We study a problem related to submodular function optimization and the exact matching problem for which we show a rather peculiar status: its natural LP-relaxation can have fractional optimal vertices, but there is always also an optimal integral vertex, which we can also compute in polynomial time. More specifically, we consider the multiplicative assignment problem with upgrades in which we are given a set of customers and suppliers and we seek to assign each customer to a different supplier. Each customer has a demand and each supplier has a regular and an upgraded cost for each unit demand provided to the respective assigned client. Our goal is to upgrade at most k suppliers and to compute an assignment in order to minimize the total resulting cost. This can be cast as the problem to compute an optimal matching in a bipartite graph with the additional constraint that we must select k edges from a certain group of edges, similar to selecting k red edges in the exact matching problem. Also, selecting the suppliers to be upgraded corresponds to maximizing a submodular set function under a cardinality constraint. Our result yields an efficient LP-based algorithm to solve our problem optimally. In addition, we also provide a purely strongly polynomial-time algorithm for it. As an application, we obtain exact algorithms for the upgrading variant of the problem to schedule jobs on identical or uniformly related machines in order to minimize their sum of completion times, i.e., where we may upgrade up to k jobs to reduce their respective processing times.

In a packing problem, we are given a collection \(I\) of \(n\) items, each one with a given profit. Our goal is to compute a maximum profit subset of these items that satisfies a given set of packing constraints which depend on the problem at hand. Several approximation algorithms for well-studied NP-hard packing problems are based on a reduction to an auxiliary (packing) problem which is then solved with a dynamic program (DP). Examples for this include approximation algorithms for Knapsack, Geometric Knapsack, Independent Set of Rectangles, and Maximum Throughput Scheduling.

We study the general scheduling problem (GSP) which generalizes and unifies several well-studied preemptive single-machine scheduling problems, such as weighted flow time, weighted sum of completion time, and minimizing the total weight of tardy jobs. We are given a set of jobs with their processing times and release times and seek to compute a (possibly preemptive) schedule for them on one machine. Each job incurs a cost that depends on its completion time in the computed schedule, as given by a separate job-dependent cost function for each job, and our objective is to minimize the total resulting cost of all jobs. The best known result for GSP is a polynomial time \(O(\log \log P)\)-approximation algorithm [Bansal and Pruhs, FOCS 2010, SICOMP 2014].

The (Non-Preemptive) Throughput Maximization problem is a natural and fundamental scheduling problem. We are given n jobs, where each job j is characterized by a processing time and a time window, contained in a global interval [0,T), during which j can be scheduled. Our goal is to schedule the maximum possible number of jobs non-preemptively on a single machine, so that no two scheduled jobs are processed at the same time. This problem is known to be strongly NP-hard. The best-known approximation algorithm for it has an approximation ratio of 1/0.6448 + ε ≈ 1.551 + ε [Im, Li, Moseley IPCO’17], improving on an earlier result in [Chuzhoy, Ostrovsky, Rabani FOCS’01]. In this paper we substantially improve the approximation factor for the problem to 4/3+ε for any constant ε>0. Using pseudo-polynomial time (nT)O(1), we improve the factor even further to 5/4+ε. Our results extend to the setting in which we are given an arbitrary number of (identical) machines.

An important objective function in the scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs, where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. In their breakthrough result, Batra, Garg, and Kumar [FOCS 2018] found the first pseudopolynomial-time constant-factor approximation algorithm for the problem, which was turned into a polynomial-time algorithm by Feige, Kulkarni, and Li [SODA 2019]. The resulting approximation ratio is a (not explicitly stated) constant which is at least 10,000. In this article, we improve this to a PTAS. 1 The algorithm by Batra et al. reduces the problem to Demand MultiCut on trees and solves the resulting instances via LP-rounding and a dynamic program. Instead, we first reduce the problem to a (different) geometric problem while losing only a factor \(1 + \varepsilon\) , and then solve its resulting instances exactly by a dynamic program. In particular, our reduction ensures certain structural properties, due to which we do not need LP-rounding techniques.

Abstract In the Time-Windows Unsplittable Flow on a Path problem ( twUFP ) we are given a resource whose available amount changes over a given time interval (modeled as the edge-capacities of a given path G ) and a collection of tasks. Each task is characterized by a demand (of the considered resource), a profit, an integral processing time, and a time window. Our goal is to compute a maximum profit subset of tasks and schedule them non-preemptively within their respective time windows, such that the total demand of the tasks using each edge e is at most the capacity of e . We prove that twUFP is $$\textsf{APX}$$ APX -hard which contrasts the setting of the problem without time windows, i.e., Unsplittable Flow on a Path, for which a PTAS was recently discovered [Grandoni, Mömke, Wiese, STOC 2022]. Then, we present a quasi-polynomial-time $$2+\varepsilon $$ 2 + ε approximation for twUFP under resource augmentation. Our approximation ratio improves to $$1+\varepsilon $$ 1 + ε if all tasks’ time windows are identical. Our $$\textsf{APX}$$ APX -hardness holds also for this special case and, hence, rules out such a PTAS (and even a QPTAS, unless $$\textsf{NP}\subseteq \textrm{DTIME}(n^{\textrm{poly}(\log n)})$$ NP ⊆ DTIME ( n poly ( log n ) ) ) without resource augmentation.

An important objective function in the scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs, where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. The currently best known polynomial time algorithm for the problem is a (2+є)-approximation by Rohwedder and Wiese [STOC 2021], which builds on the prior break-through result by Batra, Garg, and Kumar [FOCS 2018] who found the first pseudo-polynomial time constant factor approximation algorithm for the problem, and on the result by Feige, Kulkarni, and Li [SODA 2019] who turned the latter into a polynomial time algorithm. However, it remains open whether the problem admits a PTAS.