Physicists propose solution to constraint satisfaction problems

October 10, 2011, University of Notre Dame

( -- Maria Ercsey-Ravasz, a postdoctoral associate and Zoltan Toroczkai, professor of physics at the University of Notre Dame, have proposed an alternative approach to solving difficult constraint satisfaction problems.

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

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 . 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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5 / 5 (1) Oct 10, 2011
Most of the fight with the P=NP question was made through unsuccessful discrete approach. Translating it into continuous problem moves it to completely different kingdom of analysis, which is practically unexplored.
I see above article sees the problem with huge amount of constrains such approach is believed to required, like in this paper:
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 )
not rated yet Oct 10, 2011
For those who don't like the jargon: The exact(most efficient) solution to the traveling salesman problem is an example of what the authors want. They just word this differently. I don't know a description of Nature that avoids our use of constructs, especially our constructs of measure. "Constraints" for the salesman are the points he must visit.
not rated yet Oct 10, 2011
"We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic and the boundaries between the basins of attraction of the solution clusters become fractal..." - authors

Let's translate this for the salesman and tell him what the above means for him:

When the salesman has so many points to visit - too many points to visit - his movements (trajectories) appear to become chaotic and indeterminate, i.e., random.

The salesman remains undaunted by the description the authors have bestowed upon him. He knows his trajectories are simply misleading the researchers and that his travels are optimal.

The salesman can't wait to meet the researchers and tell them the optimal solution to his problem - regardless of how many points he must visit.
1 / 5 (1) Oct 10, 2011
Kind of fuzzy logic?
not rated yet Oct 10, 2011
fuzzy logic is where you take a classic problem with two solutions say is the light on -- yes / no - or 1/0 -- and the fuzzy equivalent is kind of like making logic understand terms like mostly, kind of, and slightly. so is the light on is no longer yes - it is slightly on - no longer 1 for yes but .25

Fuzzy logic allows a computer to evaluate the question - is the car going fast -- well the discrete measure of speed is 55mph and that equates to say .60 for fast but a discrete measure of 95 mph would translate into .95. The equation that translates a discrete measure to its fuzzy answer does not need to be a linear equation - but it must satisfy the condidtion of an answer between 0 - 1.
not rated yet Oct 10, 2011
The overall effectiveness of this algorithm will in the end be determined by how long it takes to compute the solutions to practical problems. After all, Quick-Sort is O(n*n) while Merge- Sort is O(n*logn), but for most problems (even very large ones) qsort with a random pivot is faster.

I solve the kinds of problems this new algorithm addresses in my every day work. I look forward to seeing what (if any) results are published on practical chip design layout optimization (for example).
not rated yet Oct 10, 2011
Then solve exactly the traveling salesman problem. And line your pocketbook with $1 million allocated to the solution.

I recommend you follow Grigori Perelman example.
Once you are chosen to accept the prize, turn the prize down.
not rated yet Oct 10, 2011
Can anyone access the article and share the PDF please? My department doesn't subscribe to Nature Physics.

Hush1: That would set a nice tradition :)
not rated yet Oct 10, 2011
Then solve exactly the traveling salesman problem. And line your pocketbook with $1 million allocated to the solution.

I recommend you follow Grigori Perelman example.
Once you are chosen to accept the prize, turn the prize down.

I never said that we solve the problems optimally. There are a large number of companies out there that solve these and other NP-complete type problems using heuristics. We can never be as optimal as we like of course.

Every printed circuit board, chip layout, and other design solution requires (as close as possible) solutions to these problems. What do you think we do? Throw our hands in the air and pronounce them unsolvable?
not rated yet Oct 10, 2011
Endorsing the share request too.


http://docs.googl..._web.pdf Optimization hardness as transient chaos in an analog approach to constraint satisfaction&hl=en&gl=us&pid=bl&srcid=ADGEESjlkfAd-ac7S2ueSunrA4Z7z2wVcOGwypIstyqcrBvkq77zEH8wAfHtHawPxWfrBkHr6xZLfgMYS61HSw2K_As2Tl4aN12GxpMUdtuMHh1MhUCgxSN8WKf5LOu2uRqEURXcsITZ&sig=AHIEtbR3DxM6edghA-oABdr8DxjlD5LGog

He is touching as many bases as possible. Unusual in theoretical physics.

not rated yet Oct 10, 2011
I encouraged you. You know exactly how close you are to the optimal in your solutions. Even if the tools - the heuristic means - don't take you to the optimal, you are close.

Never liking an exact solution for optimization?
Don't we both have to laugh at this point? lol

My words were not meant to entice, lure, or trigger a defensive rhetorical question about what you do.

Intuitively, we both know, for the salesman, there exist an exact solution. No can get this close and "Throw their(our) hands in the air and pronounce them (him) unsolvable?"

That rhetorical question answers itself. An answer we both share.

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