Finding hidden Hamiltonian cycles

Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that "hide" the cycle. Is it possible to unravel the structure, that is, to efficiently find a Hamiltonian cycle in G? We describe an O(n 3 log n) steps algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the "trouble spots" of G by a collection of disjoint paths. (This is the hard part to analyze.) Part two of A extends this cover to a full cycle by the rotation-extension technique which is already classical for such problems

Finding hidden Hamiltonian cycles

Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that “hide ” the cycle. Is it possible to unravel the structure, that is, to efficiently find a Hamiltonian cycle in G? We describe an O(n 3 log n) steps algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the “trouble spots ” of G by a collection of disjoint paths. (This is the hard part to analyze.) Part two of A extends this cover to a full cycle by the rotation-extension technique which is already classical for such problems.

On a simple randomized algorithm for finding a 2-factor in sparse graphs

We analyze the performance of a simple randomized algorithm for finding 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of degree at most three. For the directed case, the algorithm finds a 2-factor in O(n 2) expected time. The analysis of our algorithm is based on random walks on the line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [15]. For the undirected case, the algorithm finds a 2-factor in O(n 3) expected time. We also analyze random versions of these graphs and show that cycles of length Ω(n / log n) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [4] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [13]

On a simple randomized algorithm for finding long cycles in sparse graphs

We analyze the performance of a simple randomized algorithm for finding long cycles and 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of de-gree at most three. For the directed case, the algorithm finds a 2-factor in O(n²) expected time. The analysis of our algorithm is based on random walks on a line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [15]. For the undirected case, the algorithm finds a 2-factor in O(n³) expected time. We also analyze random versions of these graphs and show that cycles of length Ω(n / log n) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [4] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [13]