Weighted Treedepth is NP-complete on Graphs of Bounded Degree

A treedepth decomposition of an undirected graph $G$ is a rooted forest $F$ on the vertex set of $G$ such that every edge $uv\in E(G)$ is in ancestor-descendant relationship in $F$. Given a weight function $w\colon V(G)\rightarrow \mathbb{N}$, the weighted depth of a treedepth decomposition is the maximum weight of any path from the root to a leaf, where the weight of a path is the sum of the weights of its vertices. It is known that deciding weighted treedepth is NP-complete even on trees. We prove that weighted treedepth is also NP-complete on bounded degree graphs. On the positive side, we prove that the problem is efficiently solvable on paths and on 1-subdivided stars.

💡 Research Summary

The paper investigates the computational complexity of the weighted treedepth problem, a natural extension of the classic treedepth measure where each vertex carries a positive integer weight and the depth of a treedepth decomposition is defined as the maximum sum of weights along any root‑to‑leaf path. While it was already known that deciding weighted treedepth is NP‑complete on trees, the authors address the open question of whether the problem remains hard on graphs with bounded degree. Their main contribution is a reduction from the classic Vertex Cover problem that shows weighted treedepth is NP‑complete even on graphs whose maximum degree is at most 10. The reduction carefully assigns weights and constructs gadgets so that any treedepth decomposition of bounded weighted depth corresponds directly to a vertex cover of the original instance, while preserving the degree bound. This settles the conjecture that weighted treedepth cannot be computed in polynomial time on bounded‑degree graphs.

In addition to the hardness result, the paper presents positive algorithmic results for two very restricted graph classes. First, for weighted paths the authors give a simple dynamic‑programming algorithm that computes the optimal weighted treedepth in O(n³) time. The DP considers every sub‑path, tries each vertex as the root of the sub‑decomposition, and combines the optimal solutions of the left and right sub‑paths. Second, for 1‑subdivided stars (a central vertex a, leaf vertices b_i adjacent to a, and a pendant vertex c_i attached to each b_i) they show that an optimal decomposition can be found by evaluating only n + 1 candidate structures: either a is the root, or a is placed below a prefix of the b_i’s in the tree order. By sorting the c_i’s by decreasing weight (and breaking ties by the weight of the corresponding b_i) the algorithm can compute the weighted depth of each candidate in linear time, yielding a polynomial‑time solution.

The paper also discusses several related observations. It notes that the problem is trivially in NP because a treedepth decomposition serves as a polynomial‑size certificate. It provides an FPT algorithm parameterized by the solution value itself: by “blowing up” each vertex into a clique of size equal to its weight, the weighted treedepth of the original graph equals the (unweighted) treedepth of the blown‑up graph, which can be computed in 2^{O(td²)}·n time. This yields fixed‑parameter tractability when the weighted depth is the parameter.

Finally, the authors outline open directions. It remains unknown whether weighted treedepth is NP‑hard on graphs of maximum degree 9 or on cubic (3‑regular) graphs. The complexity on 2‑subdivided stars and on deeper tree structures is also open, as is the question of whether the unweighted version becomes tractable when parameterized by weighted treedepth. The paper thus maps the boundary between tractable and intractable instances of weighted treedepth, providing both hardness evidence and efficient algorithms for the simplest graph families.