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A variant of the online knapsack problem is considered in the setting of predictions. In Unit Profit Knapsack, the items have unit profit, i.e., the goal is to pack as many items as possible. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. The prediction available to the online algorithm is the average size of those smallest items that fit in the knapsack. For the prediction error in this hard online problem, we use the ratio r = a a ^ where a is the actual value for this average size and a ^ is the prediction. We give an algorithm which is e - 1 e -competitive, if r = 1 , and this is best possible among online algorithms knowing a and nothing else. More generally, the algorithm has a competitive ratio of e - 1 e r , if r ≤ 1 , and e - r e r , if 1 ≤ r < e . Any algorithm with a better competitive ratio for some r < 1 will have a worse competitive ratio for some r > 1 . To obtain a positive competitive ratio for all r, we adjust the algorithm, resulting in a competitive ratio of 1 2 r for r ≥ 1 and r 2 for r ≤ 1 . We show that improving the result for any r < 1 leads to a worse result for some r > 1 . [ABSTRACT FROM AUTHOR]

Online graph problems are considered in models where the irrevocability requirement is relaxed. We consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. We consider four classical graph problems: For Maximum Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, for Minimum Vertex Cover the Late Accept model is sufficient, and for Minimum Spanning Forest the Late Reject model is sufficient. The Maximum Matching problem admits constant competitive ratios in all cases. We also consider Maximum Acyclic Subgraph and Maximum Planar Subgraph, which exhibit patterns similar to Maximum Independent Set. [ABSTRACT FROM AUTHOR]

The standard measure for the quality of online algorithms is the competitive ratio. This measure is generally applicable, and for some problems it works well, but for others it fails to distinguish between algorithms that have very different performance. Thus, ever since its introduction, researchers have worked on improving the measure, defining variants, or defining measures based on other concepts to improve on the situation. Relative worst-order analysis (RWOA) is one of the most thoroughly tested such proposals. With RWOA, many separations of algorithms not obtainable with competitive analysis have been found. In RWOA, two algorithms are compared directly, rather than indirectly as is done in competitive analysis, where both algorithms are compared separately to an optimal offline algorithm. If, up to permutations of the request sequences, one algorithm is always at least as good and sometimes better than another, then the first algorithm is deemed the better algorithm by RWOA. We survey the most important results obtained with this technique and compare it with other quality measures. The survey includes a quite complete set of references. [ABSTRACT FROM AUTHOR]

The bin covering problem asks for covering a maximum number of bins with an online sequence of n items of different sizes in the range (0, 1]; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide asymptotically tight bounds of Θ (n log O P T) on the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size o (log log n) has a competitive ratio of at most 0.5. In other words, advice of size o (log log n) is useless for improving the competitive ratio of 0.5, attainable by an online algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size O (log log n) is sufficient to achieve an asymptotic competitive ratio of 0.5 3 ¯ which is strictly better than the best ratio 0.5 attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of advice bits is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. We consider important graph classes, distinguishing between connected and not necessarily connected graphs. For the classic graph classes of trees, bipartite, planar, and general graphs, we obtain tight results in almost all cases. We also derive upper and lower bounds for the class of bounded-degree graphs. From these analyses, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm's disadvantage. [ABSTRACT FROM AUTHOR]

We study two versions of graph coloring, where the goal is to assign a positive integer color to each vertex of an input graph such that adjacent vertices do not receive the same color assignment. For classic vertex coloring, the goal is to minimize the maximum color used, and for the sum coloring problem, the goal is to minimize the sum of colors assigned to all input vertices. In the offline variant, the entire graph is presented at once, and in online problems, one vertex is presented for coloring at each time, and the only information is the identity of its neighbors among previously known vertices. In batched graph coloring, vertices are presented in k batches, for a fixed integer k≥2, such that the vertices of a batch are presented as a set, and must be colored before the vertices of the next batch are presented. This last model is an intermediate model, which bridges between the two extreme scenarios of the online and offline models. We provide several results, including a general result for sum coloring and results for the classic graph coloring problem on restricted graph classes: We show tight bounds for any graph class containing trees as a subclass (forests, bipartite graphs, planar graphs, and perfect graphs, for example), and an interesting result for interval graphs and k=2, where the value of the (strict and asymptotic) competitive ratio depends on whether the graph is presented with its interval representation or not. [ABSTRACT FROM AUTHOR]

Recently, the first online complexity class, AOC, was introduced. The class consists of many online problems where each request must be either accepted or rejected, and the aim is to either minimize or maximize the number of accepted requests, while maintaining a feasible solution. All AOC-complete problems (including Independent Set, Vertex Cover, Dominating Set, and Set Cover) have essentially the same advice complexity. In this paper, we study weighted versions of problems in AOC, i.e., each request comes with a weight and the aim is to either minimize or maximize the total weight of the accepted requests. In contrast to the unweighted versions, we show that there is a significant difference in the advice complexity of complete minimization and maximization problems. We also show that our algorithmic techniques for dealing with weighted requests can be extended to work for non-complete AOC problems such as Matching in the edge arrival model (giving better results than what follow from the general AOC results) and even non- AOC problems such as scheduling. [ABSTRACT FROM AUTHOR]

Though competitive analysis is often a very good tool for the analysis of online algorithms, sometimes it does not give any insight and sometimes it gives counter-intuitive results. Much work has gone into exploring other performance measures, in particular targeted at what seems to be the core problem with competitive analysis: The comparison of the performance of an online algorithm is made with respect to a too powerful adversary. We consider a new approach to restricting the power of the adversary, by requiring that when judging a given online algorithm, the optimal offline algorithm must perform at least as well as the online algorithm, not just on the entire final request sequence, but also on any prefix of that sequence. This is limiting the adversary’s usual advantage of being able to exploit that it knows the sequence is continuing beyond the current request. Through a collection of online problems, including machine scheduling, bin packing, dual bin packing, and seat reservation, we investigate the significance of this particular offline advantage. [ABSTRACT FROM AUTHOR]

In online scenarios requests arrive over time, and each request must be serviced in an irrevocable manner before the next request arrives. Online algorithms with advice is an area of research where one attempts to measure how much knowledge of future requests is necessary to achieve a given performance level, as defined by the competitive ratio. When this knowledge, the advice, is obtainable, this leads to practical algorithms, called semi-online algorithms. On the other hand, each negative result gives robust results about the limitations of a broad range of semi-online algorithms. This survey explains the models for online algorithms with advice, motivates the study in general, presents some examples of the work that has been carried out, and includes an extensive set of references, organized by problem studied. [ABSTRACT FROM AUTHOR]

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. For hexagonal graphs, negative results trivially carry over to 3-colorable graphs, and most of our positive results do as well. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time. [ABSTRACT FROM AUTHOR]

Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin. For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all “reasonable” algorithms have a competitive ratio of 1 2 . Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restricted or uniformly distributed item sizes. In this way, our study also supplements the ongoing systematic studies of the relative strengths of various performance measures. Two classic algorithms for online bin packing that have natural dual versions are Harmonic k and Next-Fit . Even though the algorithms are quite different in nature, the dual versions are not separated by competitive analysis. We make the case that when guarantees are needed, even under restricted input sequences, dual Harmonic k is preferable. In addition, we establish quite robust theoretical results showing that if items come from a uniform distribution or even if just the ordering of items is uniformly random, then dual Next-Fit is the right choice. [ABSTRACT FROM AUTHOR]

Abstract: We study the performance of the algorithms and for two online edge coloring problems. In the min-coloring problem, all edges must be colored using as few colors as possible. In the max-coloring problem, a fixed number of colors is given, and as many edges as possible should be colored. Previous analysis using the competitive ratio has not separated the performance of and , but intuition suggests that should be better than . We compare and using the relative worst-order ratio, and show that is better than for the min-coloring problem. For the max-coloring problem, we show that and are not strictly comparable, i.e., there are graphs for which is significantly better than and graphs where is slightly better than . [Copyright &y& Elsevier]

Abstract: The relative worst-order ratio, a relatively new measure for the quality of on-line algorithms, is extended and applied to the paging problem. We obtain results significantly different from those obtained with the competitive ratio. First, we devise a new deterministic paging algorithm, Retrospective-LRU, and show that, according to the relative worst-order ratio and in contrast with the competitive ratio, it performs better than LRU. Our experimental results, though not conclusive, are slightly positive and leave it possible that Retrospective-LRU or similar algorithms may be worth considering in practice. Furthermore, the relative worst-order ratio (and practice) indicates that LRU is better than the marking algorithm FWF, though all deterministic marking algorithms have the same competitive ratio. Look-ahead is also shown to be a significant advantage with this new measure, whereas the competitive ratio does not reflect that look-ahead can be helpful. Finally, with the relative worst-order ratio, as with the competitive ratio, no deterministic marking algorithm can be significantly better than LRU, but the randomized algorithm MARK is better than LRU. [Copyright &y& Elsevier]

We define a new measure for the quality of online algorithms, the relative worst order ratio, using ideas from the max/max ratio [Ben-David and Borodin 1994] and from the random order ratio [Kenyon 1996]. The new ratio is used to compare online algorithms directly by taking the ratio of their performances on their respective worst permutations of a worst-case sequence. Two variants of the bin packing problem are considered: the classical bin packing problem, where the goal is to fit all items in as few bins as possible, and the dual bin packing problem, which is the problem of maximizing the number of items packed in a fixed number of bins. Several known algorithms are compared using this new measure, and a new, simple variant of first-fit is proposed for dual bin packing. Many of our results are consistent with those previously obtained with the competitive ratio or the competitive ratio on accommodating sequences, but new separations and easier proofs are found. [ABSTRACT FROM AUTHOR]

The applicability of the accommodating function, a relatively new measure for the quality of on-line algorithms, is extended. The standard quality measure for on-line algorithms is the competitive ratio, which is, roughly speaking, the worst case ratio of the on-line performance to the optimal off-line performance. However, for many on-line problems, the competitive ratio gives overly pessimistic results and/or fails to distinguish between algorithms that are known to perform very differently in practice. Many researchers have proposed variations on the competitive ratio to obtain more realistic results. These variations are often tailor-made for specific on-line problems. The concept of the accommodating function applies to any on-line problem with some limited resource, such as bins, seats in a train, or pages in a cache. If a limited amount n of some resource is available, the accommodating function A(α) is the competitive ratio when input sequences are restricted to those for which the amount αn of resources suffices for an optimal off-line algorithm. For all resource bounded problems, the standard competitive ratio is lim[subα→∞A(α). The accommodating function was originally used only for We focus on α < 1, observe that the function now appears interesting for a greater variety of problems, and use it to make new distinctions between known algorithms and to find new ones. [ABSTRACT FROM AUTHOR]