A New Combinatorial Approach for Sparse Graph Problems

Abstract

Abstract. We give a new combinatorial data structure for representing arbitrary Boolean matrices. After a short preprocessing phase, the data structure can perform fast vector multiplications with a given matrix, where the runtime depends on the sparsity of the input vector. The data structure can also return minimum witnesses for the matrix-vector product. Our approach is simple and implementable: the data structure works by precomputing small problems and recombining them in a novel way. It can be easily plugged into existing algorithms, achieving an asymptotic speedup over previous results. As a consequence, we achieve new running time bounds for computing the transitive closure of a graph, all pairs shortest paths on unweighted undirected graphs, and finding a maximum node-weighted triangle. Furthermore, any asymptotic improvement on our algorithms would imply a o(n 3 / log 2 n) combinatorial algorithm for Boolean matrix multiplication, a longstanding open problem in the area. We also use the data structure to give the first asymptotic improvement over O(mn) for all pairs least common ancestors on directed acyclic graphs.

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

“…Note that while this term has been used very often (e.g., [12,52,1,37,21]), it is not a well-defined term. Usually it is vaguely used to refer as an algorithm that is different from the "algebraic" approach originated by Strassen [47]; see, e.g., [9,3,4].…”

Optimal distributed all pairs shortest paths and applications

“…Also a lot of effort was spent to obtain fast sequential algorithms for various versions of computing APSP or related problems such as the diameter problem, e.g. [3,4,7,13,39,40]. These algorithms are based on fast matrix multiplication such that currently the best runtime is O(n 2.3727 ) due to [46].…”

Regularity Lemmas and Combinatorial Algorithms

“…The first deals with multiplication of an arbitrary matrix with a sparse vector, and the second deals with multiplication of a sparse matrix with another (arbitrary) matrix. Theorem 2.3 (Blelloch-Vassilevska-Williams [13]) Let B be a n × n Boolean matrix and let w be the wordsize. Let κ ≥ 1 and > κ be integer parameters.…”

“…For instance, the best known algorithm for the general all-pairs shortest paths problem is combinatorial and runs in O(n 3 poly(log log n)/ log 2 n) time [15] -essentially the same time as Four Russians(!). Some progress on special cases of BMM has been made: for instance, in the sparse case where one matrix has m << n 2 nonzeros, there is an O(mn log(n 2 /m)/(w log n)) time algorithm [23], [13]. See [38], [45], [35] for a sampling of other partial results.…”