arXiv subjects

Susanne Albers

Publications and source records attributed to Susanne Albers.

At least 19 recordsLinked to original sources

arXiv2026

Scheduling to Maximize Weighted Throughput with an Active-Time Budget

We study the active-time scheduling problem with weighted throughput maximization. In this setting, a set of $n$ jobs $J$ arrive at integer release times, each with an integer processing time and integer deadline. Jobs may be preempted at integer time slot boundaries. A schedule assigns jobs to time slots, with at most $m$ jobs assigned to the same time slot. A slot is called \emph{active} if at least one job is scheduled in it. Instead of scheduling all jobs to minimize the number of active time slots, we consider the more general variant of \emph{weighted throughput} with an active-time budget $K$, where each job $j\in J$ has a weight $w_j$. The objective is to maximize the total weight of \emph{completed} jobs using at most $K$ active time slots. This means that partially scheduled jobs do not count towards the objective. The classical active-time minimization problem is recovered by asking whether all jobs can be completed within a given active-time budget. We give hardness, approximation, and exact algorithmic results. For general intervals with unbounded parallelism, we prove NP-hardness, rule out an FPTAS unless $\mathrm{P}=\mathrm{NP}$, and give a pseudo-polynomial time $Ω(1/\log K)$-approximation. For proper intervals, we prove a canonical structural lemma and obtain an exact $(nK)^{O(m)}$-time algorithm. For laminar intervals, we give an exact $f(K,m)\cdot n^{O(1)}$-time algorithm.

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arXiv2025

Online busy time scheduling with flexible jobs

We consider the online busy time scheduling problem motivated by energy and cost minimization in cloud computing systems. The input is a set of jobs $J=\{1,\dots,n\}$ where each job $j\in J$ has a release time $r_j$, deadline $d_j$, and processing time $p_j$. $m$ homogeneous machines are given with a parallelism parameter $g\geq 1$, which is the maximal number of jobs that can be processed simultaneously on a machine. A machine is called \emph{busy} when at least one job is being processed. The objective is to find a feasible schedule for all jobs such that the sum of busy times over all machines is minimized. We consider the online setting, where a job $j\in J$ is revealed at its release time $r_j$. We show multiple algorithms in different problem variants that have a tight competitive ratio. For the busy time scheduling problem, uniform processing time jobs, and where the parallelism is unbounded ($g=\infty$), we show a $2$-competitive algorithm and an online adversary that shows that the algorithm is tight. For the setting where jobs have arbitrary processing time, agreeable deadlines, and the parallelism is unbounded, we show a different tight $2$-competitive algorithm. For machines with bounded parallelism, we show lower bounds on the competitive ratio of any online algorithm when $g$ is small. Furthermore, we improve the setting with arbitrary jobs where the algorithm is allowed lookahead.

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arXiv2025

On the 2D Demand Bin Packing Problem: Hardness and Approximation Algorithms

We study a two-dimensional generalization of the classical Bin Packing problem, denoted as 2D Demand Bin Packing. In this context, each bin is a horizontal timeline, and rectangular tasks (representing electric appliances or computational requirements) must be allocated into the minimum number of bins so that the sum of the heights of tasks at any point in time is at most a given constant capacity. We prove that simple variants of the problem are NP-hard to approximate within a factor better than $2$, namely when tasks have short height and when they are squares, and provide best-possible approximation algorithms for them; we also present a simple $3$-approximation for the general case. All our algorithms are based on a general framework that computes structured solutions for relatively large tasks, while including relatively small tasks on top via a generalization of the well-known First-Fit algorithm for Bin Packing.

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arXiv2024

Online $b$-Matching with Stochastic Rewards

The $b$-matching problem is an allocation problem where the vertices on the left-hand side of a bipartite graph, referred to as servers, may be matched multiple times. In the setting with stochastic rewards, an assignment between an incoming request and a server turns into a match with a given success probability. Mehta and Panigrahi (FOCS 2012) introduced online bipartite matching with stochastic rewards, where each vertex may be matched once. The framework is equally interesting in graphs with vertex capacities. In Internet advertising, for instance, the advertisers seek successful matches with a large number of users. We develop (tight) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms, for $b$-matching with stochastic rewards. Our bounds hold for both offline benchmarks considered in the literature. As in prior work, we first consider vanishing probabilities. We show that no randomized online algorithm can achieve a competitive ratio greater than $1-1/e\approx 0.632$, even for identical vanishing probabilities and arbitrary uniform server capacities. Furthermore, we conduct a primal-dual analysis of the deterministic \textsc{StochasticBalance} algorithm. We prove that it achieves a competitive ratio of $1-1/e$, as server capacities increase, for arbitrary heterogeneous non-vanishing edge probabilities. This performance guarantee holds in a general setting where servers have individual capacities and for the vertex-weighted problem extension. To the best of our knowledge, this is the first result for \textsc{StochasticBalance} with arbitrary non-vanishing probabilities. We remark that our impossibility result implies in particular that, for the AdWords problem, no online algorithm can be better than $(1-1/e)$-competitive in the setting with stochastic rewards.

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arXiv2022

Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities

We study the $b$-matching problem in bipartite graphs $G=(S,R,E)$. Each vertex $s\in S$ is a server with individual capacity $b_s$. The vertices $r\in R$ are requests that arrive online and must be assigned instantly to an eligible server. The goal is to maximize the size of the constructed matching. We assume that $G$ is a $(k,d)$-graph~\cite{NW}, where $k$ specifies a lower bound on the degree of each server and $d$ is an upper bound on the degree of each request. This setting models matching problems in timely applications. We present tight upper and lower bounds on the performance of deterministic online algorithms. In particular, we develop a new online algorithm via a primal-dual analysis. The optimal competitive ratio tends to~1, for arbitrary $k\geq d$, as the server capacities increase. Hence, nearly optimal solutions can be computed online. Our results also hold for the vertex-weighted problem extension, and thus for AdWords and auction problems in which each bidder issues individual, equally valued bids. Our bounds improve the previous best competitive ratios. The asymptotic competitiveness of~1 is a significant improvement over the previous factor of $1-1/e^{k/d}$, for the interesting range where $k/d\geq 1$ is small. Recall that $1-1/e\approx 0.63$. Matching problems that admit a competitive ratio arbitrarily close to~1 are rare. Prior results rely on randomization or probabilistic input models.

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arXiv2021

Scheduling in the Random-Order Model

Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to $m$ identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is $(2-1/m)$-competitive. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201. No deterministic online strategy can obtain a competitiveness smaller than 1.88. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds, for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability.

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arXiv2021

Scheduling in the Secretary Model

This paper studies Makespan Minimization in the secretary model. Formally, jobs, specified by their processing times, are presented in a uniformly random order. An online algorithm has to assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized. We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy LightLoad provides a very simple algorithm retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders. Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications. They also demonstrate the power of randomization when compared to much stronger deterministic reordering models. We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model.

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arXiv2021

Machine Covering in the Random-Order Model

In the Online Machine Covering problem jobs, defined by their sizes, arrive one by one and have to be assigned to $m$ parallel and identical machines, with the goal of maximizing the load of the least-loaded machine. In this work, we study the Machine Covering problem in the recently popular random-order model. Here no extra resources are present, but instead the adversary is weakened in that it can only decide upon the input set while jobs are revealed uniformly at random. It is particularly relevant to Machine Covering where lower bounds are usually associated to highly structured input sequences. We first analyze Graham's Greedy-strategy in this context and establish that its competitive ratio decreases slightly to $Θ\left(\frac{m}{\log(m)}\right)$ which is asymptotically tight. Then, as our main result, we present an improved $\tilde{O}(\sqrt[4]{m})$-competitive algorithm for the problem. This result is achieved by exploiting the extra information coming from the random order of the jobs, using sampling techniques to devise an improved mechanism to distinguish jobs that are relatively large from small ones. We complement this result with a first lower bound showing that no algorithm can have a competitive ratio of $O\left(\frac{\log(m)}{\log\log(m)}\right)$ in the random-order model. This lower bound is achieved by studying a novel variant of the Secretary problem, which could be of independent interest.

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arXiv2021

Explorable Uncertainty in Scheduling with Non-Uniform Testing Times

The problem of scheduling with testing in the framework of explorable uncertainty models environments where some preliminary action can influence the duration of a task. In the model, each job has an unknown processing time that can be revealed by running a test. Alternatively, jobs may be run untested for the duration of a given upper limit. Recently, Dürr et al. [5] have studied the setting where all testing times are of unit size and have given lower and upper bounds for the objectives of minimizing the sum of completion times and the makespan on a single machine. In this paper, we extend the problem to non-uniform testing times and present the first competitive algorithms. The general setting is motivated for example by online user surveys for market prediction or querying centralized databases in distributed computing. Introducing general testing times gives the problem a new flavor and requires updated methods with new techniques in the analysis. We present constant competitive ratios for the objective of minimizing the sum of completion times in the deterministic case, both in the non-preemptive and preemptive setting. For the preemptive setting, we additionally give a first lower bound. We also present a randomized algorithm with improved competitive ratio. Furthermore, we give tight competitive ratios for the objective of minimizing the makespan, both in the deterministic and the randomized setting.

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arXiv2021

Scheduling with Testing on Multiple Identical Parallel Machines

Scheduling with testing is a recent online problem within the framework of explorable uncertainty motivated by environments where some preliminary action can influence the duration of a task. Jobs have an unknown processing time that can be explored by running a test. Alternatively, jobs can be executed for the duration of a given upper limit. We consider this problem within the setting of multiple identical parallel machines and present competitive deterministic algorithms and lower bounds for the objective of minimizing the makespan of the schedule. In the non-preemptive setting, we present the SBS algorithm whose competitive ratio approaches $3.1016$ if the number of machines becomes large. We compare this result with a simple greedy strategy and a lower bound which approaches $2$. In the case of uniform testing times, we can improve the SBS algorithm to be $3$-competitive. For the preemptive case we provide a $2$-competitive algorithm and a tight lower bound which approaches the same value.

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arXiv2021

Optimal Algorithms for Right-Sizing Data Centers- Extended Version

Electricity cost is a dominant and rapidly growing expense in data centers. Unfortunately, much of the consumed energy is wasted because servers are idle for extended periods of time. We study a capacity management problem that dynamically right-sizes a data center, matching the number of active servers with the varying demand for computing capacity. We resort to a data-center optimization problem introduced by Lin, Wierman, Andrew and Thereska that, over a time horizon, minimizes a combined objective function consisting of operating cost, modeled by a sequence of convex functions, and server switching cost. All prior work addresses a continuous setting in which the number of active servers, at any time, may take a fractional value. In this paper, we investigate for the first time the discrete data-center optimization problem where the number of active servers, at any time, must be integer valued. Thereby we seek truly feasible solutions. First, we show that the offline problem can be solved in polynomial time. Second, we study the online problem and extend the algorithm {\em Lazy Capacity Provisioning\/} (LCP) by Lin et al. to the discrete setting. We prove that LCP is 3-competitive. Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than~3. In addition, we develop a randomized online algorithm that is 2-competitive against an oblivious adversary. Moreover, we prove that 2 is a lower bound for the competitive ratio of randomized online algorithms, so our algorithm is optimal. Finally, we address the continuous setting and give a lower bound of~2 on the best competitiveness of online algorithms. This matches an upper bound by Bansal et al. We prove that all lower bounds still hold in a problem variant with more restricted operating cost functions, introduced by Lin et al.

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arXiv2021

Algorithms for Energy Conservation in Heterogeneous Data Centers

Power consumption is the major cost factor in data centers. It can be reduced by dynamically right-sizing the data center according to the currently arriving jobs. If there is a long period with low load, servers can be powered down to save energy. For identical machines, the problem has already been solved optimally by Lin et al. (2013) and Albers and Quedenfeld (2018). In this paper, we study how a data-center with heterogeneous servers can dynamically be right-sized to minimize the energy consumption. There are $d$ different server types with various operating and switching costs. We present a deterministic online algorithm that achieves a competitive ratio of $2d$ as well as a randomized version that is $1.58d$-competitive. Furthermore, we show that there is no deterministic online algorithm that attains a competitive ratio smaller than $2d$. Hence our deterministic algorithm is optimal. In contrast to related problems like convex body chasing and convex function chasing, we investigate the discrete setting where the number of active servers must be integral, so we gain truly feasible solutions.

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arXiv2021

Algorithms for Right-Sizing Heterogeneous Data Centers

Power consumption is a dominant and still growing cost factor in data centers. In time periods with low load, the energy consumption can be reduced by powering down unused servers. We resort to a model introduced by Lin, Wierman, Andrew and Thereska that considers data centers with identical machines, and generalize it to heterogeneous data centers with $d$ different server types. The operating cost of a server depends on its load and is modeled by an increasing, convex function for each server type. In contrast to earlier work, we consider the discrete setting, where the number of active servers must be integral. Thereby, we seek truly feasible solutions. For homogeneous data centers ($d=1$), both the offline and the online problem were solved optimally by Albers and Quedenfeld (2018). In this paper, we study heterogeneous data centers with general time-dependent operating cost functions. We develop an online algorithm based on a work function approach which achieves a competitive ratio of $2d + 1 + ε$ for any $ε> 0$. For time-independent operating cost functions, the competitive ratio can be reduced to $2d + 1$. There is a lower bound of $2d$ shown by Albers and Quedenfeld (2021), so our algorithm is nearly optimal. For the offline version, we give a graph-based $(1+ε)$-approximation algorithm. Additionally, our offline algorithm is able to handle time-variable data-center sizes.

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arXiv2020

New Results for the $k$-Secretary Problem

Suppose that $n$ items arrive online in random order and the goal is to select $k$ of them such that the expected sum of the selected items is maximized. The decision for any item is irrevocable and must be made on arrival without knowing future items. This problem is known as the $k$-secretary problem, which includes the classical secretary problem with the special case $k=1$. It is well-known that the latter problem can be solved by a simple algorithm of competitive ratio $1/e$ which is optimal for $n \to \infty$. Existing algorithms beating the threshold of $1/e$ either rely on involved selection policies already for $k=2$, or assume that $k$ is large. In this paper we present results for the $k$-secretary problem, considering the interesting and relevant case that $k$ is small. We focus on simple selection algorithms, accompanied by combinatorial analyses. As a main contribution we propose a natural deterministic algorithm designed to have competitive ratios strictly greater than $1/e$ for small $k \geq 2$. This algorithm is hardly more complex than the elegant strategy for the classical secretary problem, optimal for $k=1$, and works for all $k \geq 1$. We derive its competitive ratios for $k \leq 100$, ranging from $0.41$ for $k=2$ to $0.75$ for $k=100$. Moreover, we consider an algorithm proposed earlier in the literature, for which no rigorous analysis is known. We show that its competitive ratio is $0.4168$ for $k=2$, implying that the previous analysis was not tight. Our analysis reveals a surprising combinatorial property of this algorithm, which might be helpful to find a tight analysis for all $k$.

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arXiv2020

Improved Online Algorithms for Knapsack and GAP in the Random Order Model

The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set. We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]. Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)].

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arXiv2020

Best Fit Bin Packing with Random Order Revisited

Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon's result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

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arXiv2020

Minimization of Weighted Completion Times in Path-based Coflow Scheduling

Coflow scheduling models communication requests in parallel computing frameworks where multiple data flows between shared resources need to be completed before computation can continue. In this paper, we introduce Path-based Coflow Scheduling, a generalized problem variant that considers coflows as collections of flows along fixed paths on general network topologies with node capacity restrictions. For this problem, we minimize the coflows' total weighted completion time. We show that flows on paths in the original network can be interpreted as hyperedges in a hypergraph and transform the path-based scheduling problem into an edge scheduling problem on this hypergraph. We present a $(2λ+ 1)$-approximation algorithm when node capacities are set to one, where $λ$ is the maximum number of nodes in a path. For the special case of simultaneous release times for all flows, our result improves to a $(2λ)$-approximation. Furthermore, we generalize the result to arbitrary node constraints and obtain a $(2λΔ+ 1)$- and a $(2λΔ)$-approximation in the case of general and zero release times, where $Δ$ captures the capacity disparity between nodes.

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arXiv2017

The Price of Uncertainty in Present-Biased Planning

The tendency to overestimate immediate utility is a common cognitive bias. As a result people behave inconsistently over time and fail to reach long-term goals. Behavioral economics tries to help affected individuals by implementing external incentives. However, designing robust incentives is often difficult due to imperfect knowledge of the parameter $β\in (0,1]$ quantifying a person's present bias. Using the graphical model of Kleinberg and Oren, we approach this problem from an algorithmic perspective. Based on the assumption that the only information about $β$ is its membership in some set $B \subset (0,1]$, we distinguish between two models of uncertainty: one in which $β$ is fixed and one in which it varies over time. As our main result we show that the conceptual loss of efficiency incurred by incentives in the form of penalty fees is at most $2$ in the former and $1 + \max B/\min B$ in the latter model. We also give asymptotically matching lower bounds and approximation algorithms.

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