We consider the following single-machine scheduling problem, which is often denoted $1||\sum f_{j}$: we are given $n$ jobs to be scheduled on a single machine, where each job $j$ has an integral processing time $p_j$, and there is a nondecreasing, nonnegative cost function $f_j(C_{j})$ that specifies the cost of finishing $j$ at time $C_{j}$; the objective is to minimize $\sum_{j=1}^n f_j(C_j)$. Bansal and Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the linear program is less than 4. Finally, we show how the technique can be adapted to yield, for any $\epsilon >0$, a polynomial time $(4+\epsilon )$-approximation algorithm for this problem.

We present the first polynomial-time approximation algorithms for single-minded envy-free profit-maximization problems (Guruswami et al., 2005) with limited supply. Our algorithms return a pricing scheme and a subset of customers that are designated the winners, which satisfy the envy-freeness constraint, whereas in our analyses, we compare the profit of our solution against the optimal value of the corresponding social-welfare-maximization (SWM) problem of finding a winner-set with maximum total value. Our algorithms take any LP-based alpha-approximation algorithm for the corresponding SWM problem as input and return a solution that achieves profit at least OPT/O (alpha ldr log umax), where OPT is the optimal value of the SWM problem, and umaxis the maximum supply of an item. This immediately yields approximation guarantees of O(radicmlog umax) for the general single-minded envy-free problem; and O(log umax) for the tollbooth and highway problems (Guruswami et al., 2005), and the graph-vertex pricing problem (Balcan and Blum, 2006) (alpha = O(1) for all the corresponding SWM problems). Since OPT is an upper bound on the maximum profit achievable by any solution (i.e., irrespective of whether the solution satisfies the envy-freeness constraint), our results directly carry over to the non-envy-free versions of these problems too. Our result also thus (constructively) establishes an upper bound of O(alpha ldr log umax) on the ratio of (i) the optimum value of the profit-maximization problem and OPT; and (ii) the optimum profit achievable with and without the constraint of envy-freeness.