Maximum Partial List H-Coloring on P_5-free graphs in polynomial time

In this article we show that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H. In particular, this implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs. This answers an open question from Agrawal, Lima, Lokshtanov, Saurabh & Sharma [SODA 2024]. This also improves the $n^{O(ω(G))}$-time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rzążewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.

💡 Research Summary

The paper studies the Maximum Partial List H‑Coloring problem, where a fixed target graph H (without loops) is given, together with an input graph G, a non‑negative weight function wt on the vertices of G, and a list function assigning each vertex a subset of V(H). The goal is to select an induced subgraph G* of G of maximum total weight such that there exists a homomorphism from G* to H respecting the lists. When H is a k‑clique, this problem coincides with the Maximum k‑Colorable Subgraph problem; for k = 2 it is equivalent to the dual of Odd Cycle Transversal.

The authors focus on the class of P₅‑free graphs (graphs that contain no induced path on five vertices). Prior work gave an algorithm for Maximum Partial H‑Coloring whose running time depends on the size ω(G) of a maximum clique in G, namely n^{O(ω(G))} (Chudnovsky et al., SIDMA 2021). An open question from SODA 2024 asked whether the special case of Maximum k‑Colorable Subgraph is polynomial‑time solvable on P₅‑free graphs. This paper resolves the question affirmatively and, more strongly, shows that for any fixed H the full Maximum Partial List H‑Coloring problem can be solved in polynomial time on P₅‑free graphs.

The algorithm builds on two structural facts about P₅‑free graphs:

1. Dominating set structure (Proposition 2.1) – Every connected P₅‑free graph has a dominating set that induces either a P₃ or a clique.
2. Maximum‑weight independent set (Proposition 2.2) – The Maximum Weight Independent Set problem can be solved in polynomial time on P₅‑free graphs (O(n^{12} m) time).

Using these, the authors design a two‑stage approach.

Stage 1 – Recursive reduction when a connected optimal solution exists.
Assume the instance (G, wt, list) admits an optimal solution C that is connected. By Proposition 2.1 we can guess a small dominating set D (size ≤ max{k,3}) of C. Removing vertices not dominated by D yields a partition of V(G) into D and neighbor sets X₁,…,X_|D|. The algorithm then iteratively refines the lists: for each pair (X_i, X_j) it identifies a tiny subset (at most two vertices) of X_i that dominates all neighbours of X_i in X_j (Claim 3.1). Vertices adjacent to these dominating vertices have their lists pruned to eliminate colors that would cause a conflict with the homomorphism constraints. After processing all pairs, any edge between different X_i’s connects vertices whose lists are disjoint, and each vertex’s list size strictly decreases whenever a conflict is detected.

If after this cleaning a vertex’s list becomes a singleton, the problem reduces to finding a maximum‑weight independent set inside the subgraph induced by vertices with that single color, which is solvable by Proposition 2.2. If some lists still contain two or more colors, the algorithm recurses on each induced subgraph G