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One-Exact Approximate Pareto Sets for APX-Hard Multiobjective Problems

Authors Fritz Bökler , Markus Chimani , Henning Jasper

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LIPIcs.ESA.2026.112.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Mathematical optimization
Keywords
  • multiobjective optimization
  • approximate Pareto sets
  • scalarization

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Abstract

There are several frameworks to compute approximate Pareto sets for multiobjective optimization (MOO) problems. An approximate Pareto set that is even precise in one specific objective is called one-exact. In such frameworks, some auxiliary single-objective problem is considered, for which a problem-specific oracle is required. However, often these oracles are required to be a PTAS or even FPTAS. As such, these frameworks are only applicable to "simple" MOO problems that allow for such strong oracles to exist. They are inapplicable whenever the auxiliary problem is APX-hard. 
We propose a general framework that, for a (possibly even non-constant) accuracy vector β = (β_2, … , β_d) and any ε > 0, computes polynomially sized, one-exact (1,(1 + ε)β_2,… ,(1 + ε)β_d)-Pareto sets for d-objective minimization problems. The framework is analogously applicable to maximization and mixed MOO problems. The running time is polynomial in the time required to solve our auxiliary problem β-RelaxedDualRestrict. Notably, these guarantees hold even if β-RelaxedDualRestrict is APX-hard. We further show that if β-RelaxedDualRestrict cannot be solved in polynomial time, then no (1,β_2, … ,β_d)-Pareto set can be computed in polynomial time. For biobjective problems, our framework even yields a (1,(1 + ε)β_2)-Pareto set of at most 𝒪(log β₂) times the size of the minimum-size one-exact (1,(1 + ε)β_2)-Pareto set. We show that this relative size guarantee is asymptotically tight. Further, we present techniques to obtain suitable oracles for β-RelaxedDualRestrict from existing (single-objective) approximation algorithms, including a general "re-randomization" method that may be of independent interest. Using these, we obtain new best approximation guarantees for several established MOO problems, including Spanner, Clique, TSP, Facility Location, and Set Cover problems.

Cite As Get BibTex

Fritz Bökler, Markus Chimani, and Henning Jasper. One-Exact Approximate Pareto Sets for APX-Hard Multiobjective Problems. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 112:1-112:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.112

Author Details

Fritz Bökler
  • Institute of Computer Science, Osnabrück University, Germany
Markus Chimani
  • Institute of Computer Science, Osnabrück University, Germany
Henning Jasper
  • Institute of Computer Science, Osnabrück University, Germany

Funding

Supported by the German Research Foundation (DFG), project #517835933.

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