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

Algorithmica ◽  
2021 ◽  
Author(s):  
Susanne Albers ◽  
Arindam Khan ◽  
Leon Ladewig
Keyword(s):  
Knapsack Problem ◽  
Competitive Ratio ◽  
Randomized Algorithm ◽  
Input Sequence ◽  
Random Order ◽  
Random Permutation ◽  
Secretary Problem ◽  
Order Model ◽  
Sequential Approach ◽  
Online Setting

AbstractThe 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. in SIAM J Comput 47(5):1939–1964, 2018). 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. in SIAM J Comput 47(5):1939–1964, 2018).

scholarly journals Scheduling in the Random-Order Model

Algorithmica ◽  
2021 ◽  
Author(s):  
Susanne Albers ◽  
Maximilian Janke
Keyword(s):  
Competitive Ratio ◽  
High Probability ◽  
Online Algorithm ◽  
Fundamental Problem ◽  
Random Order ◽  
Random Permutation ◽  
Worst Case Analysis ◽  
Makespan Minimization ◽  
Order Model ◽  
Worst Case

AbstractMakespan 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 $$(2-1/m)$$ ( 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.

scholarly journals Online Makespan Scheduling with Job Migration on Uniform Machines

Algorithmica ◽  
2021 ◽  
Author(s):  
Matthias Englert ◽  
David Mezlaf ◽  
Matthias Westermann
Keyword(s):  
Competitive Ratio ◽  
Online Algorithm ◽  
Parallel Machines ◽  
Scheduling Algorithm ◽  
Input Sequence ◽  
Job Migration ◽  
Average Completion Time ◽  
Uniform Machine ◽  
Uniform Machines ◽  
Completion Times

AbstractIn the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

2021 ◽  
Author(s):  
José Correa ◽  
Paul Dütting ◽  
Felix Fischer ◽  
Kevin Schewior
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Optimal Solution ◽  
Random Order ◽  
Secretary Problem ◽  
Maximum Value ◽  
Optimal Stopping Theory ◽  
Long Time ◽  
Object Of Study ◽  
Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

Online peak-aware energy scheduling with untrusted advice

2021 ◽  
Vol 1 (1) ◽  
pp. 59-77
Author(s):  
Russell Lee ◽  
Jessica Maghakian ◽  
Mohammad Hajiesmaili ◽  
Jian Li ◽  
Ramesh Sitaraman ◽  
...  
Keyword(s):  
Competitive Ratio ◽  
Energy Demand ◽  
Large Scale ◽  
Performance Metrics ◽  
Randomized Algorithm ◽  
Deterministic Algorithm ◽  
Pareto Optimal ◽  
Energy Prices ◽  
Worst Case ◽  
Cost Of Energy

This paper studies the online energy scheduling problem in a hybrid model where the cost of energy is proportional to both the volume and peak usage, and where energy can be either locally generated or drawn from the grid. Inspired by recent advances in online algorithms with Machine Learned (ML) advice, we develop parameterized deterministic and randomized algorithms for this problem such that the level of reliance on the advice can be adjusted by a trust parameter. We then analyze the performance of the proposed algorithms using two performance metrics: robustness that measures the competitive ratio as a function of the trust parameter when the advice is inaccurate, and consistency for competitive ratio when the advice is accurate. Since the competitive ratio is analyzed in two different regimes, we further investigate the Pareto optimality of the proposed algorithms. Our results show that the proposed deterministic algorithm is Pareto-optimal, in the sense that no other online deterministic algorithms can dominate the robustness and consistency of our algorithm. Furthermore, we show that the proposed randomized algorithm dominates the Pareto-optimal deterministic algorithm. Our large-scale empirical evaluations using real traces of energy demand, energy prices, and renewable energy generations highlight that the proposed algorithms outperform worst-case optimized algorithms and fully data-driven algorithms.

Progressive Multi-Objective Optimization

2014 ◽  
Vol 13 (05) ◽  
pp. 917-936 ◽  
Author(s):  
Kenneth Sörensen ◽  
Johan Springael
Keyword(s):  
Decision Making ◽  
Knapsack Problem ◽  
Multi Criteria Decision Making ◽  
Multi Objective Optimization ◽  
Sequential Approach ◽  
Nondominated Solutions ◽  
Multi Objective ◽  
Novel Technique ◽  
Single Solution ◽  
Set Approximation

This paper introduces progressive multi-objective optimization (PMOO), a novel technique to include the decision maker's preferences into the multi-objective optimization process. PMOO integrates a well-known method for multi-criteria decision making (PROMETHEE) into a simple multi-objective metaheuristic by maintaining and updating a small reference archive of nondominated solutions throughout the search. By applying this novel technique to a set of instances of the multi-objective knapsack problem, the superiority of PMOO over the commonly accepted sequential approach of generating a Pareto set approximation first and selecting a single solution afterwards is demonstrated.

Optimal selection based on relative ranks with a random number of individuals

1979 ◽  
Vol 11 (4) ◽  
pp. 720-736 ◽  
Author(s):  
Jacqueline Gianini-Pettitt
Keyword(s):  
Selection Procedure ◽  
Random Order ◽  
Random Variable ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Asymptotic Results ◽  
Fixed Integer ◽  
Bounded Random Variable ◽  
The Individual ◽  
Number Of Individuals

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

scholarly journals Online Roommate Allocation Problem

2017 ◽  
Author(s):  
Guangda Huzhang ◽  
Xin Huang ◽  
Shengyu Zhang ◽  
Xiaohui Bei
Keyword(s):  
Social Welfare ◽  
Competitive Ratio ◽  
Allocation Problem ◽  
Market Model ◽  
Stability Conditions ◽  
Welfare Maximization ◽  
Negative Results ◽  
Social Welfare Maximization ◽  
The Social ◽  
Online Setting

We study the online allocation problem under a roommate market model introduced in [Chan et al., 2016]. Consider a fixed supply of n rooms and a list of 2n applicants arriving sequentially in an online fashion. The problem is to assign a room to each person upon her arrival, such that after the algorithm terminates, each room is shared by exactly two people. We focus on two objectives: (1) maximizing the social welfare, which is defined as the sum of valuations that applicants have for their rooms, plus the happiness value between each pair of roommates; (2) the allocation should satisfy certain stability conditions, such that no group of people would be willing to switch roommates or rooms. We first show a polynomial-time online algorithm that achieves constant competitive ratio for social welfare maximization. We then extend it to the case where each room is assigned to c > 2 people, and achieve a competitive ratio of Ω(1/c^2). Finally, we show both positive and negative results in satisfying different stability conditions in this online setting.

scholarly journals Optimal online algorithms for the portfolio selection problem, bi-directional trading and -search with interrelated prices

2019 ◽  
Vol 53 (2) ◽  
pp. 559-576 ◽  
Author(s):  
Pascal Schroeder ◽  
Imed Kacem ◽  
Günter Schmidt
Keyword(s):  
Portfolio Selection ◽  
Online Algorithms ◽  
Competitive Ratio ◽  
Numerical Experiments ◽  
Input Sequence ◽  
Relative Price ◽  
Selection Problem ◽  
Online Search ◽  
Worst Case ◽  
Portfolio Selection Problem

In this work we investigate the portfolio selection problem (P1) and bi-directional trading (P2) when prices are interrelated. Zhang et al. (J. Comb. Optim. 23 (2012) 159–166) provided the algorithm UND which solves one variant of P2. We are interested in solutions which are optimal from a worst-case perspective. For P1, we prove the worst-case input sequence and derive the algorithm optimal portfolio for interrelated prices (OPIP). We then prove the competitive ratio and optimality. We use the idea of OPIP to solve P2 and derive the algorithm called optimal conversion for interrelated prices (OCIP). Using OCIP, we also design optimal online algorithms for bi-directional search (P3) called bi-directional UND (BUND) and optimal online search for unknown relative price bounds (RUN). We run numerical experiments and conclude that OPIP and OCIP perform well compared to other algorithms even if prices do not behave adverse.

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