K-balanced biclique partition: Kernelization and efficient algorithms

Balanced signed biclique captures cohesive friend-foe relations in social and biological networks. We initiate the study of the k-Balanced Biclique Partition (k-BBP): given a signed bipartite graph, partition its edge set into at most k balanced signed bicliques. First, we prove that deciding whether k-BBP exists is NP-hard by a polynomial reduction from Non-negative Matrix Factorization. We present the kernelization that contracts the input graph to a kernel with no more than 3^2k edges, ensuring polynomial preprocessing time. We relax the decision problem to a property-testing variant of k-BBP, designing a parameterized one-sided tester that runs in time independent of the input size. The tester accepts graphs admitting the desired partition and, with high probability, rejects graphs that are ϵ-far from having any such partition. Next, we study how to find an approximate k-BBP with a guaranteed error bound. We first recast the problem as a constrained discrete optimization problem and devise an alternating optimization algorithm with sub-exponential time complexity. We further relax the problem to a continuous optimization setting. Leveraging the multi-block convex objective, we design a linear-time approximation algorithm.