Runtime Analysis of Evolutionary Algorithms for the Knapsack Problem with Favorably Correlated Weights

Neumann, Franz‐Josef; Sutton, Andrew M.

doi:10.1007/978-3-319-99259-4_12

Cited by 8 publications

(7 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…He et al [23] proved that a multi-objective EA with helper objectives is a 1/2-approximation algorithm for the knapsack problem. Recently, Neumann and Sutton [24] analysed the running time of a variant of Global Simple Evolutionary Multiobjective Optimizer on the knapsack problem.…”

Section: Literature Reviewmentioning

confidence: 99%

Helper and Equivalent Objectives: Efficient Approach for Constrained Optimization

Shang

2022

IEEE Trans. Cybern.

View full text Add to dashboard Cite

Numerous multi-objective evolutionary algorithms have been designed for constrained optimisation in last two decades. This method is to transform a constrained optimisation problem into a multi-objective optimisation problem, then solve it by an evolutionary algorithm. In this paper, we propose a new multi-objective method for constrained optimisation, which is to convert a constrained optimisation problem into a problem with helper and equivalent objectives. An equivalent objective means that its optimal solution set is the same as that to the constrained problem but a helper objective not. Then this multiobjective optimisation problem is decomposed into a group of sub-problems using the weighted sum approach. Weights are dynamically adjusted so that each subproblem eventually tends to a problem with an equivalent objective. We theoretically analyse the computation time of the helper and equivalent objective method on a hard problem called "wide gap". The "wide gap" problem means that an algorithm needs exponential time to cross between two fitness levels. We prove that using helper and equivalent objectives may shorten the time of crossing the "wide gap". We conduct a case study for validating our method. An algorithm with helper and equivalent objectives is implemented. Experimental results show that its overall performance is ranked first when compared with other eight state-of-art evolutionary algorithms. The case study proves the efficiency of the helper and equivalent objective method for constrained optimisation.

show abstract

Section: Literature Reviewmentioning

confidence: 99%

Helper and Equivalent Objectives: Efficient Approach for Constrained Optimization

Shang

2022

IEEE Trans. Cybern.

View full text Add to dashboard Cite

show abstract

“…Recent work shows that EAs provide optimal solutions in expected FPT (fixed parameter tractable) time for the NP-hard generalised minimum spanning tree problem [29], the Euclidean and the generalised traveling salesperson problems [30,29] and approximation results for various NP-hard problems on scale-free networks [31]. EAs have been shown to be efficient also on problems in P such as the all-pairs shortest path problem [32] and some easy instance classes of the k-CNF [33] and the knapsack [34] problems. Although EAs are efficient general purpose solvers, super-polynomial lower bounds on a selection of easy problems have also recently been made available [35].…”

Section: Artificial Immune Systems and Evolutionary Algorithmsmentioning

confidence: 99%

Artificial immune systems can find arbitrarily good approximations for the NP-hard number partitioning problem

Corus

Oliveto

Yazdani

2019

Artificial Intelligence

View full text Add to dashboard Cite

Typical artificial immune system (AIS) operators such as hypermutations with mutation potential and ageing allow to efficiently overcome local optima from which evolutionary algorithms (EAs) struggle to escape. Such behaviour has been shown for artificial example functions constructed especially to show difficulties that EAs may encounter during the optimisation process. However, no evidence is available indicating that these two operators have similar behaviour also in more realistic problems. In this paper we perform an analysis for the standard NP-hard Partition problem from combinatorial optimisation and rigorously show that hypermutations and ageing allow AISs to efficiently escape from local optima where standard EAs require exponential time. As a result we prove that while EAs and random local search (RLS) may get trapped on 4/3 approximations, AISs find arbitrarily good approximate solutions of ratio (1+ ) within n( −(2/ )−1 )(1 − ) −2 e 3 2 2/ + 2n 3 2 2/ + 2n 3 function evaluations in expectation. This expectation is polynomial in the problem size and exponential only in 1/ . $ An extended abstract of this paper without proofs was presented at PPSN 2018 [1].

show abstract

“…We also consider PWT with a multi-objective algorithm using a variant of GSEMO, which uses a specific selection function to deal with the exponential size of the population (Algorithm 3). Neumann and Sutton suggested this version of GSEMO for the Knapsack Problem (KP) with correlated weights and profit to avoid an exponential population size [23]. We use the same approach since PWT easily changes to KP when R = 0.…”

Section: Algorithmsmentioning

confidence: 99%

“…If W (s 0 ) ≤ C the first phase is already complete. Therefore, let W (s 0 ) > C. Using a fitness level argument, Neumann and Sutton proved that (1+1) EA, which uses a fitness function with strictly higher priority in weight constraint satisfaction, finds a feasible solution for KP with correlated weights and profits in O(n 2 ) expected time (Theorem 3 in [23]). Their proof also holds for RLS_swap since the constraint is linear and RLS_swap is able to do a one-bit flip in O(n) expected time.…”

Section: Rls_swapmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of baseline evolutionary algorithms for the packing while travelling problem

Roostapour

Pourhassan

Neumann

2019

Proceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms

Self Cite

View full text Add to dashboard Cite

The performance of base-line Evolutionary Algorithms (EAs) on combinatorial problems has been studied rigorously. From the theoretical viewpoint, the literature extensively investigates the linear problems, while the theoretical analysis of the non-linear problems is still far behind. In this paper, variations of the Packing While Travelling (PWT) -also known as the non-linear knapsack problemare studied as an attempt to analyse the behaviour of EAs on non-linear problems from theoretical perspective. We investigate PWT for two cities and n items with correlated weights and profits, using single-objective and multi-objective algorithms. Our results show that RLS_swap, which differs from the classical RLS by having the ability to swap two bits in one iteration, finds the optimal solution in O(n 3 ) expected time. We also study an enhanced version of GSEMO, which a specific selection operator to deal with exponential population size, and prove that it finds the Pareto front in the same asymptotic expected time. In the case of uniform weights, (1+1) EA is able to find the optimal solution in expected time O(n 2 log (max{n, p max })), where p max is the largest profit of the given items. We also perform an experimental analysis to complement our theoretical investigations and provide additional insights into the runtime behavior.

show abstract

Runtime Analysis of Evolutionary Algorithms for the Knapsack Problem with Favorably Correlated Weights

Cited by 8 publications

References 26 publications

Helper and Equivalent Objectives: Efficient Approach for Constrained Optimization

Helper and Equivalent Objectives: Efficient Approach for Constrained Optimization

Artificial immune systems can find arbitrarily good approximations for the NP-hard number partitioning problem

Analysis of baseline evolutionary algorithms for the packing while travelling problem

Contact Info

Product

Resources

About