An O(n) time algorithm for finding Hamilton cycles with high probability

Abstract

We design a randomized algorithm that finds a Hamilton cycle in O(n) time with high probability in a random graph G n,p with edge probability p ≥ C log n/n. This closes a gap left open in a seminal paper by Angluin and Valiant from 1979.

A fast algorithm on average for solving the Hamilton Cycle problem

“…At the setup of Theorem 1.1 we assumed that we are given the adjacency matrix of G. In another line of research the Hamilton cycle problem with input G ∼ G(n, p) is consider at the setup where G is given in the form of randomly ordered adjacency lists. That is for every vertex v ∈ [n] we are given a random permutation of its neighbors via a list L(v) (see [2], [5], [16]). In [2] CerHam1' is presented, an algorithm that with probability 1 − o(n −7 ) solves HAM with input G ∼ G(n, p), in this model, in O(n) time, p ≥ 0.…”

Fast algorithms for solving the Hamilton Cycle problem with high probability

“…in G(n, p) for p ≥ C log n n where C is a sufficiently large constant. Their result was first improved with respect to p first by Shamir [22] and then by Bollobás, Fenner and Frieze [8] whose result is optimal with respect to p. Recently it was improved by Nenadov, Steger and Su with respect to the running time [20]. We summarize the above results at the table below (taken from [20]).…”

“…matrix Alon, Krivelevich [2] '20 O(n/p) p ≥ 70n −1/2 adj. matrix Nenadov, Steger, Su [20] '21+ O(n) p ≥ C 3 log n n adj. list…”

“…= −Case c: probability p 2,i (1 − p 2,i )(1 + O(m −1 i )). ≤ −1 + d(w, Z 2 ) + O(f(w)) Case d: otherwise(20)…”