Trimming of Graphs, with Application to Point Labeling

For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling.

💡 Research Summary

The paper introduces a novel graph‑theoretic concept called (t, g)‑trimmability. For a vertex‑weighted graph with total weight W, a subgraph is (t, g)‑trimmed if it retains at least (1 – 1/t)·W of the weight and contains no simple path longer than g edges. A family of graphs is called trimmable if, for every constant t > 0, there exists a constant g = g(t) such that every weighted graph in the family is (t, g)‑trimmable.

The authors define a new structural parameter, domino treewidth, which refines the classic treewidth by requiring each bag of a tree‑decomposition to have a “domino” shape: two adjacent clusters that intersect in at most a constant‑size separator and do not overlap otherwise. They prove that any graph class with bounded domino treewidth is trimmable. The proof proceeds by recursively decomposing the graph along its tree‑decomposition, performing a local trimming operation inside each bag that discards a small fraction of weight while guaranteeing that the remaining vertices induce only short paths. The local operation respects the domino structure, so the recursion preserves the bounded‑g property. Consequently, for any fixed t, a global constant g(t) can be derived that depends only on the domino treewidth bound.

From this structural result the paper derives a polynomial‑time approximation scheme (PTAS) for a weighted point‑labeling problem that had remained open. The labeling model is as follows: each point p in the plane carries a weight w(p) and a prescribed label length ℓ(p). Labels have unit height and may slide horizontally anywhere within an interval of length ℓ(p). Two labels must not overlap. The objective is to select a subset of points and assign a slide position to each of their labels so that the total weight of labeled points is maximized.

The key observation is that the conflict relations among points can be captured by a conflict graph G: vertices correspond to points, and an edge connects two vertices if there exists any placement of their labels that would cause an overlap. Because labels slide only horizontally and have unit height, the conflict graph is planar, has bounded degree (the bound depends on the maximum label length and the minimum separation of points), and, crucially, its domino treewidth is bounded. This follows from a geometric partition of the plane into vertical strips of width proportional to the longest label; each strip together with its neighboring strips forms a domino‑shaped region, and the induced subgraph on a strip has constant size. Hence G belongs to a trimmable family.

Applying the general trimming theorem to G yields a subgraph G′ that retains at least (1 – 1/t) of the total vertex weight and whose longest simple path has length at most g(t). Because the path length is bounded, the graph has bounded pathwidth, which enables a dynamic‑programming algorithm on the tree‑decomposition of G′ that enumerates all feasible label placements within each bag. The DP aggregates the best choices across bags, guaranteeing an overall solution whose weight is at least (1 – ε) of the optimum, where ε = 1/t can be made arbitrarily small. The running time is polynomial in the input size n, with a factor exponential only in the constant domino treewidth and in 1/ε; thus it is a PTAS.

The paper’s contributions can be summarized as follows:

1. Structural Theory – Introduction of domino treewidth and proof that bounded domino treewidth implies trimmability. This extends the known relationship between treewidth and separator‑based algorithms to a setting where the goal is to preserve most weight while eliminating long paths.

2. Algorithmic Framework – A generic trimming‑based reduction that transforms any weighted graph in a trimmable family into a bounded‑pathwidth instance, enabling exact DP solutions on the reduced graph.

3. Application to Map Labeling – Modeling the sliding‑label problem as a conflict graph with bounded domino treewidth, then applying the trimming framework to obtain a PTAS. This resolves a long‑standing open question concerning weighted point labeling with variable label lengths.

4. Broader Impact – The techniques suggest a new paradigm for approximation algorithms on weighted graph problems where the objective is to keep most of the weight while simplifying the combinatorial structure (e.g., eliminating long induced paths). Potential extensions include other geometric labeling variants, dynamic labeling scenarios, and graphs beyond planar or bounded‑degree classes.

In conclusion, by bridging a novel graph‑theoretic property with a concrete geometric optimization problem, the authors provide both deep theoretical insight and a practically relevant algorithmic tool. The PTAS for weighted sliding‑label placement not only settles a major open problem in map labeling but also opens avenues for further research on trimmability and its algorithmic exploitation.