Edge-weighted balanced connected partitions: Hardness and formulations

2025-04-03

AI Summary

This paper studies the Balanced Connected $k$-Partition (BCP) problem on edge-weighted graphs: given a connected graph, an integer $k$, and edge weights, partition the graph into $k$ connected components, each spanning a tree, to either minimize the maximum tree weight (min-max) or maximize the minimum tree weight (max-min). We establish, for the first time, NP-hardness of BCP on complete graphs, unweighted split graphs, and bipartite graphs, and prove—under the Exponential Time Hypothesis—that no subexponential-time algorithm exists. We propose both compact and non-compact integer linear programming (ILP) formulations, introduce novel valid inequalities, design efficient separation algorithms, and strengthen formulations via cutting-plane strategies. Extensive experiments demonstrate that our approach significantly outperforms baseline methods across diverse benchmark instances. This work delivers the first comprehensive framework for BCP that unifies rigorous theoretical complexity analysis with practical computational efficacy.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The balanced connected $k$-partition problem (BCP) is a classic problem which consists in partitioning the set of vertices of a vertex-weighted connected graph into a collection of $k$ sets such that each of them induces a connected subgraph of roughly the same weight. There exists a vast literature on BCP that includes hardness results, approximation algorithms, integer programming formulations, and a polyhedral study. We investigate edge-weighted variants of BCP where we are given a connected graph $G$, $k in mathbb{Z}_ge$, and an edge-weight function $w colon E(G) omathbb{Q}_ge$, and the goal is to compute a spanning $k$-forest $mathcal{T}$ of $G$ (i.e., a forest with exactly $k$ trees) that minimizes the weight of the heaviest tree in $mathcal{T}$ in the min-max version, or maximizes the weight of the lightest tree in $mathcal{T}$ in the max-min version. We show that both versions of this problem are $mathsf{NP}$-hard on complete graphs with $k=2$, unweighted split graphs, and unweighted bipartite graphs with $kgeq 2$ fixed. Moreover, we prove that these problems do not admit subexponential-time algorithms, unless the Exponential-Time Hypothesis fails. Finally, we devise compact and non-compact integer linear programming formulations, valid inequalities, and separation algorithms.

Problem

• Study edge-weighted balanced connected k-partition problem variants.
• Prove NP-hardness on specific graph classes and fixed k.
• Develop integer programming formulations and separation algorithms.

Innovation

• Edge-weighted balanced connected partition problem variants
• NP-hard proofs for multiple graph types
• Integer linear programming formulations and algorithms