On rr-Simple kk-Path

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 $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ 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 $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 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

Treatment of childhood encopresis — a review

Children with encopresis may present to a number of different professionals. The literature on different treatment methods is reviewed. The roles of verbal psychotherapy, physical treatment, behaviour therapy and mixed treatment programmes are discussed

Dynamic integration of classifiers in the space of principal components

Recent research has shown the integration of multiple classifiers to be one of the most important directions in machine learning and data mining. It was shown that, for an ensemble to be successful, it should consist of accurate and diverse base classifiers. However, it is also important that the integration procedure in the ensemble should properly utilize the ensemble diversity. In this paper, we present an algorithm for the dynamic integration of classifiers in the space of extracted features (FEDIC). It is based on the technique of dynamic integration, in which local accuracy estimates are calculated for each base classifier of an ensemble, in the neighborhood of a new instance to be processed. Generally, the whole space of original features is used to find the neighborhood of a new instance for local accuracy estimates in dynamic integration. In this paper, we propose to use feature extraction in order to cope with the curse of dimensionality in the dynamic integration of classifiers. We consider classical principal component analysis and two eigenvector-based supervised feature extraction methods that take into account class information. Experimental results show that, on some data sets, the use of FEDIC leads to significantly higher ensemble accuracies than the use of plain dynamic integration in the space of original features. As a rule, FEDIC outperforms plain dynamic integration on data sets, on which both dynamic integration works (it outperforms static integration), and considered feature extraction techniques are able to successfully extract relevant features