Analysis Of Online Algorithms Research Articles

Correction to: Exact distributional analysis of online algorithms with lookahead

Correction to: Exact distributional analysis of online algorithms with lookahead

Exact distributional analysis of online algorithms with lookahead

In online optimization, input data is revealed sequentially. Optimization problems in practice often exhibit this type of information disclosure as opposed to standard offline optimization where all information is known in advance. We analyze the performance of algorithms for online optimization with lookahead using a holistic distributional approach. To this end, we first introduce the performance measurement method of counting distribution functions. Then, we derive analytical expressions for the counting distribution functions of the objective value and the performance ratio in elementary cases of the online bin packing and the online traveling salesman problem. For bin packing, we also establish a relation between algorithm processing and the Catalan numbers. The paper shows that an exact analysis is strongly interconnected to the combinatorial structure of the problem and algorithm under consideration. Results further indicate that the value of lookahead heavily relies on the problem itself. The analysis also shows that exact distributional analysis could be used in order to discover key effects and identify related root causes in relatively simple problem settings. These insights can then be transferred to the analysis of more complex settings where the introduced performance measurement approach has to be used on an approximative basis (e.g., in a simulation-based optimization).

Distribution-free methods for multi-period, single-leg booking control

We study the multi-period, single-leg, fare class allocation problem in revenue management. Uncertainty in fare class demand is not characterized using probability distributions but by using lower and upper bounds on demand. Our multi-period model allows demand characteristics and available information to vary from period to period. Building on our previous work, static and dynamic models are developed to determine effective booking control policies based on competitive analysis of online algorithms. The underlying performance criterion used is a measure of robustness, which provides the best performance guarantee over all possible input sequences consistent with the data provided. The dynamic policies are nested by fare class in each period but the booking limits can be revised from period to period so that a fare class that is closed can be re-opened (if needed). Computational experiments compare the new policies against previously developed robust policies and also more traditional approaches. The experiments show that these new policies are effective, robust, and can provide significant gains over the existing policies.

A generalized online mirror descent with applications to classification and regression

Online learning algorithms are fast, memory-efficient, easy to implement, and applicable to many prediction problems, including classification, regression, and ranking. Several online algorithms were proposed in the past few decades, some based on additive updates, like the Perceptron, and some on multiplicative updates, like Winnow. A unifying perspective on the design and the analysis of online algorithms is provided by online mirror descent, a general prediction strategy from which most first-order algorithms can be obtained as special cases. We generalize online mirror descent to time-varying regularizers with generic updates. Unlike standard mirror descent, our more general formulation also captures second order algorithms, algorithms for composite losses and algorithms for adaptive filtering. Moreover, we recover, and sometimes improve, known regret bounds as special cases of our analysis using specific regularizers. Finally, we show the power of our approach by deriving a new second order algorithm with a regret bound invariant with respect to arbitrary rescalings of individual features.

A nonmonotone analysis with the primal–dual approach: Online routing of virtual circuits with unknown durations

We address the question of whether the primal–dual approach for the design and analysis of online algorithms can be applied to nonmonotone problems. We provide a positive answer by presenting a primal–dual analysis to the online algorithm of Awerbuch et al. [3] for routing virtual circuits with unknown durations.

Asymptotic analysis of online algorithms and improved scheme for the flow shop scheduling problem with release dates

This paper discusses the flow shop scheduling problem to minimise the total quadratic completion time (TQCT) with release dates in offline and online environments. For this NP-hard problem, the investigation is focused on the performance of two online algorithms based on the Shortest Processing Time among Available jobs rule. Theoretical results indicate the asymptotic optimality of the algorithms as the problem scale is sufficiently large. To further enhance the quality of the original solutions, the improvement scheme is provided for these algorithms. A new lower bound with performance guarantee is provided, and computational experiments show the effectiveness of these heuristics. Moreover, several results of the single-machine TQCT problem with release dates are also obtained for the deduction of the main theorem.

Online algorithms with stochastic input

The design and analysis of online algorithms, where the input to the algorithm is revealed over time and the algorithm has to make decisions immediately without knowing the future input, has received a revived interest in the last few years primarily due to their application to online advertising. The canonical problem is the Adwords problem, which is motivated by the problem of optimally allocating ad slots on search queries to budget constrained advertisers. It involves simplifications that ignore certain aspects of the actual way this allocation is done. For instance, it assumes a "first-price" pay-per-impression scheme, ignoring the game theoretic aspects, and considers only one slot per query. To be precise, the Adwords problem is as follows.

Reasoning about online algorithms with weighted automata

We describe an automata-theoretic approach for the competitive analysis of online algorithms . Our approach is based on weighted automata , which assign to each input word a cost in R ≥0 . By relating the “unbounded look ahead” of optimal offline algorithms with nondeterminism, and relating the “no look ahead” of online algorithms with determinism, we are able to solve problems about the competitive ratio of online algorithms, and the memory they require, by reducing them to questions about determinization and approximated determinization of weighted automata.

The Lazy Adversary Conjecture Fails

In the context of competitive analysis of online algorithms for the k-server problem, it has been conjectured that every randomized, memoryless online algorithm exhibits the highest competitive ratio against an offline adversary that is lazy, i.e., that will issue requests forcing it to move one of its own servers only when this is strictly necessary to force a move on the part of the online algorithm. We prove that, in general, this lazy adversary conjecture fails. Moreover, it fails in a very strong sense: there are adversaries which perform arbitrarily better than any other adversary which is even slightly “lazier.”