The Time Complexity of Fully Sparse Matrix Multiplication

What is the time complexity of matrix multiplication of sparse integer matrices with min nonzeros in the input and mout nonzeros in the output? This paper provides improved upper bounds for this question for almost any choice of min vs. mout, and provides evidence that these new bounds might be optimal up to further progress on fast matrix multiplication. Our main contribution is a new algorithm that reduces sparse matrix multiplication to dense (but smaller) rectangular matrix multiplication. Our running time thus depends on the optimal exponent ω(a, b, c) of multiplying dense n^a × n^b by n^b × n^c matrices. We discover that when mout = Θ(m^r_in) the time complexity of sparse matrix multiplication is O(m_in^σ+^ϵ), for all ϵ > 0, where σ is the solution to the equation ω(σ − 1, 2 − σ, 1 + r − σ) = σ. No matter what ω(·, ·, ·) turns out to be, and for all r ∈ (0, 2), the new bound beats the state of the art, and we provide evidence that it is optimal based on the complexity of the all-edge triangle problem. In particular, in terms of the input plus output size m = min + mout our algorithm runs in time O(m^1.3459). Even for Boolean matrices, this improves over the previous m ^{ω2+1 ω +ϵ} = O(m^1.4071) bound [Amossen, Pagh; 2009], which was a natural barrier since it coincides with the longstanding bound of all-edge triangle in sparse graphs [Alon, Yuster, Zwick; 1994]. We find it interesting that matrix multiplication can be solved faster than triangle detection in this natural setting. In fact, we establish an equivalence to a special case of the all-edge triangle problem.