Min Sum Coloring with Bundles

• The paper introduces a generalized model by integrating bundles and weighting functions to capture group performance in scheduling.
• It demonstrates that adding bundle structures increases complexity, with NP-hardness in trees and bipartite graphs under certain thresholds.
• The study presents dynamic programming, fixed-parameter, and XP algorithm techniques for structured instances and outlines future research directions.

The minimum sum coloring problem with bundles generalizes classical graph coloring and minimum sum coloring by incorporating "bundles"—subsets of vertices—whose cost is determined not by summing the colors of its vertices but by evaluating a function on the set (typically the maximum color value). This model captures a range of optimization scenarios, particularly those related to scheduling, resource allocation, and network management, where agents or systems care about performance or completion time of groups rather than individual elements.

1. Formal Problem Statement and Motivation

Let $G = (V, E)$ be an undirected graph equipped with a family $\mathcal{B}$ of bundles, $\mathcal{B} \subseteq 2^V$, where each $B \in \mathcal{B}$ may have a positive weight $w(B)$. A proper coloring is a mapping $c: V \rightarrow \mathbb{N}_+$ with $c(u) \neq c(v)$ for all $\{u, v\} \in E$. The cost for each bundle $B$ is defined as $w(B) \cdot \max\{c(v) : v \in B\}$, and the global objective is to minimize: $$F(G, \mathcal{B}, w, c) = \sum_{B \in \mathcal{B}} w(B) \cdot \max\{c(v) : v \in B\}.$$ This formulation covers classical coloring (when $\mathcal{B} = \{V\}$), minimum sum coloring (when $\mathcal{B}$ consists of all singletons and $w(B)=1$), and naturally models the total completion time in scheduling contexts where jobs are grouped and agents care about the slowest job in their group.

2. Computational Complexity

A central result is that the addition of general bundle structures renders the minimum sum coloring problem with bundles NP-hard, even on simple graph classes:

• On trees, classical coloring and minimum sum coloring are polynomial-time solvable, but introducing bundles causes a sharp increase in complexity. Specifically, when $\mathcal{B}$ partitions $V$ into independent sets and all weights are unit, the problem is NP-complete even on paths.
• For bipartite graphs, the parameterization by the number of bundles yields tight thresholds:
  • With weights, the problem is NP-hard when there are at least three bundles; polynomial-time when there are at most two.
  • In the unweighted case, NP-hardness arises with four or more bundles and remains polynomial for three or fewer.
• These results delineate a complexity frontier that depends on both the graph structure and the combinatorics of the bundle family, in contrast to classical coloring where such thresholds do not arise.

3. Algorithmic Approaches and Parameterized Results

Due to the inherent hardness for most general cases, several algorithmic results focus on restricted scenarios and parameterized settings:

• Fixed-Parameter Tractability (FPT): When the number of bundles $\ell$ is the parameter and the host graph has bounded treewidth (including trees), the problem is FPT. The color range per vertex can be bounded by $\chi(G)\cdot \ell$, and the problem can be encoded in monadic second-order logic. Dynamic programming over a tree decomposition is used, with Courcelle's theorem ensuring tractability.
• XP Algorithm (Parameterized by $n - |\mathcal{B}|$): If $\mathcal{B}$ partitions $V$ and the difference $t = n - |\mathcal{B}|$ is small (i.e., most bundles are singletons and few are nontrivial), there exists an algorithm with time complexity $O((2\ell)^t n \ell^2)$.
• Polynomial Algorithms for Structured Bundles: If bundles partition $V$ and each bundle induces a connected subgraph in a tree, a specialized dynamic programming scheme solves the problem in $O(n^6)$ time. For paths with connected bundles, the coloring range per vertex is bounded, allowing efficient solutions.

A summary of parameterized tractability across graph types:

Graph Class	Bundle Constraint	# Bundles	Complexity
Trees	Arbitrary	$\geq 2$	NP-hard
Trees	Partition, conn. subgraphs	any	Polynomial
Bipartite	Unweighted, $\leq 3$ bundles	$\leq 3$	Polynomial
Bipartite	Unweighted, $\geq 4$ bundles	$\geq 4$	NP-hard
Bipartite	Weighted, $\leq 2$ bundles	$\leq 2$	Polynomial
Bipartite	Weighted, $\geq 3$ bundles	$\geq 3$	NP-hard

4. Relationship to Scheduling and Chromatic Sum Models

When each vertex is interpreted as a job and each bundle as the jobs relevant to an agent, $\max\{c(v): v \in B\}$ models the completion time for that bundle (agent). Consequently, minimizing the sum over all bundle costs generalizes traditional scheduling objectives such as minimizing average response time or flow in multi-agent settings.

Earlier lines of research in graph coloring and scheduling have focused on chromatic sums and completion time objectives, including constant-factor approximation algorithms for minimum sum coloring and related dependent job scheduling problems. The weighted maximum-in-bundle objective in the bundled variant is a natural extension of these settings, integrating resource constraints and agent-based completion time targets.

5. Approximation Algorithms and Connections to Prior Work

While the discussed paper centers on exact tractability and hardness, there are foundational connections to classic work on approximability for sum coloring and scheduling with conflicts. Notably, techniques due to Halldórsson and Kortsarz—greedy and LP-based rounding algorithms—have yielded constant-approximation ratios in related settings. Their approaches often leverage bundle structures implicitly by grouping jobs (vertices) and bounding their completion times.

A plausible implication is that such approximation strategies, when adapted to explicitly incorporate bundle costs (the weighted maximum per bundle), may yield practical algorithms with provable bounds, especially for classes like bipartite or tree graphs where the coloring structure is well-understood. The article suggests that this remains a fertile area for future development.

6. Thresholds, Bundle Structure, and Graph Class Dependence

The complexity thresholds uncovered hinge critically on both the number of bundles and their structural properties (e.g., overlap, connectivity). The problem's tractability is sharply sensitive to these features, distinguishing it from classical coloring problems. For trees and bipartite graphs, the bundle-induced parameterization provides a framework for identifying the boundary between polynomial-time solvability and NP-hardness.

These conditional results provide guidance for modeling applications: instance structures with few bundles, high connectivity within bundles, or bounded treewidth yield feasible computational approaches; in contrast, arbitrary bundle systems generally call for parameterized or approximation strategies.

7. Open Problems and Directions

The question of approximability for the minimum sum coloring problem with bundles remains open beyond the specific cases analyzed. The development of bundle-aware rounding algorithms, generalization of metaheuristics, and integration with scheduling formulations present promising directions. Additionally, further investigation into threshold phenomena for other graph classes, the role of bundle overlaps, and parameterized complexity in broader contexts are suggested by the foundational results presented.

Overall, the minimum sum coloring problem with bundles provides a rigorous and expressive model for resource-constrained scheduling, with rich connections to graph coloring theory, combinatorial optimization, and practical applications in computer science and operations research.