On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.Comment: 16 pages, 10 figure

Majority choosability of digraphs

A \emph{majority coloring} of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ has the same color as $v$. A digraph $D$ is \emph{majority $k$-choosable} if for any assignment of lists of colors of size $k$ to the vertices there is a majority coloring of $D$ from these lists. We prove that every digraph is majority $4$-choosable. This gives a positive answer to a question posed recently by Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain this result as a consequence of a more general theorem, in which majority condition is profitably extended. For instance, the theorem implies also that every digraph has a coloring from arbitrary lists of size three, in which at most $2/3$ of the out-neighbors of any vertex share its color. This solves another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture stating that every digraph is majority $3$-colorable

Recovery Network Awareness: A Training Guide to Help Clients Choose an Aftercare Program for Sobriety

Abstract Recovery environments are a crucial role in any individual’s journey to achieve sobriety. A safe environment will allow people who have a history of substance use to work their program effectively and decrease their relapse potential. The purpose of this paper is to identify multiple recovery-based programs for new professionals and providers entering the field of substance abuse treatment that allow their clients to have the best opportunity to succeed with their personal goals. The main programs discussed in this paper include 12 step programs, SMART Recovery, Harm reduction, and Medication Assisted Therapy. Choosing the right program for clients can be a challenge and the information provided in this paper will help identify interventions that align with the client’s core beliefs for them to have the autonomy to choose what they feel is the best route for their recovery. A training (Recovery Networking) will be provided that corresponds to the topics discussed in the literature review. Keywords: recovery, aftercare, alternative

Gender Of Perpetrator, Gender Of Victim, And Relationship Between Perpetrator And Victim As Factors Influencing How Adults View Coercive Sexual Behavior In Childhood

Thesis (Ph.D.) University of Alaska Fairbanks, 2002The sexual abuse of children by adults is a serious social problem. Some sexually abused children become sexually abusive toward others. This is sometimes called coercive sexual behavior, and little is known about how adults view these acts. A better understanding of how adults view coercive sexual behavior between children is critical due to the harm it causes victims, perpetrators, and society. Also, parents are typically held legally responsible for their minor children, and it is their responsibility to intervene in this type of behavior. Three hundred and eighty-five college students participated in a study that examined descriptions of coercive sexual behavior between elementary school-aged children. This study used a 2 x 2 x 2 factorial design to examine how gender of a child perpetrator, gender of a child victim, and relationship between a child perpetrator and child victim (peer or sibling) influence how adults view coercive sexual behavior in childhood. Participants read one of eight vignettes describing an incident of coercive sexual behavior between two children and answered a twenty-eight-item questionnaire based on it. Data was analyzed using correlation coefficients, factor analysis, and multivariate analysis of variance (MANOVA). Findings from the present study suggest that the gender of the children and the relationship between them are factors influencing how adults view coercive sexual behavior in childhood

A Tight Bound for Shortest Augmenting Paths on Trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree $T=(W \uplus B, E)$ is being revealed online, i.e., in each round one vertex from $B$ with its incident edges arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis, R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with augmentations. In INFOCOM 2009] that the total length of all shortest augmenting paths found is $O(n \log n)$. In this paper, we prove a tight $O(n \log n)$ upper bound for the total length of shortest augmenting paths for trees improving over $O(n \log^2 n)$ bound [B. Bosek, D. Leniowski, P. Sankowski, and A. Zych. Shortest augmenting paths for online matchings on trees. In WAOA 2015].Comment: 22 pages, 10 figure

On the Duality of Semiantichains and Unichain Coverings

We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure

An extremal problem on crossing vectors

For positive integers $w$ and $k$, two vectors $A$ and $B$ from $\mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]\geq k$ and $B[j]-A[j]\geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $\mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $w\leq 3$ and provide weaker upper bounds for $w\geq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.Comment: Corrections and improvement