[Paper Review] A O(c^k n) 5-Approximation Algorithm for Treewidth

Hans L. Bodlaender, Pål Grønås Drange+4 more|arXiv (Cornell University)|Apr 23, 2013

Complexity and Algorithms in Graphs3 references10 citations

TL;DR

This paper presents the first constant-factor approximation algorithm for treewidth that runs in time $2^{O(k)}n$, where $k$ is the treewidth and $n$ is the number of vertices. It either certifies that the treewidth exceeds $k$ or constructs a tree decomposition of width at most $5k+4$, achieving a 5-approximation with single-exponential dependence on $k$ and linear dependence on $n$. This resolves a long-standing bottleneck in treewidth computation for dynamic programming on graphs of bounded treewidth.

ABSTRACT

We give an algorithm that for an input n-vertex graph G and integer k>0, in time 2^[O(k)]n either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k+4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single-exponential in k and linear in n. Treewidth based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single-exponential in the treewidth and linear in the input size.

Motivation & Objective

• To develop a constant-factor approximation algorithm for treewidth with running time single-exponential in $k$ and linear in $n$.
• To overcome the long-standing bottleneck in treewidth computation, where prior algorithms either had higher time complexity or did not achieve constant-factor approximation with linear $n$ dependence.
• To provide a practical and theoretically efficient subroutine for treewidth-based algorithms that require a tree decomposition of small width.

Proposed method

• The algorithm uses a recursive separator-based approach, repeatedly identifying small separators in subgraphs to guide tree decomposition construction.
• It employs a novel data structure that supports queries in $O(c^k \log n)$ time for some constant $c$, enabling efficient dynamic programming over tree decompositions.
• The method leverages a path decomposition of the tree decomposition and computes intermediate tables $D[i_t]$ along a root-to-leaf path to track valid separations and their sizes.
• It uses signature and interface-based dynamic programming to maintain consistency across tree decomposition nodes, ensuring correct propagation of separation information.
• The algorithm combines left- and right-pushed separations to detect $\frac{8}{9}$-separations in subgraphs, which are used to verify the existence of low-width decompositions.
• A dummy table is introduced at the base of the recursion to enforce terminal constraints and ensure correct initialization of separation tracking.

Experimental results

Research questions

• RQ1 Can a constant-factor approximation for treewidth be computed in time $2^{O(k)}n$?
• RQ2 Is it possible to design a treewidth approximation algorithm that is both single-exponential in $k$ and linear in $n$?
• RQ3 Can such an algorithm be used as a subroutine to speed up dynamic programming on graphs of bounded treewidth?
• RQ4 What is the best possible approximation ratio achievable under single-exponential time and linear $n$ dependence?

Key findings

• The algorithm achieves a 5-approximation for treewidth, constructing a tree decomposition of width at most $5k+4$ when the treewidth is at most $k$.
• The running time is $2^{O(k)}n$, which is the first algorithm to achieve both single-exponential dependence on $k$ and linear dependence on $n$ for a constant-factor approximation.
• An intermediate algorithm is presented that achieves a $3k+4$-approximation in time $2^{O(k)}n\log n$, demonstrating a trade-off between approximation ratio and efficiency.
• The algorithm uses a novel data structure and dynamic programming over tree decomposition paths to support efficient queries in $O(c^k \log n)$ time per query.
• The approach enables the design of faster algorithms for problems on bounded-treewidth graphs by replacing exact treewidth computation with efficient approximation.
• The result resolves a long-standing open problem in parameterized algorithms, where prior algorithms either had higher time complexity or did not achieve constant-factor approximation with linear $n$ dependence.