The Maximum Traveling Salesman Problem with Submodular Rewards

In this paper, we look at the problem of finding the tour of maximum reward on an undirected graph where the reward is a submodular function, that has a curvature of $\kappa$, of the edges in the tour. This problem is known to be NP-hard. We analyze two simple algorithms for finding an approximate solution. Both algorithms require $O(|V|^3)$ oracle calls to the submodular function. The approximation factors are shown to be $\frac{1}{2+\kappa}$ and $\max\set{\frac{2}{3(2+\kappa)},2/3(1-\kappa)}$, respectively; so the second method has better bounds for low values of $\kappa$. We also look at how these algorithms perform for a directed graph and investigate a method to consider edge costs in addition to rewards. The problem has direct applications in monitoring an environment using autonomous mobile sensors where the sensing reward depends on the path taken. We provide simulation results to empirically evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy bound with curvature

Distributed Dominating Sets on Grids

This paper presents a distributed algorithm for finding near optimal dominating sets on grids. The basis for this algorithm is an existing centralized algorithm that constructs dominating sets on grids. The size of the dominating set provided by this centralized algorithm is upper-bounded by $\lceil\frac{(m+2)(n+2)}{5}\rceil$ for $m\times n$ grids and its difference from the optimal domination number of the grid is upper-bounded by five. Both the centralized and distributed algorithms are generalized for the $k$-distance dominating set problem, where all grid vertices are within distance $k$ of the vertices in the dominating set.Comment: 10 pages, 9 figures, accepted in ACC 201

Middle and elementary school students’ changes in self-determined motivation in a basketball unit taught using the Tactical Games Model

Studies examining student motivation levels suggest that this is a significant factor in students’ engagement in physical education and may be positively affected when teachers employ alternative pedagogical models such as game-centered approaches (GCAs). The aim of this study was to investigate changes in self-determined motivation of students as they participated in a GCA-basketball unit taught using the Tactical Games Model (TGM). Participants were 173 students (84 girls), 79 middle school (45 girls) and 94 (39 girls) elementary school students from four seventh and five fourth/fifth grade co-educational classes. Two teachers taught 32 (middle) and 33 (elementary) level one TGM basketball lessons. Need satisfaction and self-determined motivation data were collected using a previously validated instrument, while lesson context and teacher behavior data were recorded using systematic observation instruments. Repeated measures MANOVAs were employed to examine pre-posttest differences. Results revealed a significant main effect for time in need satisfaction for both middle (relatedness increased) and elementary school students (autonomy decreased) and a significant main effect in self-determined motivation for middle school students only (introjected regulation, external regulation, and amotivation all increased). Approximately 48%/42% (middle/elementary) of lesson time was game play, 22%/22% skill practice, 17%/17% management, and 13%/19% knowledge. The primary teacher behaviors used were instruction, management, specific observation, corrective feedback and modelling. Results indicate that it is important for future research to pay greater attention to the contextual factors associated with the application of the TGM, such as the students’ previous exposure to TGM lessons, and the teachers’ training and experience in utilizing the TGM. Indeed, results of the present study demonstrate that a longer-term commitment to the TGM is necessary to reduce controlling teacher behaviors, which will lead to positive changes in students’ need satisfaction and self-determined motivation. Future research is therefore needed to embrace this challenge to provide an increased evidence-base for GCAs such as the TGM