The Complexity of the List Partition Problem for Graphs

The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (i≠j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete

Dear Karlin

This is a letter to the neighborhood I lived in in Prague during my Geography FSP study abroad, where I found a ‘home’ in the city

The Global Warming Case: Massachusetts v. Environmental Protection Agency

On November 29, 2006, the Supreme Court heard oral arguments in Massachusetts v. Environmental Protection Agency, the decision of which may have major implications for the regulation of carbon dioxide (“CO2”) and other greenhouse gases (“GHGs”). The case addresses whether the Environmental Protection Agency (“EPA”) has statutory authority under the Clean Air Act (“CAA”) to regulate CO2 and other GHGs emitted by new motor vehicles, and if it does, whether such authority is mandatory or discretionary. The case was brought by twelve states, three cities, an American territory, and various environmental organizations. Although the Court is unlikely to take a stand on the scientific legitimacy of climate change, its decision will have important implications for future climate-related claims, specifically regarding standing and regulatory issues