Thin graph classes and polynomial-time approximation schemes

Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques and usually focus on somewhat restricted classes of problems. We present a new type of graph decompositions (thin systems of overlays) generalizing Baker’s technique and leading to straightforward polynomial-time approximation schemes. We also show that many graph classes (all proper minor-closed classes, and all subgraph-closed classes with bounded maximum degree and strongly sublinear separators) admit such decompositions.

💡 Research Summary

The paper introduces a novel graph decomposition called a “thin system of overlays,” which generalizes Baker’s classic technique for designing polynomial‑time approximation schemes (PTAS) on planar graphs. An overlay L = (H, f, ℓ) consists of a host graph H, a homomorphism f from H to the original graph G, and a label function ℓ assigning each vertex of H a non‑negative integer at most r. The overlay is walk‑preserving: for any vertex x with ℓ(x) ≥ 1, every neighbor of f(x) in G is represented by a neighbor y of x in H with ℓ(y) ≥ ℓ(x)−1. Consequently, any walk of length at most ℓ(x) starting at f(x) can be lifted to a walk of the same length in H. An r‑neighbourhood overlay additionally guarantees that every vertex of G has at least one pre‑image with ℓ = r, i.e., the overlay faithfully reproduces the r‑neighbourhood of each vertex.

A thin system is a collection 𝓛 = {L₁,…,L_t} of such overlays whose “thickness” (the maximum number of pre‑images of any vertex of G) is bounded by a constant c. The key algorithmic insight is that, for many combinatorial optimisation problems that can be expressed in monadic second‑order logic (MSO₁) with distance predicates, the optimal solution on each overlay H_i can be found exactly in linear time because H_i has bounded treewidth. Since each vertex of G appears in at most c overlays, taking the best solution among the c possibilities yields a (1 + ε)‑approximation for the original problem, where ε can be made arbitrarily small by adjusting the parameters (the size of the “window” of consecutive layers and the spacing between selected windows). Importantly, the approximation factor influences only the constant factor in the running time, not the polynomial degree.

The authors prove that thin systems exist for two broad families of graph classes:

1. All proper minor‑closed classes – i.e., any class that excludes a fixed apex graph as a minor. Using the Robertson‑Seymour graph‑minor structure theorem together with recent results on bounded‑treewidth decompositions of apex‑minor‑free graphs, they construct r‑neighbourhood overlays of constant thickness and bounded treewidth.

2. Subgraph‑closed classes with bounded maximum degree and strongly sublinear separators – such classes have separator size O(n^α) for some α < 1. By recursively applying separator theorems, they obtain a hierarchical decomposition that yields thin systems with the same desirable properties.

With these decompositions in hand, the paper derives PTAS for a wide range of problems: maximum independent set, minimum dominating set, their distance‑restricted variants, maximum H‑matching, minimum vertex cover, and any monotone maximisation problem expressible in a restricted fragment of MSO₁ with constant‑distance predicates (including maximum c‑colourable induced subgraph for any fixed c). The approach also yields fixed‑parameter tractability when the parameter is the size of the optimal solution, because the exponential dependence appears only in the constant factor.

The paper discusses limitations. The technique requires bounded degree (or at least sublinear separators) to guarantee thinness; classes containing all 3‑regular graphs or those with exponential expansion do not admit thin systems. Moreover, the method relies on monotonicity and locality: global, non‑monotone problems such as minimum connected dominating set or minimum independent dominating set fall outside its scope, where bidimensionality theory is more appropriate. Finally, the authors note that dropping the bounded‑degree assumption for subgraph‑closed classes with sublinear separators remains an open problem.

In summary, thin systems of overlays provide a unifying and conceptually simple framework that extends Baker’s layering idea to essentially all proper minor‑closed graph families and many other sparse classes. By coupling these decompositions with exact MSO algorithms on bounded‑treewidth graphs, the authors obtain clean PTAS and FPT results for a broad spectrum of classic optimisation problems, while also clarifying the structural boundaries where the method cannot be applied.