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Where the really hard problems are
, 1991
"... It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P = NP). This paper shows that NPcomplete problems can be summarized by at least one "order parameter", and that the hard pr ..."
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Cited by 576 (1 self)
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It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P = NP). This paper shows that NPcomplete problems can be summarized by at least one "order parameter", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for Kcolorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bounds the cost). These results suggest a way of deciding if a problem is in P or NP and why they are different. 1
Can’t get no satisfaction
 AMERICAN SCIENTIST
, 1997
"... You are chief of protocol for the embassy ball. The crown prince instructs you either to invite Peru or to exclude Qatar. The queen asks you to invite either Qatar or Romania or both. The king, in a spiteful mood, wants to snub either Romania or Peru or both. Is there a guest list that will satisfy ..."
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Cited by 23 (0 self)
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You are chief of protocol for the embassy ball. The crown prince instructs you either to invite Peru or to exclude Qatar. The queen asks you to invite either Qatar or Romania or both. The king, in a spiteful mood, wants to snub either Romania or Peru or both. Is there a guest list that will satisfy the whims of the entire royal family? This contrived little puzzle is an instance of a problem that lies near the root of theoretical computer science. It is called the satisfiability problem, or SAT, and it was the first member of the notorious class known as NPcomplete problems. These are computational tasks that seem intrinsically hard, but after 25 years of effort no one has yet proved
A stochastic approach to stereo vision
 In AAAI
, 1986
"... A stochastic optimization approach to stereo matching is presented. Unlike conventional correlation matching and feature matching, the approach provides a dense array of disparities, eliminating the need for interpolation. First, the stereo matching problem is defined in terms of finding a disparit ..."
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Cited by 21 (0 self)
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A stochastic optimization approach to stereo matching is presented. Unlike conventional correlation matching and feature matching, the approach provides a dense array of disparities, eliminating the need for interpolation. First, the stereo matching problem is defined in terms of finding a disparity map that satisfies two competing constraints: (1) matched points should have similar image intensity, and (2) the disparity map should be smooth. These constraints are expressed in an ‘(energy” function that can be evaluated locally. A simulated annealing algorithm is used to find a disparity map that has very low energy (i.e., in which both constraints have simultaneously been approximately satisfied). Annealing allows the largescale structure of the disparity map to emerge at higher temperatures, and avoids the problem of converging too quickly on a local minimum. Results are shown for a sparse randomdot stereogram, a vertical aerial stereogram (shown in comparison to ground truth), and an oblique groundlevel scene with occlusion boundaries. 1
The Statistical Mechanics of kSatisfaction
 Advances in Neural Information Processing Systems
, 1994
"... The satisfiability of random CNF formulae with precisely k variables per clause ("kSAT") is a popular testbed for the performance of search algorithms. Formulae have M clauses from N variables, randomly negated, keeping the ratio ff = M=N fixed. For k = 2, this model has been proven to have a s ..."
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Cited by 14 (1 self)
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The satisfiability of random CNF formulae with precisely k variables per clause ("kSAT") is a popular testbed for the performance of search algorithms. Formulae have M clauses from N variables, randomly negated, keeping the ratio ff = M=N fixed. For k = 2, this model has been proven to have a sharp threshold at ff = 1 between formulae which are almost aways satisfiable and formulae which are almost never satisfiable as N ! 1. Computer experiments for k = 2, 3, 4, 5 and 6, (carried out in collaboration with B. Selman of ATT Bell Labs) show similar threshold behavior for each value of k. Finitesize scaling, a theory of the critical point phenomena used in statistical physics, is shown to characterize the size dependence near the threshold. Annealed and replicabased mean field theories give a good account of the results. Permanent address: IBM TJ Watson Research Center, Yorktown Heights, NY 10598 USA. (kirk@watson.ibm.com) Portions of this work were done while visiting the...
Where the REALLY Hard Problems Are
 In J. Mylopoulos and R. Reiter (Eds.), Proceedings of 12th International Joint Conference on AI (IJCAI91),Volume 1
, 1991
"... It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P 6= NP ). This paper shows that NPcomplete problems can be summarized by at least one "order parameter ", and that the hard problems ..."
Abstract

Cited by 1 (0 self)
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It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P 6= NP ). This paper shows that NPcomplete problems can be summarized by at least one "order parameter ", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for Kcolorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bound...
Evolutionary Local Search Algorithm for Portfolio Selection Problem: Spin Glass Based Approach
"... Nowadays, various imitations of natural processes are used to solve challenging optimization problems faster and more accurately. Spin glass based optimization, specifically, has shown strong local search capability and parallel processing. However, generally, spin glasses have a low rate of converg ..."
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Nowadays, various imitations of natural processes are used to solve challenging optimization problems faster and more accurately. Spin glass based optimization, specifically, has shown strong local search capability and parallel processing. However, generally, spin glasses have a low rate of convergence, since they use Monte Carlo simulation techniques such as simulated annealing (SA). Here, we investigate a new hybrid local search method based on spin glass for using adaptive distributed system capability, extremal optimization (EO) for using evolutionary locally search algorithm and SA for escaping from local optimum states. As shown in this paper, this strategy can lead to faster rate of convergence and improved performance than conventional SA and EO algorithm. The resulting are then used to solve the portfolio selection problem that is a nondeterministic polynomial complete (NPC) problem. This is confirmed by test results of five of the world's major stock markets, reliability test and phase transition diagram.