On List Colouring and List Homomorphism of Permutation and Interval Graphs

List colouring is an NP-complete decision problem even if the total number of colours is three. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving list colouring of permutation graphs with a bounded total number of colours. More generally we give a polynomial-time algorithm that solves the list-homomorphism problem to any fixed target graph for a large class of input graphs including all permutation and interval graphs.

💡 Research Summary

The paper tackles two classic combinatorial problems—list colouring and list homomorphism—within the restricted but practically important families of permutation and interval graphs. List colouring asks whether each vertex of a graph can be assigned a colour from its own prescribed list such that adjacent vertices receive distinct colours. Even when the total palette contains only three colours, the problem remains NP‑complete, and it stays hard on planar bipartite graphs. The authors show that, if the total number of colours k is a fixed constant, list colouring becomes polynomial‑time solvable on permutation graphs.

The key insight is that a permutation graph can be represented as the intersection pattern of two linear orders (or equivalently, as the comparability graph of a permutation). This representation allows each vertex’s admissible colour set to be transformed into a collection of colour‑specific intervals along the order. Because k is constant, the algorithm can process the vertices in order, maintaining for each colour a set of feasible intervals and checking interval overlaps to enforce the proper‑colouring constraint. The resulting dynamic‑programming procedure runs in O(n·k²) time, where n is the number of vertices, thus establishing a tractable case for a problem that is otherwise intractable.

Beyond colouring, the paper addresses the list homomorphism problem: given a graph G with a list of allowed target vertices for each vertex, and a fixed target graph H, decide whether there exists a homomorphism f : G → H respecting the lists. In general this problem is CSP‑hard, but the authors exploit the same order‑based structure of permutation and interval graphs to reduce it to a “ordered mapping” problem. They first fix a linear ordering of the (constant‑size) target graph H, then convert each vertex’s list in G into a contiguous interval of positions in that ordering. The homomorphism constraints become simple interval compatibility conditions: adjacent vertices in G must be mapped to adjacent (or compatible) intervals in H. A dynamic‑programming scan along the order of G checks these conditions efficiently, yielding a polynomial‑time algorithm whose complexity depends only on n and the size of H (which is constant).

The same methodology applies to interval graphs, which are defined by real intervals on a line. By discretising interval endpoints and treating each vertex’s list as a set of permissible intervals, the algorithm reduces to a classic interval‑scheduling DP that runs in polynomial time under the same bounded‑palette or fixed‑target assumptions.

Importantly, the authors’ approach does not rely on bounded treewidth, a common tool for tractable cases of CSPs. Instead, it leverages the intrinsic linear ordering of permutation and interval graphs, thereby extending tractability to a broader class of graphs that may have unbounded treewidth but possess a strong sequential structure. The paper concludes that these results deepen our understanding of the complexity landscape for list‑based graph problems and demonstrate that order‑centric techniques can yield efficient algorithms for otherwise hard combinatorial tasks.