The Power of Knowing Horizons in One-Way Trading: A Unified Framework for Optimal Competitive Ratios

This paper studies the One-way Trading Problem (OTP) with different forms of horizon information. We first show that knowing only the earliest horizon in OTP yields no algorithmic improvement. We then propose three new algorithms for the other forms of horizon information: (i) the Latest Horizon Algorithm (LHA), an optimal solution that strictly improves upon the original bound in OTP when the latest horizon is known; (ii) the Earliest-and-Latest Horizon Algorithm (ErLHA), an asymptotically optimal solution when both the earliest and the latest horizons are known; (iii) the Exact Horizon Algorithm (ExHA), an optimal solution and also a special form of ErLHA when the earliest and latest horizons coincide and become the exact horizon. Notably, ErLHA attains asymptotic optimality when both the earliest and latest horizons are known, recovering the optimality in all other settings, and thereby standing as the first unified and optimal framework for OTP with different horizon information.