Minimizing the sum of weighted completion times in a concurrent open shop

We study minimizing the sum of weighted completion times in a concurrent open shop. We give a primal–dual 2-approximation algorithm for this problem. We also show that several natural linear programming relaxations for this problem have an integrality gap of 2. Finally, we show that this problem is inapproximable within a factor strictly less than 6/5 if P ≠ NP , or strictly less than 4/3 if the Unique Games Conjecture also holds.