The 11/6 supremum of the Wang-Sitters rounding scheme for graph balancing

Abstract

Wang and Sitters'11/6-approximation for graph balancing is not one algorithm but a set of permitted executions: Step 1 may return any feasible solution of the relaxation and Step 3 any of the many ways to match the remaining jobs into the slots the rounding opens. We determine exactly what that latitude permits: ratios arbitrarily close to 11/6, and none reaching it, so 11/6 is the least constant that bounds every permitted run, and no run attains it. We then determine the worst-case guarantee as a function of the big-job threshold beta, measured against the optimum itself. On Wang and Sitters'own range 1/2<beta<1 the guarantee is exactly max{3/2 + beta/2, 5/2 - beta}. We then extend the same eligibility rule to 0<beta<= 1/2 -- outside the range they state, and where a big job's two shares can both reach the threshold, so Step 2 acquires a third choice -- and determine the guarantee there as well: exactly 3/2 + (1-beta)floor(1/beta), hence unbounded as beta falls. The worst-case ratio is therefore known at every threshold in (0,1), and attained at none. Consequently 2/3 is the unique optimal threshold, and the guarantee jumps at one half rather than degrading smoothly. A companion note asks what does not fix the constant.