Investigation of land use of northern megalopolis using ERTS-1 imagery

Primary objective was to produce a color-coded land use map and digital data base for the northern third of Megalopolis. Secondary objective was to investigate possible applications of ERTS products to land use planning. Many of the materials in this report already have received national, dissemination as a result of unexpected interest in land use surveys from ERTS. Of special historical interest is the first comprehensive urban-type land use map from space imagery, which covered the entire state of Rhode Island and was made from a single image taken on 28 July 1972

Random Costs in Combinatorial Optimization

The random cost problem is the problem of finding the minimum in an exponentially long list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. This explains the bad performance of heuristic approaches to the number partitioning problem and allows us to calculate the probability distributions of the optimum and sub-optimum costs.Comment: 4 pages, Revtex, 2 figures (eps), submitted to PR

Number partitioning as random energy model

Number partitioning is a classical problem from combinatorial optimisation. In physical terms it corresponds to a long range anti-ferromagnetic Ising spin glass. It has been rigorously proven that the low lying energies of number partitioning behave like uncorrelated random variables. We claim that neighbouring energy levels are uncorrelated almost everywhere on the energy axis, and that energetically adjacent configurations are uncorrelated, too. Apparently there is no relation between geometry (configuration) and energy that could be exploited by an optimization algorithm. This ``local random energy'' picture of number partitioning is corroborated by numerical simulations and heuristic arguments.Comment: 8+2 pages, 9 figures, PDF onl

Analysis of the Karmarkar-Karp Differencing Algorithm

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling $n^{-c\ln n}$, where $c=1/(2\ln2)=0.7213...$. Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.Comment: 9 pages, 8 figures; minor change

Landscape Encodings Enhance Optimization

Hard combinatorial optimization problems deal with the search for the minimum cost solutions (ground states) of discrete systems under strong constraints. A transformation of state variables may enhance computational tractability. It has been argued that these state encodings are to be chosen invertible to retain the original size of the state space. Here we show how redundant non-invertible encodings enhance optimization by enriching the density of low-energy states. In addition, smooth landscapes may be established on encoded state spaces to guide local search dynamics towards the ground state

Easily Searched Encodings for Number Partitioning

Can stochastic search algorithms outperform existing deterministic heuristics for the NP-hard problem if given a su#cient, but practically realizable amount of time? In a thorough empirical investigation using a straightforward implementation of one such algorithm, simulated annealing, Johnson et al. #1991# concluded tentatively that the answer is #no." In this paper we show that the answer can be #yes" if attention is devoted to the issue of problem representation #encoding#. We present results from empirical tests of several encodings of Number Partitioning with problem instances consisting of multiple-precision integers drawn from a uniform probability distribution. With these instances and with an appropriate choice of representation, stochastic and deterministic searches can---routinely and in a practical amount of time---#nd solutions several orders of magnitude better than those constructed by the best heuristic known #Karmarkar and Karp, 1982#, which does not employ searching. The choice of encoding is found to be more important than the choice of search technique in determining search e#cacy. Three alternative explanations for the relative performance of the encodings are tested experimentally. The best encodings tested are found to contain a high proportion of good solutions; moreover, in those encodings, the solutions are organized into a single #bumpy funnel" centered at a known position in the search space. This is likely to be the only relevant structure in the search space because a blind search performs as well as any other search technique tested when the search space is restricted to the funnel tip. To appear in the Journal of Optimization Theory and Applications, 1995. This work may not be copied or reproduced in whole or in part for any commercial ..