Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai’s study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.

💡 Research Summary

This paper investigates the kernelization complexity of graph coloring problems when the instances are parameterized by structural measures of the input graph, rather than by the number of colors q. The authors adopt the framework introduced by Cai (DAM 2003), where a graph class F is a family of graphs on which coloring can be solved in polynomial time, and a parameter k is the size of a modulator X (a set of vertices, edges, or both) whose removal brings the input graph G into F. The focus is on q‑Coloring (the decision problem whether χ(G) ≤ q) with a fixed q, and on the related q‑List‑Coloring problem.

The main contributions are twofold: (i) positive results showing that for many natural graph classes F, q‑Coloring on F + k_v (graphs that become members of F after deleting at most k vertices) admits a polynomial kernel; and (ii) negative results proving that for several other parameters, a polynomial kernel would imply unlikely complexity collapses (NP ⊆ coNP/poly). The authors also explore domination‑related parameters and the relationship between the size of “no‑certificates” for q‑List‑Coloring on F and kernel sizes for q‑Coloring on F + k_v.

General kernelization theorem.
The authors prove that if every NO‑instance of q‑List‑Coloring on graphs from F contains a small “obstruction” of at most g(q) vertices (i.e., the non‑list‑colorability is “local”), then q‑Coloring on F + k_v admits a kernel with O(k·q·g(q)) vertices. The proof proceeds by enumerating all q^k possible colorings of the modulator X, and for each coloring checking whether the remaining graph G − X can be list‑colored respecting the pre‑assigned colors. Because any NO‑instance can be reduced to at most g(q) vertices, the reduction can be performed by keeping only those vertices that belong to some obstruction for some coloring of X. This yields a kernel whose size depends polynomially on k and q, provided g(q) is a function of q alone.

Applications of the positive theorem.

• Cographs + k_v. Cographs are P₄‑free graphs. The authors show that any NO‑instance of q‑List‑Coloring on a cograph has an obstruction of size O(q), so g(q)=Θ(q). Consequently, q‑Coloring on Cograph + k_v has a kernel with O(k·q²) vertices.
• S‑Cochordal + k_v. The class S‑Cochordal consists of disjoint unions of co‑chordal graphs (graphs whose complements are chordal). Again, the obstruction size is bounded by a function of q, leading to a polynomial kernel for q‑Coloring on this class.
• Independent + k_v (feedback vertex set). Here g(q)=q, giving a kernel with O(k·q²) vertices, which can even be encoded in O(k·q²) bits using a compact representation derived from a reduction to Not‑All‑Equal q‑SAT.

Lower bounds (non‑existence of kernels).
The paper strengthens earlier results that 3‑Coloring parameterized by treewidth has no polynomial kernel unless all coNP‑complete problems admit distillation. The authors prove a stronger statement: 3‑Coloring on Path + k_v (graphs that become a single path after deleting k vertices) does not admit a polynomial kernel unless NP ⊆ coNP/poly. Since Path + k_v is a stricter parameter than Forest + k_v or Interval + k_v, this lower bound automatically excludes polynomial kernels for those larger parameters as well.

A key technical tool is the concept of an irreducible NO‑instance for (q − 2)-List‑Coloring: a NO‑instance on t vertices such that deleting any vertex turns it into a YES‑instance. The authors show that if such an instance exists, then any kernel for q‑Coloring on F + k_v must have size at least Ω(k^t). Hence, unless NP ⊆ coNP/poly, kernels of size O(k^{t−ε}) are impossible. This explains why the exponent t (the size of the smallest obstruction) appears in the exponent of the kernel size in the positive theorem.

Domination‑related parameters.
When a dominating set X of size k is given together with the graph, the authors present an O*(3^k) algorithm for 3‑Coloring, exploiting the fact that each vertex outside X has at most three possible colors once the colors of X are fixed. However, they also prove that 3‑Coloring parameterized solely by the size of a dominating set does not admit a polynomial kernel unless NP ⊆ coNP/poly. To obtain a kernel, they consider the class Dominated + k_v, where each connected component contains a dominating vertex. For this class, they construct a polynomial kernel for 3‑Coloring, showing that the additional structural restriction (a guaranteed local dominator) is crucial.

Methodological framework.
The paper relies heavily on the machinery of parameterized complexity: polynomial‑parameter transformations, cross‑compositions, and kernel lower bounds based on the non‑existence of compressions for q‑CNF‑SAT (Dell‑van Melkebeek). The authors also use classic graph‑theoretic concepts (split graphs, chordal graphs, co‑chordal graphs, cographs, forests, intervals) and hereditary properties to organize the hierarchy of parameters (Fig. 1 in the paper). By treating the parameters as a hierarchy rather than isolated cases, they can transfer results up and down the hierarchy: a lower bound for a “smaller” parameter automatically yields lower bounds for all “larger” parameters.

Overall significance.
The work provides a comprehensive map of which structural parameterizations admit efficient preprocessing (polynomial kernels) for q‑Coloring and which do not. It establishes a clear link between the size of minimal obstructions for list‑coloring on the base class F and the achievable kernel size for the corresponding parameterized problem. Moreover, the negative results delineate the limits of kernelization, showing that even modest structural parameters such as distance to a path already preclude polynomial kernels under standard complexity assumptions. The domination‑based results further illustrate that the presence of a global structural feature (a dominating set) is insufficient for kernelization, whereas a local domination property (each component having a dominating vertex) restores it.

In summary, the paper advances the theory of kernelization for graph coloring by (1) presenting a general positive kernelization theorem tied to obstruction size, (2) applying it to several natural graph classes, (3) proving strong lower bounds for a range of parameters, and (4) exploring the nuanced role of domination in kernelization. These contributions deepen our understanding of how structural graph properties influence the feasibility of data reduction for coloring problems.