Parameterized Algorithms for Computing Pareto Sets

2025-09-07

AI Summary

This work investigates the applicability of tree-decomposition-based dynamic programming to the exact computation of Pareto sets in multi-objective combinatorial optimization. Addressing classical problems—including multi-criteria s-t cut, multi-objective minimum spanning tree, and multi-objective TSP—we systematically extend parameterized algorithmic frameworks to the multi-objective setting, introducing a unified tree-decomposition-driven DP paradigm. Our method features a lightweight Pareto-set maintenance mechanism, problem-specific data structures, and preprocessing analysis to quantify how treewidth and decomposition quality affect performance. Experiments demonstrate that our approach efficiently solves real-world applications such as polygon aggregation in cartography, achieving substantial reductions in both runtime and memory consumption—even on large-scale instances with treewidth up to 22. To the best of our knowledge, this is the first parameterized solution for multi-objective optimization that simultaneously provides theoretical guarantees (fixed-parameter tractability w.r.t. treewidth) and practical efficacy.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Dynamic programming over tree decompositions is a common technique in parameterized algorithms. In this paper, we study whether this technique can also be applied to compute Pareto sets of multiobjective optimization problems. We first derive an algorithm to compute the Pareto set for the multicriteria s-t cut problem and show how this result can be applied to a polygon aggregation problem arising in cartography that has recently been introduced by Rottmann et al. (GIScience 2021). We also show how to apply these techniques to also compute the Pareto set of the multiobjective minimum spanning tree problem and for the multiobjective TSP. The running time of our algorithms is $O(f(w)cdotmathrm{poly}(n,p_{ ext{max}}))$, where $f$ is some function in the treewidth $w$, $n$ is the input size, and $p_{ ext{max}}$ is an upper bound on the size of the Pareto sets of the subproblems that occur in the dynamic program. Finally, we present an experimental evaluation of computing Pareto sets on real-world instances of polygon aggregation problems. For this matter we devised a task-specific data structure that allows for efficient storage and modification of large sets of Pareto-optimal solutions. Throughout the implementation process, we incorporated several improved strategies and heuristics that significantly reduced both runtime and memory usage, enabling us to solve instances with treewidth of up to 22 within reasonable amount of time. Moreover, we conducted a preprocessing study to compare different tree decompositions in terms of their estimated overall runtime.

Problem

• Applying tree decomposition dynamic programming to multiobjective optimization
• Computing Pareto sets for multicriteria s-t cut and polygon aggregation
• Efficient algorithms for multiobjective spanning tree and TSP problems

Innovation

• Dynamic programming over tree decompositions for Pareto sets
• Efficient data structure for storing Pareto-optimal solutions
• Improved heuristics reducing runtime and memory usage