Parameterized Complexity Analysis of Randomized Search Heuristics

Neumann, Franz‐Josef; Sutton, Andrew M.

doi:10.1007/978-3-030-29414-4_4

Cited by 7 publications

(4 citation statements)

References 50 publications

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“…Applying parameterized complexity analysis to the run time analysis of evolutionary algorithms is useful when one would like to gain direct insight into how problem instance structure influences run time on NP-hard problems [23,30]. For a randomized search heuristic, the optimization time is characterized as a random variable T that measures the number of fitness function evaluations until an optimal solution is first visited.…”

Section: Fixed-parameter Tractable Easmentioning

confidence: 99%

Focused Jump-and-Repair Constraint Handling for Fixed-Parameter Tractable Graph Problems Closed Under Induced Subgraphs

Branson¹,

Sutton²

2022

Preprint

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Repair operators are often used for constraint handling in constrained combinatorial optimization. We investigate the (1+1) EA equipped with a tailored jump-and-repair operation that can be used to probabilistically repair infeasible offspring in graph problems. Instead of evolving candidate solutions to the entire graph, we expand the genotype to allow the (1+1) EA to develop in parallel a feasible solution together with a growing subset of the instance (an induced subgraph). With this approach, we prove that the EA is able to probabilistically simulate an iterative compression process used in classical fixed-parameter algorithmics to obtain a randomized FPT performance guarantee on three NP-hard graph problems. For k-VertexCover, we prove that the (1+1) EA using focused jump-and-repair can find a k-cover (if one exists) in O(2 k n 2 log n) iterations in expectation. This leads to an exponential (in k) improvement over the best-known parameterized bound for evolutionary algorithms on VertexCover. For the k-FeedbackVertexSet problem in tournaments, we prove that the EA finds a feasible feedback set in O(2 k k!n 2 log n) iterations in expectation, and for OddCycleTransversal, we prove the optimization time for the EA is O(3 k kmn 2 log n). For the latter two problems, this constitutes the first parameterized result for any evolutionary algorithm. We discuss how to generalize the framework to other parameterized graph problems closed under induced subgraphs and report experimental results that illustrate the behavior of the algorithm on a concrete instance class.

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Section: Fixed-parameter Tractable Easmentioning

confidence: 99%

Focused Jump-and-Repair Constraint Handling for Fixed-Parameter Tractable Graph Problems Closed Under Induced Subgraphs

Branson¹,

Sutton²

2022

Preprint

Self Cite

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“…Evolutionary algorithms that approximate the optimum are also known in the subfield of fixed-parameter tractability. While most of these results prove an approximation within a constant factor or growing slowly with the problem dimension, there are also statements similar to approximation schemes for the vertex cover problem (Neumann and Sutton, 2020). However, in general it is safe to say that there are only few results in the literature that characterize very simple randomized search heuristics like the (1 + 1) EA and SA as polynomial-time approximation schemes for classical (non-noisy) combinatorial optimization problems.…”

Section: Previous Workmentioning

confidence: 99%

Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

Doerr,

Rajabi,

Witt

2022

Preprint

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We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (2005). More precisely, denoting by n, m, w max , and w min the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T 0 ≥ w max and multiplicative cooling schedule with factor 1 − 1/ℓ, where ℓ = ω(mn ln(m)), with probability at least 1 − 1/m computes in time O(ℓ(ln ln(ℓ) + ln(T 0 /w min ))) a spanning tree with weight at most 1 + κ times the optimum weight, where 1 + κ = (1+o(1)) ln(ℓm) ln(ℓ)−ln(mn ln(m)) . Consequently, for any ε > 0, we can choose ℓ in such a way that a (1+ε)-approximation is found in time O((mn ln(n)) 1+1/ε+o(1) (ln ln n+ln(T 0 /w min ))) with probability at least 1 − 1/m. In the special case of so-called (1 + ε)separated weights, this algorithm computes an optimal solution (again in time O((mn ln(n)) 1+1/ε+o(1) (ln ln n + ln(T 0 /w min )))), which is a significant speed-up over Wegener's runtime guarantee of O(m 8+8/ε ).

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“…Evolutionary algorithms that approximate the optimum are also known in the subfield of fixed-parameter tractability. While most of these results prove an approximation within a constant factor or growing slowly with the problem dimension, there are also statements similar to approximation schemes for the vertex cover problem [16]. However, in general it is safe to say that there are only few results in the literature that characterize very simple randomized search heuristics like the (1 + 1) EA and SA as polynomial-time approximation schemes for classical (non-noisy) combinatorial optimization problems.…”

Section: Previous Workmentioning

confidence: 99%

Simulated annealing is a polynomial-time approximation scheme for the minimum spanning tree problem

Doerr

Rajabi

Witt

2022

Proceedings of the Genetic and Evolutionary Computation Conference

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We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (2005). More precisely, denoting by 𝑛, 𝑚, 𝑤 max , and 𝑤 min the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature 𝑇 0 ≥ 𝑤 max and multiplicative cooling schedule with factor 1 − 1/ℓ, where ℓ = 𝜔 (𝑚𝑛 ln(𝑚)), with probability at least 1 − 1/𝑚 computes in time 𝑂 (ℓ (ln ln(ℓ) + ln(𝑇 0 /𝑤 min ))) a spanning tree with weight at most 1 + 𝜅 times the optimum weight, where 1 + 𝜅 = (1+𝑜 (1)) ln(ℓ𝑚) ln(ℓ)−ln(𝑚𝑛 ln(𝑚)) . Consequently, for any 𝜀 > 0, we can choose ℓ in such a way that a (1 + 𝜀)-approximation is found in time 𝑂 ((𝑚𝑛 ln(𝑛)) 1+1/𝜀+𝑜 (1) (ln ln 𝑛 + ln(𝑇 0 /𝑤 min ))) with probability at least 1 − 1/𝑚. In the special case of so-called (1 + 𝜀)-separated weights, this algorithm computes an optimal solution (again in time 𝑂 ((𝑚𝑛 ln(𝑛)) 1+1/𝜀+𝑜 (1) (ln ln 𝑛 + ln(𝑇 0 /𝑤 min )))), which is a significant speed-up over Wegener's runtime guarantee of 𝑂 (𝑚 8+8/𝜀 ). CCS CONCEPTS• Theory of computation → Theory of randomized search heuristics.

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Parameterized Complexity Analysis of Randomized Search Heuristics

Cited by 7 publications

References 50 publications

Focused Jump-and-Repair Constraint Handling for Fixed-Parameter Tractable Graph Problems Closed Under Induced Subgraphs

Focused Jump-and-Repair Constraint Handling for Fixed-Parameter Tractable Graph Problems Closed Under Induced Subgraphs

Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

Simulated annealing is a polynomial-time approximation scheme for the minimum spanning tree problem

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