A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs

Given a graph $G$, the longest path problem asks to compute a simple path of $G$ with the largest number of vertices. This problem is the most natural optimization version of the well known and well studied Hamiltonian path problem, and thus it is NP-hard on general graphs. However, in contrast to the Hamiltonian path problem, there are only few restricted graph families such as trees and some small graph classes where polynomial algorithms for the longest path problem have been found. Recently it has been shown that this problem can be solved in polynomial time on interval graphs by applying dynamic programming to a characterizing ordering of the vertices of the given graph \cite{longest-int-algo}, thus answering an open question. In the present paper, we provide the first polynomial algorithm for the longest path problem on a much greater class, namely on cocomparability graphs. Our algorithm uses a similar - but essentially simpler - dynamic programming approach, which is applied to a Lexicographic Depth First Search (LDFS) characterizing ordering of the vertices of a cocomparability graph. Therefore, our results provide evidence that this general dynamic programming approach can be used in a more general setting, leading to efficient algorithms for the longest path problem on greater classes of graphs. LDFS has recently been introduced in \cite{Corneil-LDFS08}. Since then, a similar phenomenon of extending an existing interval graph algorithm to cocomparability graphs by using an LDFS preprocessing step has also been observed for the minimum path cover problem \cite{Corneil-MPC}. Therefore, more interestingly, our results also provide evidence that cocomparability graphs present an interval graph structure when they are considered using an LDFS ordering of their vertices, which may lead to other new and more efficient combinatorial algorithms.

💡 Research Summary

The paper tackles the classic Longest Path Problem (LPP), which asks for a simple path containing the maximum number of vertices in a given undirected graph G. While LPP is NP‑hard on arbitrary graphs, polynomial‑time solutions are known only for a few restricted families such as trees and, more recently, interval graphs. The authors extend the interval‑graph result to the much larger class of cocomparability graphs by exploiting a recently introduced vertex ordering technique called Lexicographic Depth‑First Search (LDFS).

Key Contributions

1. LDFS‑Based Characterisation – The authors show that when a cocomparability graph is processed by LDFS, the resulting vertex order σ = (v₁,…,vₙ) enjoys two crucial properties: (i) it is “umbrella‑free”, meaning that for any i < j < k, if (vᵢ, vₖ) is an edge while (vᵢ, vⱼ) is not, then (vⱼ, vₖ) must be present; and (ii) the set of right‑hand neighbours of each vertex, N⁺(vᵢ) = {vⱼ | j > i and (vᵢ, vⱼ)∈E}, forms a contiguous interval of indices. These properties are exactly the structural hallmarks of interval graphs under their right‑endpoint ordering. Consequently, an LDFS‑ordered cocomparability graph behaves like an interval graph from the algorithmic point of view.

2. Dynamic‑Programming Scheme – Leveraging the interval‑like structure, the authors define a DP table DP