We consider the following single-machine scheduling problem, which is often denoted 1|| 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 n j=1 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 ǫ > 0, a (4+ǫ)-approximation algorithm for this problem.
We present the first polynomial-time approximation algorithms for single-minded envy-free profit-maximization problems [13] 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 α-approximation algorithm for the corresponding SWM problem as input and return a solution that achieves profit at least OPT /O(α · log u max ), where OPT is the optimal value of the SWM problem, and u max is the maximum supply of an item. This immediately yields approximation guarantees of O( √ m log u max ) for the general single-minded envy-free problem; and O(log u max ) for the tollbooth and highway problems [13], and the graph-vertex pricing problem [3] (α = 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(α · log u max ) 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.
The joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management, logistics, and maintenance scheduling. In this problem, there are multiple item types, each having a given time-dependent sequence of demands that need to be satisfied. In order to satisfy demand, orders of the item types must be placed in advance of the due dates for each demand. Every time an order of item types is placed, there is an associated joint setup cost depending on the subset of item types ordered. This ordering cost can be due to machine, transportation, or labor costs, for example. In addition, there is a cost to holding inventory for demand that has yet to be served. The overall goal is to minimize the total ordering costs plus inventory holding costs.In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types. For this general problem, we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands. Then we consider three special cases of submodular functions which we call the laminar, tree, and cardinality cases, each of which can model real world scenarios that previously have not been captured. For each of these cases, we provide a constant factor
We consider the following single-machine scheduling problem, which is often denoted 1|| 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 n j=1 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 ǫ > 0, a (4+ǫ)-approximation algorithm for this problem.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.