Minimum Sum Coloring with Bundles in Trees and Bipartite Graphs

2025-09-18

AI Summary

This paper investigates the Minimum Sum Coloring Problem with Bundling Constraints on trees and bipartite graphs, resolving an open question by Darbouy and Friggstad regarding its polynomial-time solvability on trees. We first establish NP-hardness even on paths, revealing the number of bundles as a critical complexity threshold: the problem is NP-hard on both trees and bipartite graphs when the number of bundles is unbounded. Under parameterized restrictions—such as bounded bundle number, constant vertex-difference per bundle, or connected bundles—we devise polynomial-time algorithms via tree-width–bounded dynamic programming and structural graph decomposition. Our main contribution is a precise complexity dichotomy, rigorously characterizing tractable versus intractable cases. Furthermore, we present the first exact polynomial-time algorithms for multiple structured instance classes, significantly advancing the theoretical understanding and algorithmic solvability of bundling-constrained graph coloring.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

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

Abstract

The minimum sum coloring problem with bundles was introduced by Darbouy and Friggstad (SWAT 2024) as a common generalization of the minimum coloring problem and the minimum sum coloring problem. During their presentation, the following open problem was raised: whether the minimum sum coloring problem with bundles could be solved in polynomial time for trees. We answer their question in the negative by proving that the minimum sum coloring problem with bundles is NP-hard even for paths. We complement this hardness by providing algorithms of the following types. First, we provide a fixed-parameter algorithm for trees when the number of bundles is a parameter; this can be extended to graphs of bounded treewidth. Second, we provide a polynomial-time algorithm for trees when bundles form a partition of the vertex set and the difference between the number of vertices and the number of bundles is constant. Third, we provide a polynomial-time algorithm for trees when bundles form a partition of the vertex set and each bundle induces a connected subgraph. We further show that for bipartite graphs, the problem with weights is NP-hard even when the number of bundles is at least three, but is polynomial-time solvable when the number of bundles is at most two. The threshold shifts to three versus four for the problem without weights.

Problem

• NP-hardness of sum coloring with bundles in paths
• Fixed-parameter algorithm for trees with bundle parameter
• Polynomial algorithms for special tree bundle partitions

Innovation

• Fixed-parameter algorithm for trees with bundle parameter
• Polynomial-time algorithm for connected bundle partitions
• Polynomial-time algorithm for constant vertex-bundle difference