Efficient Polynomial-Time Approximation Scheme (EPTAS)

• EPTAS is a framework of approximation algorithms that compute solutions within a (1+ε) factor for NP-hard problems while isolating the exponential dependency on ε.
• It exploits structural properties like bounded treewidth through the removal of a small vertex set, enabling efficient dynamic programming on the remaining graph.
• EPTAS unifies classical PTAS techniques, broadening applications to various problems including packing, covering, and connectivity on planar and H-minor-free graphs.

An Efficient Polynomial-Time Approximation Scheme (EPTAS) is a family of approximation algorithms for NP-hard optimization problems that, for any fixed accuracy parameter $\varepsilon > 0$, computes a solution whose cost is within a factor of $(1+\varepsilon)$ (for minimization) or $(1-\varepsilon)$ (for maximization) of the optimum, with a running time of the form $f(1/\varepsilon) \cdot n^{O(1)}$, where $f$ is a function exponential (or worse) in $1/\varepsilon$, but polynomial in the size $n$ of the instance. EPTASs are distinguished from polynomial-time approximation schemes (PTASs) in that all exponential dependence on $1/\varepsilon$ is isolated in a multiplicative factor outside the input size exponent, making EPTASs significantly more practical for fixed values of $\varepsilon$.

1. Core Principles and Frameworks for EPTAS Construction

The construction of EPTASs typically relies on recognizing and exploiting special structural properties in problem instances that permit efficient reduction of complexity after carefully removing or modifying a small fraction of the optimal solution set.

A prominent example is the bidimensionality theory applied to graph optimization problems on $H$-minor-free graphs (notably planar and apex-minor-free graphs), where a key insight is that removing a carefully chosen "small" subset of vertices, proportional in size to $\varepsilon\cdot \mathrm{OPT}$, reduces the graph to bounded treewidth, enabling powerful algorithmic techniques such as dynamic programming and meta-algorithmic theorems (e.g., Courcelle's theorem) (1005.5449). The central decomposition lemma formalizes this by stating:

• For any $\varepsilon>0$, one can find in polynomial time a vertex set $X$ of size at most $\varepsilon\cdot \mathrm{OPT}$ so that the treewidth of $G \setminus X$ is bounded by a function $f(\varepsilon)$.

The framework is structured around two properties for problems:

• Reducibility: The ability to relate the value of an optimal solution to that on a bounded-treewidth graph with only a small loss, once a small set is removed.
• $\eta$-Transversability: The existence of a small vertex set (the "transversal") whose removal makes the remaining graph have treewidth at most $\eta$.

This bidimensional framework, originally designed for parameterized algorithms, is rearchitected and unified to provide streamlined construction of EPTASs for a wide variety of problems, transcending limitations of classical separator-based or layering techniques.

2. Algorithmic Strategies and Decomposition Lemma

The technical heart of most EPTAS constructions is a reduction step that shrinks the "hardness" of the problem instance to a manageable size, followed by exact or efficiently optimal solution methods on the simplified instance.

The decomposition lemma (1005.5449) guarantees that for any hereditary graph class of truly sublinear treewidth and input subset $X$ (whose removal reduces treewidth), one can:

• Compute $X'\subseteq V(G)$ with $|X'| \leq \varepsilon |X|$
• Ensure any connected component $C$ of $G\setminus X'$ satisfies $|C\cap X|,\,|N(C)|\leq \gamma(\varepsilon)$.

By recursive or single-stage removal of these small sets, one ultimately reduces the input to a bounded-treewidth instance, on which powerful methods (including dynamic programming, CMSO logic, Baker's technique, or separator-based recursion) become feasible and highly efficient.

The EPTAS approach typically proceeds in the following steps:

1. Identify and remove a small set $X$ whose presence is responsible for the nonlocal structure/complexity—$|X|\leq \varepsilon\cdot \mathrm{OPT}$.
2. Solve the reduced, bounded-treewidth instance to exact optimality using dynamic programming or meta-algorithmic logic.
3. Combine the partial solution to the bounded-treewidth instance with an optimal (or near-optimal) solution on the small set $X$.

Formally, if $\mathcal{P}$ is the class of instances, and, for any $G\in\mathcal{P}$, we produce $X$ as above, then we find an optimal solution $S_1$ on $G\setminus X$, and an optimal solution $S_2$ restricted to $X$, combining to form a global solution with objective value at most $(1+\varepsilon)\cdot \mathrm{OPT}$.

3. Unification of Planar Graph PTAS Strategies

Classic PTAS construction in planar and minor-free graphs proceeded along two dominant tracks:

• Lipton–Tarjan separator-based approach: Recursively divides the graph using small separators, partitioning the optimal solution.
• Baker's technique: Decomposes the graph into layers, solves subinstances via dynamic programming on bounded treewidth layers, and combines using local shifts.

The EPTAS framework unifies and improves upon these by using the decomposition lemma to:

• Remove a small explicitly constructed set once, converting the residual into a bounded-treewidth graph on which dynamic programming is tractable.
• Avoid the overhead of recursive partitioning or repeated layering; instead, operate directly using only the minimum fragmentation necessary.

This unification has several advantages:

• Applies directly to packing, covering, and partial covering problems.
• Handles problems not closed under taking minors or contractions, expanding applicability to new classes of combinatorial problems.
• Simplifies the verification of problem compatibility with the EPTAS framework via explicit, checkable criteria (reducibility and $\eta$-transversability).

4. Problem Classes and Applications

The EPTAS framework based on bidimensionality and the decomposition lemma yields efficient approximation schemes for many previously intractable or only PTAS-amenable problems in planar and $H$-minor-free graphs. Notable applications include:

• Packing Problems: Cycle packing, vertex-$\mathcal{H}$-packing (finding the maximum number of vertex-disjoint subgraphs from a family $\mathcal{H}$, each containing a planar graph as a minor). No EPTAS was previously known for these problems on planar graphs.
• Connected Dominating Set and Feedback Vertex Set: Problems involving connectivity or feedback minimization, for which only PTASs (or complex routines) were previously available.
• Maximum Leaf Spanning Tree and Partial $r$-Dominating Set: New EPTAS results for these problems on planar graphs.

The central technical tool is the existence of small transversals, and the key procedural step is the use of bounded treewidth after deletion of a small set—enabling dynamic programming, color-coding, or MSO logic.

Problems that are not minor-closed, not contraction-closed, or have complex global constraints can still fit within this unified framework, making it extremely broad in impact.

5. Theoretical Impact and Implications

The EPTAS methodology places a large class of combinatorial optimization problems on sparse graphs under a common algorithmic paradigm, where intractability is effectively "neutralized" by a local-to-global reduction underpinned by bounded treewidth reduction and low-depth structural properties (1005.5449).

Key theoretical consequences include:

• Provides running times of $2^{O(1/\varepsilon)} n + n^{O(1)}$ for many NP-hard graph problems where previously only quasi-polynomial or high-degree polynomial-time approximation was known.
• Supplies a conceptual explanation—via truly sublinear treewidth and the presence of small, "boundary" vertex sets—of why sparse, minor-closed graph classes admit efficient approximation for many hard problems.
• Generalizes and simplifies beyond known kernelization and parameterized algorithms, linking deep combinatorial properties (bidimensionality, excluded grid theorem) with practical approximation.

The EPTAS paradigm thus extends beyond traditional local problems to global, packing, or partial covering problems, including for graph classes not always closed under minors or contractions.

6. Implementation and Practical Considerations

Implementation of an EPTAS in the bidimensionality framework typically entails the following:

• Input: An $H$-minor-free graph $G$ and instance of a compatible (i.e., reducible and $\eta$-transversable) problem.
• Treewidth Reduction: Identify, in polynomial time, a set $X$ of at most $\varepsilon \cdot \mathrm{OPT}$ vertices whose removal yields $G\setminus X$ with treewidth at most $f(\varepsilon)$.
• Dynamic Programming: Solve the problem exactly on $G\setminus X$ via an algorithm exploiting the bounded treewidth, e.g., using dynamic programming on a tree decomposition, or algorithms for MSO-definable problems.
• Solution Synthesis: Combine this partial solution with optimal or near-optimal choices on $X$ (which, due to its small size, can be handled efficiently by brute force or enumeration).

Resource requirements are governed by the dependence of $f(1/\varepsilon)$ on the error parameter; for bounded $1/\varepsilon$, the running time is polynomial in $n$. The approach is practical for moderate values of $\varepsilon$ due to the absence of $1/\varepsilon$ in the exponent of $n$.

A simplified pseudocode illustrating the overall structure is:

def EPTAS(G, problem, epsilon): # Compute small set X, |X| <= epsilon * OPT, such that removing X leaves bounded treewidth X = find_transversal(G, epsilon) # Solve exactly on the bounded-treewidth graph solution_main = DP_on_treewidth_bounded(G - X, problem) # Solve exactly or approximately on X (brute-force or via small problem instance) solution_X = best_solution_on_X(X, problem) # Combine both solutions return combine(solution_main, solution_X)

The precise subroutines depend on the problem specification and the class of minor-free graphs under consideration.

7. Influence on Approximation Algorithm Design

The EPTAS framework anchored in bidimensionality and small-set removal has catalyzed a series of advances in the design of efficient approximation schemes for graph and network optimization problems. The imported concepts of reducibility, $\eta$-transversability, and truly sublinear treewidth are widely applicable.

This unification and extensibility inspire broader research:

• Extension to new problem types and graph classes (e.g., apex-minor-free, bounded-genus, or locally bounded treewidth graphs).
• Translation of meta-algorithmic insights to more direct, problem-specific approximations.
• The practical use of treewidth-reduction and dynamic programming across diverse areas—from network design to computational biology and social networks.

The EPTAS framework continues to inform both theoretical developments and practically efficient algorithms for large-scale, complex combinatorial problems on sparse graph classes.