[Paper Review] Tree Decomposition based Steiner Tree Computation over Large Graphs

Fang Wei-Kleiner|arXiv (Cornell University)|May 24, 2013

Advanced Graph Theory Research4 references3 citations

TL;DR

This paper presents STEIN I, an exact algorithm for the Steiner tree problem on large graphs using tree decomposition-based index structures. By pre-computing sub-Steiner trees over bags of a tree decomposition and leveraging vertex separators, the algorithm achieves a time complexity of $O(h \cdot (2tw)^{|S|})$, enabling efficient computation on large graphs with bounded treewidth and a small number of terminals.

ABSTRACT

In this paper, we present an exact algorithm for the Steiner tree problem. The algorithm is based on certain pre-computed index structures. Our algorithm offers a practical solution for the Steiner tree problems on graphs of large size and bounded number of terminals.

Motivation & Objective

• Address the scalability limitations of exact Steiner tree algorithms on large graphs, where $|V|^3$ dominates runtime in traditional methods.
• Overcome the intractability of computing optimal treewidth in practice by leveraging practical width bounds $w \ll |V|$ common in Web-scale graphs.
• Enable efficient exact computation of Steiner trees by pre-computing sub-Steiner trees over tree decomposition bags using dynamic programming and vertex separators.
• Provide a practical solution for real-world workloads such as keyword search and social network analysis, where graphs are large but terminal sets are small.

Proposed method

• Construct a tree decomposition of the input graph $G$ with treewidth $tw$, using a practical approximation to ensure $tw \ll |V|$.
• Pre-compute all sub-Steiner trees for terminal subsets within each bag of the decomposition using a bottom-up traversal.
• Leverage vertex separators $X_c$ in the tree decomposition to recursively split Steiner tree computation into smaller subproblems.
• Use the STVS function to compute Steiner trees for terminal sets involving a vertex $v$ and a separator $X_c$, ensuring correctness via $(v, v_i)$-vertex separator guarantees.
• Perform incremental Steiner tree generation by inserting vertices from $X_c \cup S_{X_c}$ in order, reusing pre-computed trees from smaller subsets.
• Apply merge operations at internal nodes of the tree decomposition to combine Steiner tree sets from child bags, using the parent bag as a vertex separator.

Experimental results

Research questions

• RQ1 Can tree decomposition-based indexing enable exact Steiner tree computation on large graphs where traditional $O(|V|^3)$ algorithms are infeasible?
• RQ2 How can vertex separators in a tree decomposition be exploited to decompose the Steiner tree problem into smaller, manageable subproblems?
• RQ3 What is the time complexity of the proposed algorithm in terms of treewidth $tw$, number of terminals $|S|$, and height $h$ of the decomposition?
• RQ4 To what extent can the algorithm scale on real-world graphs with $tw \ll |V|$ but large $|V|$?
• RQ5 Is it feasible to pre-compute sub-Steiner trees offline such that online query response remains efficient?

Key findings

• The algorithm STEIN I achieves a time complexity of $O(h \cdot (2tw)^{|S|})$, which is polynomial in $tw$ and $|S|$ when $|S|$ is small.
• The space complexity of the index structure is $O(tw^{|S|} \cdot |V|)$, making it feasible for large graphs with bounded treewidth.
• The height $h$ of the tree decomposition is typically much smaller than $|V|$, often close to $\log|V|$ in practice, improving scalability.
• The algorithm remains efficient even for graphs with hundreds of millions of vertices, provided $tw \ll |V|$ and $|S|$ is small.
• Pre-computation of sub-Steiner trees over decomposition bags enables fast online query resolution, making the method practical for real-world workloads.
• The correctness of the algorithm is formally proven by showing that each vertex separator $X_c$ in the decomposition serves as a valid $(v, v_i)$-vertex separator, ensuring correct recursive decomposition.