Minimum-weight triangulation is NP-hard

A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures. This revision contains a few improvements in the expositio

The Cowles Commission and Foundation for Research in Economics

Founded in 1932 by a newspaper heir disillusioned by the failure of forecasters to predict the Great Crash, the Cowles Commission promoted the use of formal mathematical and statistical methods in economics, initially through summer research conferences in Colorado and through support of the Econometric Society (of which Alfred Cowles was secretary-treasurer for decades). After moving to the University of Chicago in 1939, the Cowles Commission sponsored works, many later honored with Nobel Prizes but at the time out of the mainstream of economics, by Haavelmo, Hurwicz and Koopmans on econometrics, Arrow and Debreu on general equilibrium, Yntema and Mosak on general equilibrium in international trade theory, Arrow on social choice, Koopmans on activity analysis, Klein on macroeconometric modelling, Lange, Marschak and Patinkin on macroeconomic theory, and Markowitz on portfolio choice, but came into intense methodological, ideological and personal conflict with the emerging “Chicago school.” This conflict led the Cowles Commission to move to Yale in 1955 as the Cowles Foundation, directed by James Tobin (who had declined to move to Chicago to direct it). The Cowles Foundation remained a leader in the more technical areas of economics, notably with Tobin’s “Yale school” of monetary theory, Scarf’s computable general equilibrium, Shubik in game theory, and later Phillips and Andrews in econometric theory but as formal methods in economic theory and econometrics pervaded the discipline of economics, Cowles (like the Econometric Society) became less distinct from the rest of economics. This entry is part of an archivally-based history of the Cowles Commission and Foundation commissioned by the Cowles Foundation. This paper is the entry on “The Cowles Commission and Foundation for Research in Economics” in The New Palgrave Online https://link.springer.com/referencework/10.1057/978-1-349-95121-5 and is included as a Cowles Foundation Discussion Paper by the kind permission of Springer Nature

Private epistemic virtue, public vices : moral responsibility in the policy sciences

In this chapter we address what we call "The-Everybody-Did-It" (TEDI) Syndrome, a symptom for collective negligence. Our main thesis is that the character of scientific communities can be evaluated morally and be found wanting in terms of moral responsibility. Even an epistemically successful scientific community can be morally responsible for consequences that were unforeseen by it and its members and that follow from policy advice given by its individual members. We motivate our account by a critical discussion of a recent proposal by Heather Douglas. We offer three, related criticisms of Douglas's account. First, she assumes that scientific fields are communicative communities. Second, in a system where the scientific community autonomously sets standards, there is a danger of self-affirming reasoning. Third, she ignores that the character of a scientific community is subject to moral evaluation. We argue that these omissions in Douglas's theory leave it with no adequate response to TEDI Syndrome. Moreover, we deny that science ought to be characterized by unanimity of belief among its competent practitioners, this leads easily to the vices of close-mindedness and expert-overconfidence. If a scientific community wishes to avoid these vices it should create conditions for an active pluralism when it and its members aspire to the position of rational policy decision-making