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# Combinatorial Optimization Meets Calculus: Scaling Exponents for TSP

David J. Aldous and Allon G. Percus

Freshman calculus tells us how to find a minimum of a smooth function : set the derivative and check . The related series expansion tells us, for points near to , how the distance relates to the difference in -values: scales as . This scaling exponent persists for functions defined on multidimensional space: if is a local minimum and , then scales as for a generic smooth function .

Combinatorial optimization, exemplified by the traveling salesman problem (TSP: find the minimum length tour through points in the plane), is traditionally viewed as a quite distinct subject, with theoretical analysis focussed on the number of steps that algorithms require to find the optimal solution. To make a connection, compare an arbitrary tour with the optimal (minimum-length) tour by the two quantities

where is the length of the minimum length tour. Now define to be the minimum value of over all tours for which . Although the function will depend on and the problem instance, we anticipate that for typical instances drawn from a suitable probability model it will converge in the limit to some deterministic function . The universality paradigm from statistical physics [1] suggests there might be a scaling exponent defined by

and that the exponent should be robust under model details.

Our study of the TSP yields strong evidence that the scaling exponent is . This is based on exact analytic methods in a mean-field model of interpoint distances (distances between pairs of points are random, independent for different pairs, thus ignoring geometric constraints) and on Monte Carlo simulations for random points in , and dimensional space. The new analytic results, to be detailed elsewhere, build upon a recent probabilistic reinterpretation [2] of work of Krauth and Mézard [3] establishing the average length of mean-field TSP tours. Parallel but mathematically easier results show that for the minimum spanning tree problem, a standard algorithmically easy problem, the scaling exponent equals .

 Click for figure

These results are intriguing for two reasons. For a combinatorial optimization problem, a larger exponent means that there are more near-optimal solutions, suggesting that the algorithmic problem of finding the optimal solution is intrinsically harder. So scaling exponents may serve to separate combinatorial optimization problems of an appropriate type into a small set of classes of increasing difficulty. For instance, the minimum matching and minimum Steiner tree problems are expected to have scaling exponent , and thus be in the same class as TSP in a quantitative way, as distinct from their qualitative similarity as NP-complete problems under worst-case inputs. This parallels the notion of universality class in statistical physics. Secondly, in the best-understood statistical physics areas such as percolation, the mean-field scaling exponents are exact for -dimensional problems only for larger than some critical dimension, which typically is at least [4]. The fact that for TSP the mean field exponent is either exactly or a close approximation to the correct exponent in all dimensions is therefore quite unexpected, and a convincing conceptual explanation remains to be discovered.

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David & 2005-12-09