Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Air Pollution and Mortality for 60 U.S. Cities in 1960. Data

Data includes measurements on mortality rate and explanatory variables (air-pollution, socio-economic and meteorological) for 60 US cities in 1960. This data was originally published in McDonald, G.C. and Schwing, R.C. (1973) 'Instabilities of regression estimates relating air pollution to mortality', Technometrics, vol.15, 463-482. It was redistributed through Carnegie Mellon University's StatLib (lib.stat.cmu.edu). Brief codebook notes that it is heavily used for research in ridge regression.PREC (Average annual precipitation in inches) -- JANT (Average January temperature in degrees F) -- JULT(Average July temperature in degrees F) -- OVR65 (Percentage of 1960 SMSA population aged 65 or older) POPN (Average household size) -- EDUC (Median school years completed by those over 22) -- HOUS (Percentage of housing units which are found and with all facilities) -- DENS (Population per sq. mile in urbanized areas, 1960) -- NONW (Percentage of non-white population in urbanized areas, 1960) -- WWDRK (Percentage employed in white collar occupations) -- POOR (Percentage of families with income <$3000) -- HC (Relative hydrocarbon pollution potential) -- NOX (Relative nitric oxide pollution potential) -- SO@ (Relative sulphur dioxide pollution potential) -- HUMID (Annual average percentage relative humidity at 1 p.m.) -- MORT (Total age-adjusted mortality rate per 100,000)Available as Microsoft Excel workbook (xslx); comma separated values (csv) and original plain text.Data set was downloaded with permission from the Carnegie Mellon StatLib (http://lib.stat.cmu.edu/datasets/pollution)Data set includes original text manifest and a small codebook explaining the 16 variables

Effect of gamma radiation on various petrochemical reactions : annual report, August 1955 to July, 1956

http://deepblue.lib.umich.edu/bitstream/2027.42/6468/5/bac8651.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/6468/4/bac8651.0001.001.tx

Effect of gamma radiation on various petrochemical reactions : semi-annual report, July, 1956 to January, 1957

http://deepblue.lib.umich.edu/bitstream/2027.42/6469/5/bac8652.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/6469/4/bac8652.0001.001.tx

Effect of radiation on chemical reactions.

http://deepblue.lib.umich.edu/bitstream/2027.42/6471/5/bac8631.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/6471/4/bac8631.0001.001.tx

THE USE OF RISK ANALYSIS IN FEDERAL REGULATORY AGENCIES: AN EXPLORATION -super-1

This paper reports on an exploratory study of the role of risk analysis in seven federal agencies concerned with health, safety, and environmental regulation. Interviews with key decision-makers suggest that the use of risk analysis has increased in recent years, but that its employment in risk management decision-making varies considerably both within given agencies and across agencies and policy areas. Despite methodological and institutional constraints, risk analysis is likely to play a greater role in the future as demands for comparative assessment of technological risks increase. Copyright 1982 by The Policy Studies Organization.