A general approximation method for bicriteria minimization problems

Halffmann, Pascal; Ruzika, Stefan; Thielen, Clemens; Willems, David

doi:10.1016/j.tcs.2017.07.003

Cited by 10 publications

(12 citation statements)

References 23 publications

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“…For biobjective minimization problems, Halffmann et al [17] show how to obtain a (σ •(1+2ε), σ •(1+ 2 ε ))-approximation for any given 0 < ε ≤ 1 if a polynomial-time σ -approximation algorithm for the weighted sum scalarization is given.…”

Section: Previous Workmentioning

confidence: 99%

The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

et al. 2021

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We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

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Section: Previous Workmentioning

confidence: 99%

The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

et al. 2021

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“…We showed how human expertise can successfully be combined with decision-making tools, and received positive feedback from our collaborators. We hope that this work inspires new real-world applications involving multiple objectives, and will be used in multi-objective optimisation [15,16,23,29], planning [4,41,48,53], and RL [2,33,42,52] alike.…”

Section: Discussionmentioning

confidence: 99%

Ordered Preference Elicitation Strategies for Supporting Multi-Objective Decision Making

Zintgraf,

Roijers,

Linders

et al. 2018

Preprint

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In multi-objective decision planning and learning, much attention is paid to producing optimal solution sets that contain an optimal policy for every possible user preference profile. We argue that the step that follows, i.e, determining which policy to execute by maximising the user's intrinsic utility function over this (possibly infinite) set, is under-studied. This paper aims to fill this gap. We build on previous work on Gaussian processes and pairwise comparisons for preference modelling, extend it to the multi-objective decision support scenario, and propose new ordered preference elicitation strategies based on ranking and clustering. Our main contribution is an in-depth evaluation of these strategies using computer and human-based experiments. We show that our proposed elicitation strategies outperform the currently used pairwise methods, and found that users prefer ranking most. Our experiments further show that utilising monotonicity information in GPs by using a linear prior mean at the start and virtual comparisons to the nadir and ideal points, increases performance. We demonstrate our decision support framework in a real-world study on traffic regulation, conducted with the city of Amsterdam.

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“…Most notably, they show that, for p-objective minimization problems, for any ε > 0, a (p + ε)-approximation can be computed using the weighted sum scalarization. A specific algorithm using the weighted sum scalarization for computing approximations in biobjective minimization problems is given in [11]. For biobjective optimization problems with convex feasible sets and linear objective functions, an efficient algorithm for computing (1 + ε)-approximations is studied in [4].…”

Section: Related Workmentioning

confidence: 99%

Approximating Biobjective Minimization Problems Using General Ordering Cones

Herzel¹,

Helfrich²,

Ruzika³

et al. 2021

Preprint

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This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex ordering cones of a fixed inner angle γ ∈ [ π 2 , π], an approximation guarantee between α and 2α is achieved, which depends continuously on γ. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation.Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.

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A general approximation method for bicriteria minimization problems

Cited by 10 publications

References 23 publications

The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

Ordered Preference Elicitation Strategies for Supporting Multi-Objective Decision Making

Approximating Biobjective Minimization Problems Using General Ordering Cones

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