Popularity at minimum cost

We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph $G = (\mathcal{A}\cup\mathcal{B},E)$ , where $\mathcal{A}$ is a set of people, $\mathcal{B}$ is a set of items, and each person $a \in\mathcal{A}$ ranks a subset of items in order of preference, with ties allowed. The popular matching problem seeks to compute a matching M ^? between people and items such that there is no matching M where more people are happier with M than with M ^?. Such a matching M ^? is called a popular matching. However, there are simple instances where no popular matching exists. Here we consider the following natural extension to the above problem: associated with each item $b \in\mathcal{B}$ is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to “augment” G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of $\sqrt{n_{1}}/2$ , where n ₁ is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn ₁) time, where m is the number of edges.     

Max-coloring paths: tight bounds and extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G=(V,E) with vertex weights w such that $\sum_{i=1}^{k}\max_{v\in C_{i}}w(v_{i})$ is minimized, where C ₁,??,C _k are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V|+time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of ??(|V|log?|V|) in the algebraic computation tree model.     

Negotiating reform: young people's leadership in the educational arena

Youth organizing within the institutional context of community-based organizations has grown exponentially. Drawing on interviews with more than eighty organizers, youth, and educators, this article examines young people's experiences as they organize to expand educational opportunities for themselves and their peers in urban school districts. The authors explore educator responses to youth organizing and analyze how race- and class-based assumptions about youth leadership, as well as differing cultural norms between schools and youth organizing groups, pose challenges for young people fighting for education reform. The authors describe three strategies youth organizing groups use to address these challenges: intensive leadership development, targeted relationship building with district administrators, and alliance building. Implications for both educators and youth organizing groups are discussed.