We consider the problem of online preemptive scheduling on a single machine to minimize the total flow time. In clairvoyant scheduling, where job processing times are revealed upon arrival, the Shortest Remaining Processing Time (SRPT) algorithm is optimal. In practice, however, exact processing times are often unknown. At the opposite extreme, non-clairvoyant scheduling, in which processing times are revealed only upon completion, suffers from strong lower bounds on the competitive ratio. This motivates the study of intermediate information models. We introduce a new model in which processing times are revealed gradually during execution. Each job consists of a sequence of operations, and the processing time of an operation becomes known only after the preceding one completes. This models many scheduling scenarios that arise in computing systems. Our main result is a deterministic O(m²)-competitive algorithm, where m is the maximum number of operations per job. More specifically, we prove a refined competitive ratio in O(m₁ ⋅ m₂), where m₁ and m₂ are instance-dependent parameters describing the operation size structure. Our algorithm and analysis build on recent advancements in robust flow time minimization (SODA '26), where jobs arrive with estimated sizes. However, in our setting we have no bounded estimate on a job’s processing time. Thus, we design a highly adaptive algorithm that gradually explores a job’s operations while working on them, and groups them into virtual chunks whose size can be well-estimated. This is a crucial ingredient of our result and requires a much more careful analysis compared to the robust setting. We also provide lower bounds showing that our bounds are essentially best possible. For the special case of scheduling with uniform obligatory tests, we show that SRPT at the operation level is 2-competitive, which is best possible.

The 2-Edge-Connected Spanning Subgraph Problem (2ECSS) is a fundamental problem in survivable network design. Given an undirected 2-edge-connected graph, the goal is to find a 2-edge-connected spanning subgraph with the minimum number of edges; a graph is 2-edge-connected if it is connected after the removal of any single edge. 2ECSS is APX-hard and has been extensively studied in the context of approximation algorithms. Very recently, Bosch-Calvo, Garg, Grandoni, Hommelsheim, Jabal Ameli, and Lindermayr showed the currently best-known approximation ratio of \(^5\!/\!_4\) [STOC 2025]. This factor is tight for many of their techniques and arguments, and it was not clear whether \(^5\!/\!_4\) can be improved.

Abstract. The polytope scheduling problem (PSP) was introduced by Im, Kulkarni, and Munagala [ J. ACM, 65 (2018), pp. 3:1–3:33] as a very general abstraction of resource allocation over time, where jobs can receive processing rates subject to arbitrary packing constraints. It captures many well-studied problems, including classical unrelated machine scheduling, multidimensional scheduling, and broadcast scheduling. An elegant and well-known algorithm for instantaneous rate allocation with good fairness and efficiency properties is the proportional fairness (PF) algorithm, which was analyzed for PSP by Im, Kulkarni, and Munagala. We drastically improve the analysis of PF for both the general PSP and several of its important special cases subject to the objective of minimizing the sum of weighted completion times. We reduce the upper bound on the competitive ratio from 128 to 27 for general PSP and to 4 for the prominent class of monotone PSP. For certain heterogeneous machine environments, we even close the substantial gap to the lower bound of 2 for nonclairvoyant scheduling. Our analysis also gives the first polynomial-time improvement over the nearly 30-year-old bounds on the competitive ratio of the doubling framework, which was introduced by Hall, Shmoys, and Wein (SODA 1996) for clairvoyant online preemptive scheduling on unrelated machines. Somewhat surprisingly, we achieve this improvement by a nonclairvoyant algorithm, thereby demonstrating that nonclairvoyance is not a (significant) hurdle. Our improvements are based on exploiting monotonicity properties of PSP, providing tight dual fitting arguments on structured instances, and showing new algebraic properties of the optimal objective value for scheduling on unrelated machines. Finally, we establish new connections between PF and matching markets and thereby provide new insights on equilibria and their computational complexity.

We revisit the classical problem of minimizing the total flow time of jobs on a single machine in the online setting where jobs arrive over time. It has long been known that the Shortest Remaining Processing Time (SRPT) algorithm is optimal (i.e., 1-competitive) when the job sizes are known upfront [Schrage, 1968]. But in the non-clairvoyant setting where job sizes are revealed only when the job finishes, no algorithm can be constant-competitive [Motwani, Phillips, and Torng, 1994]. We consider the $\varepsilon$-clairvoyant setting, where $\varepsilon \in[0,1]$, and each job’s processing time becomes known once its remaining processing time equals an $\varepsilon$ fraction of its processing time. This captures settings where the system user uses the initial $(1-\varepsilon)$ fraction of a job’s processing time to learn its true length, which it can then reveal to the algorithm. The model was proposed by Yingchareonthawornchai and Torng (2017), and it smoothly interpolates between the clairvoyant setting (when $\varepsilon=1$) and the non-clairvoyant setting (when $\varepsilon=0$). In a concrete sense, we are asking: how much knowledge is required to circumvent the hardness of this problem? We show that a little knowledge is enough, and that a constant competitive algorithm exists for every constant $\varepsilon\gt 0$. More precisely, for all $\varepsilon \in(0,1)$, we present a deterministic $\left\lceil\frac{1}{\varepsilon}\right\rceil$-competitive algorithm, which is optimal for deterministic algorithms. We also present a matching lower bound (up to a constant factor) for randomized algorithms. Our algorithm to achieve this bound is remarkably simple and applies the “optimism in the face of uncertainty” principle. For each job, we form an optimistic estimate of its length, based on the information revealed thus far and run SRPT on these optimistic estimates. The proof relies on maintaining a matching between the jobs in OPT’s queue and the algorithm’s queue, with small prefix expansion. We achieve this by carefully choosing a set of jobs to arrive earlier than their release times without changing the algorithm, and possibly helping the adversary. These early arrivals allow us to maintain structural properties inductively, giving us the tight guarantee.

In non-clairvoyant scheduling, the goal is to minimize the total job completion time without prior knowledge of individual job processing times. This classical online optimization problem has recently gained attention through the framework of learning-augmented algorithms. We introduce a natural setting in which the scheduler receives continuous feedback in the form of progress bars—estimates of the fraction of each job completed over time. We design new algorithms for both adversarial and stochastic progress bars and prove strong competitive bounds. Our results in the adversarial case surprisingly induce improved guarantees for learning-augmented scheduling with job size predictions. We also introduce a general method for combining scheduling algorithms, yielding further insights in scheduling with predictions. Finally, we propose a stochastic model of progress bars as a more optimistic alternative to conventional worst-case models, and present an asymptotically optimal scheduling algorithm in this setting.

The Polytope Scheduling Problem (PSP) was introduced by Im, Kulkarni, and Munagala (JACM 2018) as a very general abstraction of resource allocation over time and captures many well-studied problems including classical unrelated machine scheduling, multidimensional scheduling, and broadcast scheduling. In PSP, jobs with different arrival times receive processing rates that are subject to arbitrary packing constraints. An elegant and well-known algorithm for instantaneous rate allocation with good fairness and efficiency properties is the Proportional Fairness algorithm (PF), which was analyzed for PSP by Im et al.

Abstract We study the online k -server problem in a learning-augmented setting. While in the traditional online model, an algorithm has no information about the request sequence, we assume that there is given some advice (for example, machine-learned predictions) on an algorithm’s decision. There is, however, no guarantee on the quality of the prediction, and it might be far from being correct. Our main result is a learning-augmented variation of the well-known Double Coverage algorithm for k -server on the line (Chrobak et al. in SIAM J Discret Math 4(2):172–181, 1991) in which we integrate predictions as well as our trust into their quality. We give an error-dependent worst-case performance guarantee, which is a function of a user-defined confidence parameter, and which interpolates smoothly between an optimal performance in case that all predictions are correct, and the best-possible performance regardless of the prediction quality. When given good predictions, we improve upon known lower bounds for online algorithms without advice. We further show that our algorithm achieves for any k almost optimal guarantees, within a class of deterministic learning-augmented algorithms respecting local and memoryless properties. Our algorithm outperforms a previously proposed (more general) learning-augmented algorithm. It is noteworthy that the previous algorithm crucially exploits memory, whereas our algorithm is memoryless. Finally, we demonstrate in experiments the practicability and the superior performance of our algorithm on real-world data.

Querying complex models for precise information (e.g. traffic models, database systems, large ML models) often entails intense computations and results in long response times. Thus, weaker models which give imprecise results quickly can be advantageous, provided inaccuracies can be resolved using few queries to a stronger model. In the fundamental problem of computing a maximum-weight basis of a matroid, a well-known generalization of many combinatorial optimization problems, algorithms have access to a clean oracle to query matroid information. We additionally equip algorithms with a fast but dirty oracle. We design and analyze practical algorithms which only use few clean queries w.r.t. the quality of the dirty oracle, while maintaining robustness against arbitrarily poor dirty oracles, approaching the performance of classic algorithms for the given problem. Notably, we prove that our algorithms are, in many respects, best-possible. Further, we outline extensions to other matroid oracle types, non-free dirty oracles and other matroid problems.

In this paper we study the relation of two fundamental problems in scheduling and fair allocation: makespan minimization on unrelated parallel machines and max-min fair allocation, also known as the Santa Claus problem. For both of these problems the best approximation factor is a notorious open question; more precisely, whether there is a better-than-2 approximation for the former problem and whether there is a constant approximation for the latter.

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d, k decides in time f( k, d) · n c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k.

We consider non-clairvoyant scheduling with online precedence constraints, where an algorithm is oblivious to any job dependencies and learns about a job only if all of its predecessors have been completed. Given strong impossibility results in classical competitive analysis, we investigate the problem in a learning-augmented setting, where an algorithm has access to predictions without any quality guarantee. We discuss different prediction models: novel problem-specific models as well as general ones, which have been proposed in previous works. We present lower bounds and algorithmic upper bounds for different precedence topologies, and thereby give a structured overview on which and how additional (possibly erroneous) information helps for designing better algorithms. Along the way, we also improve bounds on traditional competitive ratios for existing algorithms.

We consider online scheduling on unrelated (heterogeneous) machines in a speed-oblivious setting, where an algorithm is unaware of the exact job-dependent processing speeds. We show strong impossibility results for clairvoyant and non-clairvoyant algorithms and overcome them in models inspired by practical settings: (i) we provide competitive learning-augmented algorithms, assuming that (possibly erroneous) predictions on the speeds are given, and (ii) we provide competitive algorithms for the speed-ordered model, where a single global order of machines according to their unknown job-dependent speeds is known. We prove strong theoretical guarantees and evaluate our findings on a representative heterogeneous multi-core processor. These seem to be the first empirical results for scheduling algorithms with predictions that are evaluated in a non-synthetic hardware environment.

Exploring unknown environments is a fundamental task in many domains, e.g., robot navigation, network security, and internet search. We initiate the study of a learning-augmented variant of the classical, notoriously hard online graph exploration problem by adding access to machine-learned predictions. We propose an algorithm that naturally integrates predictions into the well-known Nearest Neighbor (NN) algorithm and significantly outperforms any known online algorithm if the prediction is of high accuracy while maintaining good guarantees when the prediction is of poor quality. We provide theoretical worst-case bounds that gracefully degrade with the prediction error, and we complement them by computational experiments that confirm our results. Further, we extend our concept to a general framework to robustify algorithms. By interpolating carefully between a given algorithm and NN, we prove new performance bounds that leverage the individual good performance on particular inputs while establishing robustness to arbitrary inputs.

We introduce a novel measure for quantifying the error in input predictions. The error is based on a minimum-cost hyperedge cover in a suitably defined hypergraph and provides a general template which we apply to online graph problems. The measure captures errors due to absent predicted requests as well as unpredicted actual requests; hence, predicted and actual inputs can be of arbitrary size. We achieve refined performance guarantees for previously studied network design problems in the online-list model, such as Steiner tree and facility location. Further, we initiate the study of learning-augmented algorithms for online routing problems, such as the online traveling salesperson problem and the online dial-a-ride problem, where (transportation) requests arrive over time (online-time model). We provide a general algorithmic framework and we give error-dependent performance bounds that improve upon known worst-case barriers, when given accurate predictions, at the cost of slightly increased worst-case bounds when given predictions of arbitrary quality.

In non-clairvoyant scheduling, the task is to find an online strategy for scheduling jobs with a priori unknown processing requirements with the objective to minimize the total (weighted) completion time. We revisit this well-studied problem in a recently popular learning-augmented setting that integrates (untrusted) predictions in online algorithm design. While previous works used predictions on processing requirements, we propose a new prediction model, which provides a relative order of jobs which could be seen as predicting algorithmic actions rather than parts of the unknown input. We show that these predictions have desired properties, admit a natural error measure as well as algorithms with strong performance guarantees and that they are learnable in both, theory and practice. We generalize the algorithmic framework proposed in the seminal paper by Kumar et al. (NeurIPS'18) and present the first learning-augmented scheduling results for weighted jobs and unrelated machines. We demonstrate in empirical experiments the practicability and superior performance compared to the previously suggested single-machine algorithms.

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d and k decides in time f(k,d)⋅ n^c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k.