[Paper Review] Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Stefan Fafianie, Hans L. Bodlaender+1 more|arXiv (Cornell University)|May 31, 2013

Algorithms and Data Compression3 citations

TL;DR

This paper evaluates the rank-based dynamic programming approach—using representative sets via Gaussian elimination—for the Steiner Tree problem on tree decompositions. It demonstrates that the method significantly reduces table sizes and speeds up computation, outperforming classic dynamic programming even on small treewidth instances, confirming its practical viability beyond theoretical gains.

ABSTRACT

Dynamic programming on tree decompositions is a frequently used approach to solve otherwise intractable problems on instances of small treewidth. In recent work by Bodlaender et al., it was shown that for many connectivity problems, there exist algorithms that use time, linear in the number of vertices, and single exponential in the width of the tree decomposition that is used. The central idea is that it suffices to compute representative sets, and these can be computed efficiently with help of Gaussian elimination. In this paper, we give an experimental evaluation of this technique for the Steiner Tree problem. A comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs the table reduction shows that the new approach gives significant improvements on the running time of the algorithm and the size of the tables computed by the dynamic programming algorithm, and thus that the rank based approach from Bodlaender et al. does not only give significant theoretical improvements but also is a viable approach in a practical setting, and showcases the potential of exploiting the idea of representative sets for speeding up dynamic programming algorithms.

Motivation & Objective

• To evaluate the practical performance of the rank-based dynamic programming approach for the Steiner Tree problem.
• To compare the runtime and table size efficiency of classic dynamic programming (CDP) versus the representative set-based algorithm (RBA).
• To assess whether theoretical improvements in time complexity translate into measurable speedups in real-world instances.
• To analyze the impact of tree decomposition structure on algorithm performance beyond treewidth alone.
• To explore the feasibility of applying representative set techniques to exact algorithms for NP-hard problems.

Proposed method

• The rank-based approach uses representative sets to maintain only the most promising partial solutions during dynamic programming on tree decompositions.
• It applies Gaussian elimination on a matrix encoding feasible extensions of partial solutions to compute minimal, representative sets of rows.
• The method leverages the concept of representativity and linear algebra to reduce table sizes while preserving optimality.
• For each bag in the tree decomposition, the algorithm computes a reduced table using a basis of the extension matrix, minimizing redundant entries.
• The approach is implemented and compared against classic dynamic programming (CDP) and two variants (RBA, RBC) on benchmark and random graphs.
• Performance is measured by runtime, table size, and success rate within time limits, with detailed tracking of time spent on table filling and Gaussian elimination.

Experimental results

Research questions

• RQ1 Does the rank-based approach with representative sets provide measurable runtime improvements over classic dynamic programming for the Steiner Tree problem?
• RQ2 How does the size of intermediate tables compare between classic and representative set-based dynamic programming?
• RQ3 To what extent does the structure of the tree decomposition (e.g., number of join bags) affect the performance of the rank-based algorithm?
• RQ4 Is the time spent on Gaussian elimination for table reduction outweighed by the gains from smaller table sizes?
• RQ5 Can the rank-based method solve instances where classic dynamic programming times out, especially as treewidth increases?

Key findings

• The rank-based approach (RBA) achieves significant speedups over classic dynamic programming (CDP), with the performance gain increasing with treewidth.
• Table sizes are drastically reduced—e.g., for treewidth 11, CDP stores 860,867 partial solutions, while RBA reduces this to 80,694 in the es500fst05 instance.
• The time spent on Gaussian elimination is minimal compared to the time saved by smaller tables, making the overhead negligible.
• For the i080-004 instance, the rank-based approach succeeded where CDP failed within the time limit, especially due to the high number of join bags in the decomposition.
• The reduction in table size is most pronounced at higher treewidths, confirming the method’s scalability and practical advantage.
• The RBA and RBC variants show identical performance, indicating that the choice of basis representation has little impact on runtime in practice.