Fast Hamiltonicity Checking Via Bases of Perfect Matchings

Abstract

For an even integer t ≥ 2, the Matching Connectivity matrix H t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph on t vertices; an entry H t [ M 1 , M 2 ] is 1 if M 1 and M 2 form a Hamiltonian cycle and 0 otherwise. Motivated by applications for the Hamiltonicity problem, we show that H t has rank exactly 2 t /2−1 over GF(2). The upper bound is established by an explicit factorization of H t as the product of two submatrices; the matchings labeling columns and rows, respectively, of the submatrices therefore form a basis X t of H t . The lower bound follows because the 2 t /2−1 × 2 t /2−1 submatrix with rows and columns labeled by X t can be seen to have full rank. We obtain several algorithmic results based on the rank of H t and the particular structure of the matchings in X t . First, we present a 1.888 n n O (1) time Monte Carlo algorithm that solves the Hamiltonicity problem in directed bipartite graphs. Second, we give a Monte Carlo algorithm that solves the problem in (2 + √ 2) pw n O (1) time when provided with a path decomposition of width pw for the input graph. Moreover, we show that this algorithm is best possible under the Strong Exponential Time Hypothesis, in the sense that an algorithm with running time (2 + √2 − ϵ) pw n O (1) , for any ϵ > 0, would imply the breakthrough result of a (2 − ϵ ′ ) n -time algorithm for CNF-Sat for some ϵ ′ > 0.