1.4 already provides a strong inapproximability result (no (1+ε)-approximation for ε < 3), the paper retains Conjecture 3.2 from the earlier version for the record, asserting a* ≥ 4 and motivating stronger lower bounds grounded in direct 3SAT reductions.