An Exponentially Smaller Kernel for Exact Weighted Clique Decomposition.

Mining groups of genes that consistently co-express is an important problem in biomedical research, where it is critical for applications such as drug-repositioning and designing new disease treatments. Recently, Cooley et al. modeled this problem as Exact Weighted Clique Decomposition (EWCD) in which, given an edge-weighted graph G and a positive integer κ, the goal is to decompose G into at most κ (overlapping) weighted cliques so that an edge's weight is exactly equal to the sum of weights for cliques it participates in. They show that EWCD is fixed-parameter-tractable, giving a 4κ-kernel alongside a backtracking algorithm (together called cricca) to iteratively build a decomposition. Unfortunately, because of inherent exponential growth in the space of potential solutions, cricca is typically able to decompose graphs only when κ ≤ 11. In this work, we establish reduction rules that exponentially decrease the size of the kernel (from 4κ to κ2κ) for EWCD. In addition, we use insights about the structure of potential solutions to give new search rules that speed up the decomposition algorithm. At the core of our techniques is a result from combinatorial design theory called Fisher's inequality characterizing set systems with restricted intersections. Experimental evaluation of our kernelization and decomposition algorithms (together called DeCAF) on a corpus of biologically-inspired data showed that in most cases DeCAF leads to over 80% reduction in the size of the kernel and orders of magnitude improvement in the time required to obtain a decomposition. As a result, DeCAF scales to instances with κ ≥ 17.* This work was supported by the NIH R01 HG010067, the Gordon & Betty Moore Foundation under award GBMF4560, and the National Science Foundation under Grant # 2127309 to the Computing Research Association for the CIFellows 2021 Project. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reect the views of the National Science Foundation or the Computing Research Association.