[Paper Review] Combining tabu search and graph reduction to solve the maximum balanced biclique problem

Yi Zhou, Jin‐Kao Hao|arXiv (Cornell University)|May 20, 2017

Plant biochemistry and biosynthesis3 citations

TL;DR

This paper proposes TSGR-MBBP, a novel hybrid algorithm combining constraint-based tabu search with two graph reduction techniques to solve the Maximum Balanced Biclique Problem (MBBP). It improves best-known results on 10 of 30 random benchmarks and proves optimality for 14 out of 25 large real-life graphs from KONECT, significantly accelerating convergence through effective pruning and unbalance-constrained search.

ABSTRACT

The Maximum Balanced Biclique Problem is a well-known graph model with relevant applications in diverse domains. This paper introduces a novel algorithm, which combines an effective constraint-based tabu search procedure and two dedicated graph reduction techniques. We verify the effectiveness of the algorithm on 30 classical random benchmark graphs and 25 very large real-life sparse graphs from the popular Koblenz Network Collection (KONECT). The results show that the algorithm improves the best-known results (new lower bounds) for 10 classical benchmarks and obtains the optimal solutions for 14 KONECT instances.

Motivation & Objective

• To address the computational challenge of solving the NP-hard Maximum Balanced Biclique Problem (MBBP) on both dense random graphs and large real-world sparse networks.
• To develop a heuristic algorithm that efficiently finds high-quality bicliques while maintaining balance between vertex sets in U and V.
• To integrate graph reduction techniques to shrink the search space and accelerate convergence in large-scale instances.
• To outperform existing state-of-the-art methods, including evolutionary algorithms and exact solvers like CPLEX, in both solution quality and runtime.
• To validate the effectiveness of unbalance-constrained tabu search and dual reduction rules on diverse graph types.

Proposed method

• The algorithm uses a Constraint-Based Tabu Search (CBTS) procedure that explores relaxed search space including unbalanced bicliques, with a specific unbalance constraint to guide the search toward balanced solutions.
• The CBTS employs a 'push' operator to perform 'add' and 'swap' moves, dynamically adjusting the current solution while avoiding cycles via tabu tenure.
• Two graph reduction techniques are applied after each improvement: (1) degree-based pruning of low-degree vertices, and (2) exact search on small subgraphs to detect and remove non-promising components.
• The reduction rules are applied iteratively to prune vertices and subgraphs, significantly reducing the search space for subsequent iterations.
• The algorithm maintains a lower bound on the maximum balanced biclique size and uses it to trigger reductions, ensuring global optimality when the graph is reduced to trivial size.
• The method is evaluated on 30 random dense graphs and 25 large sparse graphs from the KONECT collection, using both synthetic and real-world benchmarks.

Experimental results

Research questions

• RQ1 Can combining tabu search with graph reduction techniques significantly improve solution quality and convergence speed for the Maximum Balanced Biclique Problem?
• RQ2 How effective are the two proposed reduction rules—degree-based pruning and exact subgraph detection—in shrinking the search space for large sparse graphs?
• RQ3 Does the unbalance-constrained tabu search outperform standard tabu or greedy approaches in finding high-quality balanced bicliques?
• RQ4 Can the proposed algorithm discover new best-known solutions or prove optimality for previously unsolved instances in the KONECT benchmark set?
• RQ5 How does the performance of TSGR-MBBP compare to state-of-the-art methods like EA/SM and CPLEX in terms of solution quality and computational time?

Key findings

• TSGR-MBBP improved the best-known solution (new lower bound) for 10 out of 30 classical random benchmark instances.
• The algorithm proved optimal solutions for 14 out of 25 real-life sparse graphs from the KONECT collection, which had not been achieved before.
• On average, the variant using both reduction rules ('Reduction 1&2') converged faster and found higher-quality solutions earlier than the variant without reduction or with only one rule.
• For two instances—'dblp-author' and 'wiki-en-cat'—the algorithm reached the optimal solution in the very first iteration due to early graph pruning to trivial size.
• The algorithm outperformed CPLEX 12.6.1 in both solution quality and runtime, especially on large sparse graphs.
• The integration of graph reduction techniques led to significant vertex pruning, particularly on large-scale instances, confirming their effectiveness in reducing search space.