Weak adversaries for the k-server problem

We study the k-server problem when the offline algorithm has fewer than k servers. We give two upper bounds of the cost WFA(ρ) of the Work Function Algorithm. The first upper bound is kOPT_h(ρ)+(h-1)OPT_k(ρ), where OPT_m (ρ) denotes the optimal cost to service ρ by m servers. The second upper bound is 2hOPTh(ρ)-OPT_k(ρ) for h⩽k. Both bounds imply that the Work Function Algorithm is (2k-1)-competitive. Perhaps more important is our technique which seems promising for settling the k-server conjecture. The proofs are simple and intuitive and they do not involve potential functions. We also apply the technique to give a simple condition for the Work Function Algorithm to be k-competitive; this condition results in a new proof that the k-server conjecture holds for k=2