A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

TL;DR

This paper introduces a primal-dual approximation algorithm for the single-machine scheduling problem minimizing sum of nondecreasing cost functions, improving the approximation ratio and utilizing knapsack-cover inequalities.

Contribution

It presents a novel primal-dual pseudo-polynomial-time algorithm with a 4-approximation guarantee for the problem, enhancing previous results.

Findings

Achieves a 4-approximation ratio for the scheduling problem.

Demonstrates the tightness of the 4-approximation bound for their algorithm.

Extends the technique to obtain a (4+ε)-approximation for any ε > 0.

Abstract

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 \& 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 LP is less than 4. Finally, we show how the technique can be adapted to yield, for any $\epsilon >0$, a $(4+\epsilon )$-approximation algorithm for this problem.