Hitting-time and occupation-time bounds implied by drift analysis with applications

Hajek, Bruce

doi:10.2307/1426671

Cited by 368 publications

(259 citation statements)

References 19 publications

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“…Hajek introduced drift analysis to provide a flexible technique for proving the stability of processes frequently encountered in queuing systems [5]. This technique also turned out to be useful in the analysis of the computational complexity of simulated annealing for the maximum matching problem [13].…”

Section: Drift Analysismentioning

confidence: 99%

“…Hajek's previous work [5], He and Yao derived two conditions for proving exponential lower bounds on the expected runtime of an evolutionary algorithm [7]. Later on, Giel and Wegener noticed that from He and Yao's proof, a statement on the success probability could be obtained rather than just a bound on the expected runtime [4].…”

Section: Theorem 1 (Drift Theorem For Upper Bounds)mentioning

confidence: 99%

“…Interestingly, in essence, there is only a single inequality, namely the bound on the moment-generating function of the one-step drift, to be checked for the final statement of the theorem to hold. By carefully looking at the original presentation by Hajek [5], it follows that the remaining conditions can be either rephrased or removed. In particular, there is no need for the following values λ( ) and D( ) to be constant or p( ) to be polynomial.…”

Section: The Simplified Drift Theoremmentioning

confidence: 99%

“…Moreover, we set g := id. The following argumentation is also inspired by Hajek's work [5]. Fix t ≥ 0 and some i such that a < i < b.…”

Section: The Simplified Drift Theoremmentioning

confidence: 99%

“…the introduction in [5]), was introduced for the analysis of EAs by He and Yao [7,8]. The authors concentrated on the obtainment of both lower and upper bounds on the expected runtime of EAs.…”

mentioning

confidence: 99%

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Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

2010

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Drift analysis is a powerful tool used to bound the optimization time of evolutionary algorithms (EAs). Various previous works apply a drift theorem going back to Hajek in order to show exponential lower bounds on the optimization time of EAs. However, this drift theorem is tedious to read and to apply since it requires two bounds on the moment-generating (exponential) function of the drift. A recent work identifies a specialization of this drift theorem that is much easier to apply. Nevertheless, it is not as simple and not as general as possible. The present paper picks up Hajek's line of thought to prove a drift theorem that is very easy to use in evolutionary computation. Only two conditions have to be verified, one of which holds for virtually all EAs with standard mutation. The other condition is a bound on what is really relevant, the drift. Applications show how previous analyses involving the complicated theorem can be redone in a much simpler and clearer way. In some cases even improved results may be achieved. Therefore, the simplified theorem is also a didactical contribution to the runtime analysis of EAs.

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Section: Drift Analysismentioning

confidence: 99%

Section: Theorem 1 (Drift Theorem For Upper Bounds)mentioning

confidence: 99%

Section: The Simplified Drift Theoremmentioning

confidence: 99%

“…Moreover, we set g := id. The following argumentation is also inspired by Hajek's work [5]. Fix t ≥ 0 and some i such that a < i < b.…”

Section: The Simplified Drift Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

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Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

2010

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Coupling vs. conductance for the Jerrum-Sinclair chain

Kumar

Ramesh

2000

Random Struct. Alg.

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We address the following question: is the Causal Coupling method as strong as the Conductance method in showing rapid mixing of Markov Chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum-Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments.

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Phase transitions of the Moran process and algorithmic consequences

Goldberg

Lapinskas

Richerby

2019

Random Struct Algorithms

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The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ϵ>0, we show that the expected absorption time on an n‐vertex graph is o(n3+ϵ). Specifically, it is at most n3eOfalse(false(loglognfalse)3false), and there is a family of graphs where it is Ω(n3). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.

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Hitting-time and occupation-time bounds implied by drift analysis with applications

Cited by 368 publications

References 19 publications

Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

Coupling vs. conductance for the Jerrum-Sinclair chain

Phase transitions of the Moran process and algorithmic consequences

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