An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4 log k/ log r. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex.We then give a randomized algorithm that runs in time poly(n) · 2 O(k·log r/r) that solves the r-SIMPLE k-PATH on a graph with n vertices with one-sided error. We also show that a randomized algorithm with running time poly(n) · 2 (c/2)k/r with c < 1 gives a randomized algorithm with running time poly(n) · 2 cn for the Hamiltonian path problem in a directed graph -an outstanding open problem. So in a sense our algorithm is optimal up to an O(log r) factor.

In this paper, we study the problem of learning a monotone DNF with at most s terms of size (number of variables in each term) at most r (s term r-MDNF) from membership queries. This problem is equivalent to the problem of learning a general hypergraph using hyperedge-detecting queries, a problem motivated by applications arising in chemical reactions and genome sequencing.We first present new lower bounds for this problem and then present deterministic and randomized adaptive algorithms with query complexities that are almost optimal. All the algorithms we present in this paper run in time linear in the query complexity and the number of variables n. In addition, all of the algorithms we present in this paper are asymptotically tight for fixed r and/or s.

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