Faster Decomposition of Weighted Graphs into Cliques using Fisher's Inequality

Abstract

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 $k$, the goal is to decompose $G$ into at most $k$ (overlapping) weighted cliques so that an edge's weight is exactly equal to the sum of weights for cliques it participates in. They show EWCD is fixed-parameter-tractable, giving a $4^k$-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 $k \leq 11$. In this work, we establish reduction rules that exponentially decrease the size of the kernel (from $4^k$ to $k2^k$) 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. We deploy our kernelization and decomposition algorithms (together called DeCAF) on a corpus of biologically-inspired data and obtain over two orders of magnitude speed-up over cricca. As a result, DeCAF scales to instances with $k \geq 17$.