Deterministic Online Algorithm Research Articles

Deterministic primal-dual algorithms for online k-way matching with delays

In this paper, we study the Min-cost Perfect k-way Matching with Delays (k-MPMD), recently introduced by Melnyk et al. In the problem, m requests arrive one-by-one over time in a metric space. At any time, we can irrevocably make a group of k requests who arrived so far, that incurs the distance cost among the k requests in addition to the sum of the waiting cost for the k requests. The goal is to partition all the requests into groups of k requests, minimizing the total cost. The problem is a generalization of the min-cost perfect matching with delays (corresponding to 2-MPMD). It is known that no online algorithm for k-MPMD can achieve a bounded competitive ratio in general, where the competitive ratio is the worst-case ratio between its performance and the offline optimal value. On the other hand, k-MPMD is known to admit a randomized online algorithm with competitive ratio O(k5log⁡n) for a certain class of k-point metrics called the H-metric, where n is the size of the metric space. In this paper, we propose a deterministic online algorithm with a competitive ratio of O(mk2) for the k-MPMD in H-metric space. Furthermore, we show that the competitive ratio can be improved to O(m+k2) if the metric is given as a diameter on a line.

Competitive Data-Structure Dynamization

Data-structure dynamization is a general approach for making static data structures dynamic. It is used extensively in geometric settings and in the guise of so-called merge (or compaction) policies in big-data databases such as LevelDB and Google Bigtable. Previous theoretical work is based on worst-case analyses for uniform inputs—insertions of one item at a time and non-varying read rate. In practice, merge policies must not only handle batch insertions and varying read/write ratios, they can take advantage of such non-uniformity to reduce cost on a per-input basis. To model this, we initiate the study of data-structure dynamization through the lens of competitive analysis via two new online set-cover problems. For each, the input is a sequence of disjoint sets of weighted items. The sets are revealed one at a time. The algorithm must respond to each with a set cover that covers all items revealed so far. It obtains the cover incrementally from the previous cover by adding one or more sets and optionally removing existing sets. For each new set the algorithm incurs build cost equal to the weight of the items in the set. In the first problem the objective is to minimize total build cost plus total query cost , where the algorithm incurs a query cost at each time \(t\) equal to the current cover size. In the second problem, the objective is to minimize the build cost while keeping the query cost from exceeding \(k\) (a given parameter) at any time. We give deterministic online algorithms for both variants, with competitive ratios of \(\Theta(\log^{*}n)\) and \(k\) , respectively. The latter ratio is optimal for the second variant.

Behavioral insights into compression: a study of Move-to-Front-or-Middle deterministic online algorithm through sequence classification and characterization

Behavioral insights into compression: a study of Move-to-Front-or-Middle deterministic online algorithm through sequence classification and characterization

Almost tight bounds for online hypergraph matching

In the online hypergraph matching problem, hyperedges of size k over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A naïve greedy algorithm for this problem achieves a competitive ratio of 1k. We show that no (randomized) online algorithm has competitive ratio better than 2+o(1)k. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio 1−o(1)ln⁡(k) and show that no online algorithm can have competitive ratio strictly better than 1+o(1)ln⁡(k). Lastly, we give a 1−o(1)ln⁡(k) competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.

Optimal algorithms for solving green online reheat furnace scheduling problems with deterioration effect

Optimizing reheating furnace scheduling in terms of energy saving and reheating time reduction is a vital component in improving competitive advantage of the steel enterprises. However, current reheating furnace scheduling models are still limited since (1) most of the scheduling problems involving reheating furnace suppose that the steel orders and their information are known in advance and (2) the reheating time of a slab in reheating operation is a constant in the majority of current reheating furnace scheduling models. To address these issues, three online reheat furnace scheduling models with deterioration effect are proposed in order to minimize the maximum residence time, the maximum energy consumption as well as total energy consumption of all slabs in reheating operation, respectively. Furthermore, for these online scheduling problems, deterministic online algorithms are presented and shown to be optimal, respectively. In addition, the computational experiments confirm that the three online algorithms developed in this paper are particularly efficient and effective in practice.

Online class cover problem

In this paper, we study the online class cover problem where a (finite or infinite) family F of geometric objects and a set Pr of red points in Rd are given a prior, and blue points from Rd arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from F that do not cover any points of Pr. The objective of the problem is to place a minimum number of objects. When F consists of axis-parallel unit squares in R2, we prove that the competitive ratio of any deterministic online algorithm is Ω(log⁡|Pr|), and also propose an O(log⁡|Pr|)-competitive deterministic algorithm for the problem.

Online Conversion with Switching Costs: Robust and Learning-Augmented Algorithms

We introduce and study online conversion with switching costs, a family of online problems that capture emerging problems at the intersection of energy and sustainability. In this problem, an online player attempts to purchase (alternatively, sell) fractional shares of an asset during a fixed time horizon with length T. At each time step, a cost function (alternatively, price function) is revealed, and the player must irrevocably decide an amount of asset to convert. The player also incurs a switching cost whenever their decision changes in consecutive time steps, i.e., when they increase or decrease their purchasing amount. We introduce competitive (robust) threshold-based algorithms for both the minimization and maximization variants of this problem, and show they are optimal among deterministic online algorithms. We then propose learning-augmented algorithms that take advantage of untrusted black-box advice (such as predictions from a machine learning model) to achieve significantly better average-case performance without sacrificing worst-case competitive guarantees. Finally, we empirically evaluate our proposed algorithms using a carbon-aware EV charging case study, showing that our algorithms substantially improve on baseline methods for this problem.

The Online Pause and Resume Problem: Optimal Algorithms and An Application to Carbon-Aware Load Shifting

We introduce and study the online pause and resume problem. In this problem, a player attempts to find the k lowest (alternatively, highest) prices in a sequence of fixed length T, which is revealed sequentially. At each time step, the player is presented with a price and decides whether to accept or reject it. The player incurs a switching cost whenever their decision changes in consecutive time steps, i.e., whenever they pause or resume purchasing. This online problem is motivated by the goal of carbon-aware load shifting, where a workload may be paused during periods of high carbon intensity and resumed during periods of low carbon intensity and incurs a cost when saving or restoring its state. It has strong connections to existing problems studied in the literature on online optimization, though it introduces unique technical challenges that prevent the direct application of existing algorithms. Extending prior work on threshold-based algorithms, we introduce double-threshold algorithms for both variants of this problem. We further show that the competitive ratios achieved by these algorithms are the best achievable by any deterministic online algorithm. Finally, we empirically validate our proposed algorithm through case studies on the application of carbon-aware load shifting using real carbon trace data and existing baseline algorithms.

Semi-online scheduling on two uniform parallel machines with initial lookahead

This work studies the problem of semi-online scheduling on two uniform parallel machines with speeds 1 and s (≥2), respectively. We introduce a novel concept of initial lookahead in which any deterministic online algorithm has the full knowledge of the first k jobs at the beginning, while the remaining jobs are released one-by-one in the online over-list mode. The objective of the considered problem is to minimize the makespan. We focus on the case where the first k jobs are of a total processing time not less than (s + 1)Δ where Δ is the largest job length, and it is assumed that s is an integer. We prove a lower bound of (s2+s+1)/(s2+s) , and propose a deterministic semi-online algorithm with competitive ratio of (s+1)2/s2+s+1. The ratio is at most 9/7 and much less than that of 1.618 for the corresponding case without initial lookahead (Cho and Sahni, SIAM J. Comput. 9 (1980) 91–103). Our results demonstrate that a finite ability of initial lookahead can greatly improve the competitiveness of online algorithms.

Improved Online Scheduling of Moldable Task Graphs under Common Speedup Models

We consider the online scheduling problem of moldable task graphs on multiprocessor systems for minimizing the overall completion time (or makespan). Moldable job scheduling has been widely studied in the literature, in particular when tasks have dependencies (i.e., task graphs) or when tasks are released on-the-fly (i.e., online). However, few studies have focused on both (i.e., online scheduling of moldable task graphs). In this article, we design a new online scheduling algorithm for this problem and derive constant competitive ratios under several common yet realistic speedup models (i.e., roofline, communication, Amdahl, and a general combination). These results improve the ones we have shown in the preliminary version of the article. We also prove, for each speedup model, a lower bound on the competitiveness of any online list scheduling algorithm that allocates processors to a task based only on the task’s parameters and not on its position in the graph. This lower bound matches exactly the competitive ratio of our algorithm for the roofline, communication, and Amdahl’s model, and is close to the ratio for the general model. Finally, we provide a lower bound on the competitive ratio of any deterministic online algorithm for the arbitrary speedup model, which is not constant but depends on the number of tasks in the longest path of the graph.

An Online Fairness-Aware Task Planning Approach for Spatial Crowdsourcing

Spatial crowdsourcing, a human-centric paradigm for performing spatial tasks, has drawn rising attention. Task assignment and worker scheduling are basic issues in spatial crowdsourcing, which has a high demand on the timeliness of tasks and fairness among workers. In this paper, we propose Utility-Fairness Index to evaluate the performance of the crowdsourcing platform and introduce the Fairness-Aware Task Planning problem that maximizes the utility while well-maintaining fairness during task assignment. We prove that its offline version is NP-hard, and clarify that there is no deterministic online algorithm with a constant competitive ratio for it. Two greedy-based online algorithms, Traversal Search and Advanced Nearest Neighbor are designed to solve the problem. We make optimization on running time with recurrence method to reach linear complexity, and use Gaussian Mixture Model to reveal the distribution regularity of tasks in these algorithms. Experiments on both synthetic and real datasets validate the efficiency and effectiveness of our algorithms.

The Online Pause and Resume Problem: Optimal Algorithms and An Application to Carbon-Aware Load Shifting

We introduce and study the online pause and resume problem. In this problem, a player attempts to find the k lowest (alternatively, highest) prices in a sequence of fixed length T, which is revealed sequentially. At each time step, the player is presented with a price and decides whether to accept or reject it. The player incurs aswitching cost whenever their decision changes in consecutive time steps, i.e., whenever they pause or resume purchasing. This online problem is motivated by the goal of carbon-aware load shifting, where a workload may be paused during periods of high carbon intensity and resumed during periods of low carbon intensity and incurs a cost when saving or restoring its state. It has strong connections to existing problems studied in the literature on online optimization, though it introduces unique technical challenges that prevent the direct application of existing algorithms. Extending prior work on threshold-based algorithms, we introducedouble-threshold algorithms for both the minimization and maximization variants of this problem. We further show that the competitive ratios achieved by these algorithms are the best achievable by any deterministic online algorithm. Finally, we empirically validate our proposed algorithm through case studies on the application of carbon-aware load shifting using real carbon trace data and existing baseline algorithms.

Toward Online Mobile Facility Location on General Metrics

We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. (SODA 258–267 2007)). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of k mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics. A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the k-server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.

A good balance for an online scheduling problem considering both learning and deteriorating effects in the handicraft process

The paper considers an online scheduling problem with the effects of both learning and deterioration to minimize the total completion time. More specifically, we assume that the actual processing length of job Jj is p jr = p j r b + at , where pj is the initial processing time of Jj , t is the starting time of Jj , r is the seating arrangement position of Jj , b is the learning factor and a is the deterioration factor, respectively. For this problem, we show that the performance ratio of any deterministic online algorithm is not <2 and provide a best possible online algorithm DSBPT with a competitive ratio of 2. Furthermore, we also present a concise computational simulation study to verify the effectiveness and efficiency of the proposed algorithm DSBPT, as well as the management implications provided for decision-makers to production optimization.

An online joint replenishment problem combined with single machine scheduling

This paper considers a combination of the joint replenishment problem with single machine scheduling. There is a single resource, which is required by all the unit-time jobs, and a job can be started at time point t on the machine if and only if the machine does not process another job at t, and the resource is replenished between its release date and t. Each replenishment has a cost, which is independent of the amount replenished. The objective is to minimize the total replenishment cost plus the maximum flow time of the jobs. We consider the online variant of the problem, where the jobs are released over time, and once a job is inserted into the schedule, its starting time cannot be changed. We propose a deterministic 2-competitive online algorithm for the general input. Moreover, we show that for a certain class of inputs (so-called p-bounded input), the competitive ratio of the algorithm tends to sqrt{2} as the number of jobs tends to infinity. We also derive several lower bounds for the best competitive ratio of any deterministic online algorithm under various assumptions.

Cost-Efficient Sharing Algorithms for DNN Model Serving in Mobile Edge Networks

With the fast growth of mobile edge computing (MEC), the deep neural network (DNN) has gained more opportunities in application to various mobile services. Given the tremendous number of learning parameters and large model size, the DNN model is often trained in cloud center and then dispatched to end devices for inference via edge network. Therefore, maximizing the cost-efficiency of learned model dispatch in the edge network would be a critical problem for the model serving in various application contexts. To reach this goal, in this paper we focus mainly on reducing the total model dispatch cost in the edge network while maintaining the efficiency of the model inference. We first study this problem in its off-line form as a baseline where a sequence of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> requests can be pre-defined in advance and exploit dynamic programming techniques to obtain a fast optimal algorithm in time complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(m^{2}n)$</tex-math></inline-formula> under a semi-homogeneous cost model in a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -sized network. Then, we design and implement a 2.5-competitive algorithm for its online case with a provable lower bound of 2 for any deterministic online algorithm. We verify our results through careful algorithmic analysis and validate their actual performance via a trace-based study based on a public open international mobile network dataset.

An Improved Algorithm for Online Min-Sum Set Cover

We study a fundamental model of online preference aggregation, where an algorithm maintains an ordered list of n elements. An input is a stream of preferred sets R_1, R_2, ..., R_t, ... Upon seeing R_t and without knowledge of any future sets, an algorithm has to rerank elements (change the list ordering), so that at least one element of R_t is found near the list front. The incurred cost is a sum of the list update costs (the number of swaps of neighboring list elements) and access cost (the position of the first element of R_t on the list). This scenario occurs naturally in applications such as ordering items in an online shop using aggregated preferences of shop customers. The theoretical underpinning of this problem is known as Min-Sum Set Cover. Unlike previous work that mostly studied the performance of an online algorithm ALG in comparison to the static optimal solution (a single optimal list ordering), in this paper, we study an arguably harder variant where the benchmark is the provably stronger optimal dynamic solution OPT (that may also modify the list ordering). In terms of an online shop, this means that the aggregated preferences of its user base evolve with time. We construct a computationally efficient randomized algorithm whose competitive ratio (ALG-to-OPT cost ratio) is O(r^2) and prove the existence of a deterministic O(r^4)-competitive algorithm. Here, r is the maximum cardinality of sets R_t. This is the first algorithm whose ratio does not depend on n: the previously best algorithm for this problem was O(r^(3/2) * n^(1/2))-competitive and Ω(r) is a lower bound on the performance of any deterministic online algorithm.

PackCache: An Online Cost-Driven Data Caching Algorithm in the Cloud

In this paper, we study a data caching problem in the cloud environment, where multiple frequently co-utilised data items could be packed as a single item being transferred to serve a sequence of data requests dynamically with reduced cost. To this end, we propose an online algorithm with respect to a homogeneous cost model, called <i>PackCache</i> , that can leverage the FP-Tree technique to mine those frequently co-utilised data items for packing whereby the incoming requests could be cost-effectively served online by exploiting the concept of anticipatory caching. We show the algorithm is <inline-formula><tex-math notation="LaTeX">$2/\alpha$</tex-math></inline-formula> competitive, reaching the lower bound of the competitive ratio for any deterministic online algorithm on the studied caching problem, and also time and space efficient to serve the requests. Finally, we evaluate the performance of the algorithm via experimental studies to show its actual cost-effectiveness and scalability.

Semi-Online Scheduling on Two Identical Parallel Machines with Initial-Lookahead Information

This work investigates a new semi-online scheduling problem with lookahead. We focus on job scheduling on two identical parallel machines, where deterministic online algorithms only know the information of [Formula: see text] initial jobs (i.e., the initial-lookahead information), while the following jobs still arrive one-by-one in an over-list fashion. We consider makespan minimization as the objective. The study aims at revealing the value of knowing [Formula: see text] initial jobs, which are used to improve the competitive performance of those online algorithms without such initial-lookahead information. We provide the following findings: (1) For the scenario where the [Formula: see text] initial jobs are all the largest jobs with length [Formula: see text], we prove that the classical LIST algorithm is optimal with competitive ratio [Formula: see text]; (2) For the scenario where the total length of these [Formula: see text] jobs is at least [Formula: see text], we show that any online algorithm has a competitive ratio at least 3/2, implying that the initial-lookahead knowledge is powerless since there exists a 3/2-competitive online algorithm without such information; (3) For the scenario where the total length of these [Formula: see text] jobs is at least [Formula: see text] ([Formula: see text]), we propose an online algorithm, named as LPT-LIST, with competitive ratio of [Formula: see text], implying that the initial-lookahead information indeed helps to improve the competitiveness of those online algorithms lacking such information.

Online Routing Over Parallel Networks: Deterministic Limits and Data-driven Enhancements

Over the past decade, GPS-enabled traffic applications such as Google Maps and Waze have become ubiquitous and have had a significant influence on billions of daily commuters’ travel patterns. A consequence of the online route suggestions of such applications, for example, via greedy routing, has often been an increase in traffic congestion since the induced travel patterns may be far from the system optimum. Spurred by the widespread impact of traffic applications on travel patterns, this work studies online traffic routing in the context of capacity-constrained parallel road networks and analyzes this problem from two perspectives. First, we perform a worst-case analysis to identify the limits of deterministic online routing. Although we find that deterministic online algorithms achieve finite, problem/instance-dependent competitive ratios in special cases, we show that for a general setting the competitive ratio is unbounded. This result motivates us to move beyond worst-case analysis. Here, we consider algorithms that exploit knowledge of past problem instances and show how to design data-driven algorithms whose performance can be quantified and formally generalized to unseen future instances. We then present numerical experiments based on an application case for the San Francisco Bay Area to evaluate the performance of the proposed data-driven algorithms compared with the greedy algorithm and two look-ahead heuristics with access to additional information on the values of time and arrival time parameters of users. Our results show that the developed data-driven algorithms outperform commonly used greedy online-routing algorithms. Furthermore, our work sheds light on the interplay between data availability and achievable solution quality. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Funding: This work was supported by National Science Foundation (NSF) Award 1830554 and by the German Research Foundation (DFG) under [Grant 449261765]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2023.1275 .