Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles

On the Autoreducibility of Random Sequences

A language A f0; 1g is called i.o. autoreducible if A is Turing-reducible to itself via a machine M such that, for infinitely many input words w, M does not query its oracle A about w. We examine the question if algorithmically random languages in the sense of Martin-Löf are i.o. autoreducible. We obtain the somewhat counterintuitive result that every algorithmically random language is polynomial-time i.o. autoreducible where the autoreducing machine poses its queries in a "quasi-nonadaptive" way; however, if in the above definition the "infinitely many" is replaced by "almost all," then every algorithmically random language is not autoreducible in this stronger sense. Further results obtained give upper and lower bounds on the number of queries of the autoreducing machine M and the number of inputs w for which M does not query the oracle about w

Sparse Hard Sets for P

Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for low-level complexity classes. In particular, the Hartmanis' sparseness conjectures for P and NL have been resolved. This article overviews the history of sparse hard set problems and exposes some of the recent results