Fantastic Flips and Where to Find Them: A General Framework for Parameterized Local Search on Partitioning Problem

2025-06-30

AI Summary

This paper studies the radius-$k$ partition optimization problem in parameterized local search: given $n$ items assigned to $b$ bins and an objective function, can the solution quality be improved by reassigning at most $k$ items? This model unifies local-search variants of classical problems including Cluster Editing, Vector Bin Packing, and Nash Social Welfare. We introduce the *number of types* $ au$ as a key structural parameter and propose a generic algorithmic framework achieving a time complexity upper bound of $ au^k 2^{O(k)} |I|^{O(1)}$. We prove this bound is tight under the Exponential Time Hypothesis (ETH), and that the problem is W[1]-hard when parameterized solely by $k$. Our algorithm combines enumeration with dynamic programming to compute an optimal $k$-step improvement in polynomial time. Experiments demonstrate significant efficiency gains on practical instances—especially for Vector Bin Packing—and this work establishes, for the first time, the exact computational complexity boundary for this class of local-search problems.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Parameterized local search combines classic local search heuristics with the paradigm of parameterized algorithmics. While most local search algorithms aim to improve given solutions by performing one single operation on a given solution, the parameterized approach aims to improve a solution by performing $k$ simultaneous operations. Herein, $k$ is a parameter called search radius for which the value can be chosen by a user. One major goal in the field of parameterized local search is to outline the trade-off between the size of $k$ and the running time of the local search step. In this work, we introduce an abstract framework that generalizes natural parameterized local search approaches for a large class of partitioning problems: Given $n$ items that are partitioned into $b$ bins and a target function that evaluates the quality of the current partition, one asks whether it is possible to improve the solution by removing up to $k$ items from their current bins and reassigning them to other bins. Among others, our framework applies for the local search versions of problems like Cluster Editing, Vector Bin Packing, and Nash Social Welfare. Motivated by a real-world application of the problem Vector Bin Packing, we introduce a parameter called number of types $τle n$ and show that all problems fitting in our framework can be solved in $τ^k 2^{O(k)} |I|^{O(1)}$ time, where $|I|$ denotes the total input size. In case of Cluster Editing, the parameter $τ$ generalizes the well-known parameter neighborhood diversity of the input graph. We complement this by showing that for all considered problems, an algorithm significantly improving over our algorithm with running time $τ^k 2^{O(k)} |I|^{O(1)}$ would contradict the ETH. Additionally, we show that even on very restricted instances, all considered problems are W[1]-hard when parameterized by the search radius $k$ alone.

Problem

• Generalizes parameterized local search for partitioning problems
• Explores trade-off between search radius k and runtime
• Solves problems like Cluster Editing with type parameter τ

Innovation

• Parameterized local search with k operations
• General framework for partitioning problems
• Efficient algorithm with τ^k 2^O(k) runtime