# Exhaustive Search, Combinatorial Optimization and Enumeration: Exploring the Potential of Raw Computing Power

## Abstract

For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve “simple” (wellde fined and well-structured) problems has dominated algorithm design. Over the same time period, both processingan d storage capacity of computers have increased by roughly a factor of a million. The next few decades may well give us a similar rate of growth in raw computing power, due to various factors such as continuingmi niaturization, parallel and distributed computing. If a quantitative change of orders of magnitude leads to qualitative advances, where will the latter take place? Only empirical research can answer this question.

Asymptotic complexity theory has emerged as a surprisingly effective tool for predictingru n times of polynomial-time algorithms. For NPhard problems, on the other hand, it yields overly pessimistic bounds. It asserts the non-existence of algorithms that are efficient across an entire problem class, but ignores the fact that many instances, perhaps includingt hose of interest, can be solved efficiently. For such cases we need a complexity measure that applies to problem instances, rather than to over-sized problem classes.

Combinatorial optimization and enumeration problems are modeled by state spaces that usually lack any regular structure. Exhaustive search is often the only way to handle such “combinatorial chaos”. Several general purpose search algorithms are used under different circumstances. We describe reverse search and illustrate this technique on a case study of enumerative optimization: enumeratingt he *k* shortest Euclidean spanning trees.

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### References

- 1.K. Appel and W. Haken. The Solution of the Four-Color-Map Problem.
*Scientific American*, pages 108–121, October 1977. 34Google Scholar - 2.D. Avis and K. Fukuda. Reverse Search for Enumeration.
*Discrete Apllied Mathematics*, 65:21–46, 1996. 25, 31, 32MATHCrossRefMathSciNetGoogle Scholar - 3.A Bruengger.
*Solving hard combinatorial optimization problems in parallel. Two case studies*. PhD thesis, ETH Zurich, 1997. 34Google Scholar - 4.R. Gasser.
*Harnessing computational resources for efficient exhaustive search*. PhD thesis, ETH Zurich, 1995. 34Google Scholar - 5.J. Horgan. The Death of Proof.
*Scientific American*, pages 74–82, 1993. 34Google Scholar - 6.D. H. Lehmer. The machine tools of combinatorics. In E. F. Beckenbach, editor,
*Applied combinatorial mathematics*, chapter 1, pages 5–31. Wiley, NY, edition, 1964. 34Google Scholar - 7.A Marzetta.
*ZRAM: A library of parallel search algorithms and its use in enumeration and combinatorial optimization*. PhD thesis, ETH Zurich, 1998. 34Google Scholar - 8.A. Marzetta and J. Nievergelt. Enumerating the
*k*best plane spanning trees. In*Computational Geometry—Theory and Application*, 2000. To appear. 31Google Scholar - 9.H. Maurer. Forecasting: An impossible necessity. In
*Symposium Computer and Information Technology*, http://www.inf.ethz.ch/latsis2000/, Invited Talk. ETH Zurich. 2000. 21 - 10.J. Nievergelt, R. Gasser, F. Mäser, and C. Wirth. All the needles in a haystack: Can exhaustive search overcome combinatorial chaos? In J. van Leeuwen, editor,
*Computer Science Today*, Lecture Notes in Computer Science LNCS 1000, pages 254–274. Springer, 1995. 35CrossRefGoogle Scholar - 11.K. Thomson. Retrograde analysis of certain endgames.
*ICCA J.*, 9(3):131–139, 1986. 34Google Scholar - 12.H. van Houten. The physical basis of digital computing. In
*Symposium Computer and Information Technology*, http://www.inf.ethz.ch/latsis2000/, Invited Talk. ETH Zurich. 2000. 19 - 13.C. Wirth and J. Nievergelt. Exhaustive and heuristic retrograde analysis of the KPPKP endgame.
*ICCA J.*, 22(2):67–81, 1999 34Google Scholar