Online hypergraph coloring with rejection

Abstract In this paper we investigate the online hypergraph coloring problem with rejection, where the algorithm is allowed to reject a vertex instead of coloring it but each vertex has a penalty which has to be paid if it is not colored. The goal is to minimize the sum of the number of the used colors for the accepted vertices and the total penalty paid for the rejected ones. We study the online problem which means that the algorithm receives the vertices of the hypergraph in some order v1, . . . , vn and it must decide about vi by only looking at the subhypergraph Hi = (Vi, Ei) where Vi = {v1, . . . , vi} and Ei contains the edges of the hypergraph which are subsets of Vi. We consider two models: in the full edge model only the edges where each vertex is accepted must be well-colored, in the trace model the subsets of the edges formed by the accepted vertices must be well colored as well. We consider proper and conflict free colorings. We present in each cases optimal online algorithms in the sense that they achieve asymptotically the smallest possible competitive ratio.

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Online Graph Coloring Against a Randomized Adversary

International Journal of Foundations of Computer Science ◽

10.1142/s0129054118410058 ◽

2018 ◽

Vol 29 (04) ◽

pp. 551-569 ◽

Cited By ~ 1

Author(s):

Elisabet Burjons ◽

Juraj Hromkovič ◽

Rastislav Královič ◽

Richard Královič ◽

Xavier Muñoz ◽

...

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Online Algorithms ◽

Graph Coloring ◽

Competitive Ratio ◽

Upper And Lower Bounds ◽

Coloring Problem ◽

Single Instance ◽

Graph Coloring Problem

We consider an online model where an adversary constructs a set of [Formula: see text] instances [Formula: see text] instead of one single instance. The algorithm knows [Formula: see text] and the adversary will choose one instance from [Formula: see text] at random to present to the algorithm. We further focus on adversaries that construct sets of [Formula: see text]-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in [Formula: see text] and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.

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Best-Possible Online Algorithms for Single Machine Scheduling to Minimize the Maximum Weighted Completion Time

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595918500483 ◽

2018 ◽

Vol 35 (06) ◽

pp. 1850048

Author(s):

Xing Chai ◽

Lingfa Lu ◽

Wenhua Li ◽

Liqi Zhang

Keyword(s):

Online Algorithms ◽

Single Machine ◽

Competitive Ratio ◽

Completion Time ◽

Online Algorithm ◽

Optimal Schedule ◽

Machine Scheduling ◽

Single Machine Scheduling ◽

Asia Pacific ◽

Weighted Completion Time

In this paper, we consider the online single machine scheduling problem to minimize the maximum weighted completion time of the jobs. For the preemptive problem, we show that the LW (Largest Weight first) rule yields an optimal schedule. For the non-preemptive problem, Li [Li, W (2015). A best possible online algorithm for the parallel-machine scheduling to minimize the maximum weighted completion time. Asia-Pacific Journal of Operational Research, 32(4), 1550030 (10 pages)] presented a lower bound 2, and then provided an online algorithm with a competitive ratio of 3. In this paper, we present two online algorithms with the best-possible competitive ratio of [Formula: see text] for the non-preemptive problem.

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

RAIRO - Operations Research ◽

10.1051/ro/2018064 ◽

2019 ◽

Vol 53 (2) ◽

pp. 559-576 ◽

Cited By ~ 3

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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ONLINE ALGORITHMS FOR SCHEDULING WITH MACHINE ACTIVATION COST

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595907001231 ◽

2007 ◽

Vol 24 (02) ◽

pp. 263-277 ◽

Cited By ~ 2

Author(s):

YONG HE ◽

SHUGUANG HAN ◽

YIWEI JIANG

Keyword(s):

Lower Bound ◽

Online Algorithms ◽

Competitive Ratio ◽

Online Algorithm ◽

Machine Scheduling ◽

Parallel Machine Scheduling ◽

Parallel Machine ◽

Scheduling Problem ◽

Identical Machines ◽

Parallel Machine Scheduling Problem

In this paper, we consider a variant of the classical parallel machine scheduling problem. For this problem, we are given m potential identical machines to non-preemptively process a sequence of independent jobs. Machines need to be activated before starting to process, and each machine activated incurs a fixed machine activation cost. No machines are initially activated, and when a job is revealed the algorithm has the option to activate new machines. The objective is to minimize the sum of the makespan and activation cost of machines. We first present two optimal online algorithms with competitive ratios of 3/2 and 5/3 for m = 2, 3 cases, respectively. Then we present an online algorithm with a competitive ratio of at most 2 for general m ≥ 4, while the lower bound is 1.88.

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Quantum and Classical Log-Bounded Automata for the Online Disjointness Problem

Mathematics ◽

10.3390/math10010143 ◽

2022 ◽

Vol 10 (1) ◽

pp. 143

Author(s):

Kamil Khadiev ◽

Aliya Khadieva

Keyword(s):

Online Algorithms ◽

Minimization Problem ◽

Competitive Ratio ◽

Communication Complexity ◽

Point Of View ◽

Online Version ◽

Query Complexity

We consider online algorithms with respect to the competitive ratio. In this paper, we explore one-way automata as a model for online algorithms. We focus on quantum and classical online algorithms. For a specially constructed online minimization problem, we provide a quantum log-bounded automaton that is more effective (has less competitive ratio) than classical automata, even with advice, in the case of the logarithmic size of memory. We construct an online version of the well-known Disjointness problem as a problem. It was investigated by many researchers from a communication complexity and query complexity point of view.

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Online Algorithms for Scheduling Unit Length Jobs on Unbounded Parallel-Batch Machines with Linearly Lookahead

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595919500246 ◽

2019 ◽

Vol 36 (05) ◽

pp. 1950024

Author(s):

Chengwen Jiao ◽

Jinjiang Yuan ◽

Qi Feng

Keyword(s):

Online Algorithms ◽

Competitive Ratio ◽

Online Algorithm ◽

Positive Root ◽

Unit Length ◽

Online Scheduling ◽

Time Interval ◽

Steady Growth ◽

Scheduling Model

In this paper, we propose a new online scheduling model with linear lookahead intervals, which has the character that at any time [Formula: see text], one can foresee the jobs that will coming in the time interval [Formula: see text] in which [Formula: see text]. In this new lookahead model, the length of the lookahead intervals are variable as the time going on and the number of jobs increasing, and has the tend of steady growth. In this paper, we consider online scheduling of unit length jobs on [Formula: see text] identical parallel-batch machines under this new lookahead model to minimize makespan. The batch capacity is unbounded, that is [Formula: see text]. We present an optimal online algorithm for [Formula: see text], and provide a best possible online algorithm of competitive ratio [Formula: see text] for [Formula: see text], where [Formula: see text] is the positive root of [Formula: see text].

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Online Preemptive Hierarchical Scheduling on Two Uniform Machines with Rejection

Asia Pacific Journal of Operational Research ◽

10.1142/s021759591550027x ◽

2015 ◽

Vol 32 (04) ◽

pp. 1550027

Author(s):

Xiao Min ◽

Jing Liu ◽

Yanxia Dong ◽

Ming Jiang

Keyword(s):

Lower Bound ◽

Online Algorithms ◽

Upper Bound ◽

Competitive Ratio ◽

Scheduling Problem ◽

Hierarchical Scheduling ◽

Uniform Machines ◽

Level 1

This paper studies the online hierarchical scheduling problem on two uniform machines with rejection. Two uniform machines M1, M2 run at the speeds of s ∈ (0, +∞), 1 separately; and they are provided with different capabilities. Each machine has a certain GOS level 1 or 2 and every job is also associated with a hierarchy 1 or 2. The job can only be assigned to the machine whose GOS level does not exceed the job's hierarchy. Preemption is permitted but idle is not introduced. Jobs come one by one over list. When a job arrives, it can be accepted and scheduled on some machine or rejected by paying its penalty. The objective is to minimize the sum of makespan yielded by accepted jobs and total penalties of all rejected jobs. For this problem, we propose a family of several online algorithms according to the range of s and the related lower bound is also obtained. These algorithms achieve optimal competitive ratio when s ∈ (0, 1) ∪ [1.618, +∞), but have a small gap between upper bound and lower bound in interval [1, 1.618).

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Online Bi-Criteria Scheduling on Batch Machines with Machine Costs

Mathematics ◽

10.3390/math7100960 ◽

2019 ◽

Vol 7 (10) ◽

pp. 960 ◽

Cited By ~ 1

Author(s):

Wenhua Li ◽

Weina Zhai ◽

Xing Chai

Keyword(s):

Online Algorithms ◽

Competitive Ratio ◽

Processing Time ◽

Positive Root ◽

Batch Scheduling ◽

Cost Functions ◽

Fixed Cost ◽

Machine Cost ◽

The Cost ◽

Over Time

We consider online scheduling with bi-criteria on parallel batch machines, where the batch capacity is unbounded. In this paper, online means that jobs’ arrival is over time. The objective is to minimize the maximum machine cost subject to the makespan being at its minimum. In unbounded parallel batch scheduling, a machine can process several jobs simultaneously as a batch. The processing time of a job and a batch is equal to 1. When job J j is processed on machine M i , it results cost c i j . We only consider two types of cost functions: c i j = a + c j and c i j = a · c j , where a is the fixed cost of machines and c j is the cost of job J j . The number of jobs is n and the number of machines is m. For this problem, we provide two online algorithms, which are showed to be the best possible with a competitive ratio of ( 1 + β m , ⌈ n m ⌉ ) , where β m is the positive root of the equation ( 1 + β m ) m + 1 = β m + 2 .

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