# Physicists propose solution to constraint satisfaction problems

Their paper, "Optimization hardness as transient chaos in an analog approach to constraint satisfaction," was published this week in the journal *Nature Physics*.

The approach proposed by Ercsey-Ravasz and Toroczkai involves Boolean satisfiability (k-SAT), one of the most studied optimization problems. It applies to a vast range of decision-making, scheduling and operations research problems from drawing delivery routes to the placement of circuitry elements in microchip design. The formal proof for the existence or non-existence of fast and efficient (polynomial-time) algorithms to solve such problems constitutes the famous P vs. NP problem. It is one of the six greatest unsolved problems of mathematics, called Millennium Prize Problems, with $1 million allocated to the solution of each problem, awarded by the Clay Institute of Mathematics.

The paper proposes a mapping of k-SAT into a deterministic continuous-time (that is, analog) dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. It shows that as the constraints are increased, i.e., as the problems become harder, the analog trajectories of the system become transiently chaotic, signaling the appearance of optimization hardness.

The proposed dynamical system always finds solutions if they exist, including for problems considered among the hardest algorithmic benchmarks. It finds these solutions in polynomial continuous-time, however, at the cost of trading search times for unbounded fluctuations in the system’s energy function. The authors establish a fundamental link between optimization hardness and chaotic behavior, suggesting new ways to approach hard optimization problems both theoretically, using nonlinear dynamical systems methods, and practically, via special-purpose built analog devices.

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**More information:**www.nature.com/nphys/journal/v … /full/nphys2105.html

**Citation**: Physicists propose solution to constraint satisfaction problems (2011, October 10) retrieved 20 June 2019 from https://phys.org/news/2011-10-physicists-solution-constraint-satisfaction-problems.html

I see above article sees the problem with huge amount of constrains such approach is believed to required, like in this paper:

http://citeseerx.....122.726

But it occurred that constrains are not required: k-SAT can be translated into just optimization of nonnegative polynomial, for example (x OR y) can be changed into optimizing

((x-1)^2 plus y^2)((x-1)^2 plus (y-1)^2)(x^2 plus (y-1)^2)

and correspondingly 14 degree polynomial for alternatives of 3 variables. Zeros of sum of such polynomials for 3-SAT terms corresponds to the solution. Its gradient flow is dynamical system like it the article, but without constrains.

( http://www.usenet...&p=0 )