## Abstract

In this paper the utility and the difficulties of probabilistic analysis for optimization algorithms are discussed. Such an analysis is expected to deliver valuable criteria-better than the worst-case complexity-for the efficiency of an algorithm in practice.

The author has done much work of that kind in the field of linear programming. Based on that experience he gives some insight into the general principles for such an approach. He reports on some typical and representative attempts to analyze algorithms, resp. problems, of linear and combinatorial optimization. For each case he describes the problem, the stochastic model under consideration, the algorithm, the results, and tries to give a brief idea of the way these results could be obtained. He concludes with a discussion of some drawbacks and difficulties in that field of research. Among these are the strong sensibility with respect to the chosen model, the restriction of results to the asymptotic case, the restriction to somehow inefficient algorithms, etc. These points are the reasons why probabilistic analysis is of limited value for practice today. On the other hand, they show which principal problems should be attacked in the future to obtain the desired utility.

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## References

Adler I., Karp R., and Shamir R.: A family of simplex variants solving an

*m x d*linear program in expected number of pivot steps depending on*d*only, University of California, Computer Science Division, Berkeley, December 1983.Adler I., Karp R., and Shamir R.: A simplex variant solving an

*m x d*linear program in*O*(min^{2},*d*^{2})) expected number of pivot steps, University of California, Computer Science Division, Berkeley, 1983.Adler I. and Megiddo N.: A simplex algorithm where the average number of steps is bounded between two quadratic functions of the smaller dimension, Dept. of Industrial Engineering and Operations Research, University of California, Berkeley, California, December 1983.

Adler, I.: The expected number of Pivots needed to solve parametric linear programs and the efficiency of the self-dual-simplex method, University of California, Dept. of IE and OR, 1983.

Aho A. V., Hopcroft J. E., and Ullmann J. D.:

*The Design and Analysis of Computer Algorithms*, Addison-Wesley, Reading, Mass, 1974.Artstein Z. and Hart S.: Law of large numbers for random sets and allocation processes,

*Math. Operations Res.***6**(1981), 485–492.Bachem A. and Euler R.: Recent trends in combinatorial optimization,

*OR-Spektrum***6**(1984), 1–22.Benveniste R.: Evaluating computational efficiency: a stochastic approach,

*Mathematical Programming***22**(1982), 261–287.Berenguer S. E. and Smith R. L.: The expected number of extreme points of a random linear program,

*Mathematical Programming Study***35**(1985), 1–6.Blair C.: Random linear programs with many variables and few constraints, College of Commerce and Business Administration, University of Illinois at Urbana, Champaign, Aprill 1983.

Borgwardt, K. H.: Untersuchungen zur Asymptotik der mittleren Schrittzahl von Simplexverfahren in der linearen Optimierung, Dissertation, Universität Kaiserslautern, 1977.

Borgwardt K. H.: Some distribution-independent results about the asymptotic order of the average number of pivot steps of the simplex method,

*Math. Operations Res.***7**(1982), 441–462.Borgwardt K. H.: The average number of pivot steps required by the simplex method is polynomial,

*Zeit. Operations Res.***26**(1982), 157–177.Borgwardt, K. H.: The Simplex Method — A Probabilistic Analysis, Springer Verlag (1986).

Brown, G. G. and Schubert, B. O.: On random binary trees,

*Math. Operations Res.*No. 1 (1984), 43–65.Burkard R. E. and Fincke U.: On random quadratic bottleneck assignment problems,

*Mathematical Programming***23**(1982), 227–232.Burkard R. E. and Fincke U.: The asymptotic probabilistic behaviour of quadratic sum assignment problems,

*Zeit. Operations Res.***27**(1983), 73–81.Burkard R. and Fincke U.: Probabilistic asymptotic properties of some combinatorial optimization problems,

*Discrete Appl. Math.***12**(1985), 21–29.Carson J. S. and Law A. M.: A note on SPIRA'S algorithm for the all-pairs shortest-path problem,

*SIAM J. Computation***6**(1977), 696–699.D'Atri G. and Puech C.: Probabilistic analysis of the subset-sum problem,

*Discrete Appl. Math.***4**(1982), 329–334.Dempster M. A. H., Fisher M. L., Jansen L., Lageweg B. J., Lenstra J. K., and Rinnooy Kan A. H. G.: Analysis of heuristics for stochastic programming:

*Results for Hierarchical Scheduling Problems***8**(1983), 525–537.Dijkstra E. W.: A note on two problems in connexion with graphs,

*Numerische Mathematik***1**(1959), 269–271.Erdös P. and Renyi A.: On random graphs I,

*Publ. Math. Debrecan***6**(1959), 290–297.Erdös P.: Graph theory and probability I,

*Canadian J. Math.***11**(1959), 34–38.Erdös P.: Graph theory and probability II,

*Canadian J. Math.***13**(1961), 346–352.Fisher M. L. and Hochbaum D. S.: Probabilistic analysis of the planar K-MEDIAN problem,

*Math. Operations Res.***5**(1980), 27–34.Fisher M. L. and Krieger Abba M.: Analysis of a linearization heuristic for single-machine scheduling to maximize profit,

*Mathematical Programming***28**(1984), 218–225.Fox B. L., Lenstra J. K., Rinnooy Kan A. H. G. and Schrage L. E.: Branching from the largest upper bound: folklore and facts, Report 7722, Erasmus University, Rotterdam, 1977.

Frenk J. B. G., Van Houweninge M. and Rinnooy Kan A. H. G.: Asymptotic properties of the quadratic assignment problem,

*Math. Operation Res.***10**(1982), 100–116.Frieze A. M.: On the exact solution of random travelling salesman problems with medium size integer coefficients, Dept. of Comp. Science & Statistics, Queen Mary College, London, England, 1985.

Gale D.: How to solve linear inequalities,

*Am. Math. Monthly***76**(1969), 589–599.Garey M. R. and Johnson D. S.:

*Computers and Intractability: A Guide to the Theory of NP-Completeness*, Freeman, San Francisco, 1979.Gazmuri P. G.: Probabilistic analysis of a machine scheduling problem

*Math. Operations Res.***10**(1985), 328–339.Grötschel M., Lovász L., and Schrijver A.: The ellipsoid method and its consequences in combinatorial optimization,

*Combinatorica***1**(1981), 169–197.Grötschel, M.: Developments in combinatorial optimization, Universität Augsburg, Preprint No. 26, 1984.

Grötschel, M.: Operations research I, Skripten zur Vorlesung SS 1985, Universität Augsburg, 1985.

Haimovich M.: The simplex algorithm is very good! — On the expected number of pivot steps and related properties of random linear programs, Columbia University, New York, 1983.

Halton J. H. and Terada R.: A fast algorithm for the Euclidean travelling salesman problem optimal with probability one,

*SIAM J. Computation***11**(1982), 28–46.Hochbaum D. S. and Shmoys D. B.: A best possible heuristic for the

*k*-center problem,*Math. Operations Res.***10**(1985), 180–184.Karp R. M. and Steele J. M.: Probabilistic analysis of heuristics, in Lawler E. G., Lenstra J. K., Rinnooy Kan A. H. G., and Schmoys D. (eds.),

*The Travelling Salesman Problem*, John Wiley, New York, 1985, pp. 181–205.Karp R. M.: The probabilistic analysis of some combinatorial search algorithms, in J. F. Traub (ed.),

*Algorithms and Complexity*, Academic Press, New York, 1976, pp. 1–19.Kemp R.:

*Fundamentals of the Average Case Analysis of Particular Algorithms*, Teubner, John Wiley, New York, 1984.Klee V.: Combinatorial optimization: what is the state of the art?

*Math. Operations Res.***5**(1980), 1–26.Klee, V. and Minty, G. J.: How good is the simplex algorithm? In O. Shisha (ed.) Inequalities III, Academic Press, New York, pp. 159–175.

Korte B. and Hausmann D.: An analysis of the Greedy Heuristic for independence systems,

*Ann. Discrete Math.***2**(1978), 65–74.Lawler E. L.: Fast approximation algorithms for knapsack problems,

*Math. Operations Res.***4**(1979), 339–356.Lenstra J. K., Rinnooy Kan A. H. G., and Van Emde Boas P.: An appraisal of computational complexity for operations researchers, Erasmus University, Rotterdam, Econometric Institute, Report 8211/0, 1982.

Lenstra J. K. and Rinnooy Kan A. H. G. U.: Computational complexity of discrete optimization problems, Erasmus University, Rotterdam, Report 7727/0, 1977.

Lieberherr K.: Probabilistic combinatiorial optimization,

*Lecture Notes in Computer Sci.***118**(1981), 423–432.Lindberg P. O. and Olafsson S.: On the lengths of simplex paths: the assignment case, Royal Institute of Technology, Stockholm, TRITA-MAI 1980–29, 1980.

Lindberg P. O. and Olafsson S.: On the lengths of simplex paths: the transportation case, Royal Institute of Technology, Stockholm, TRITA-MAI 1983–6, 1983.

Lifschitz V.: The efficiency of an algorithm of integer programming: a probabilistic analysis,

*Proc. AMS***79**(1980), 72–76.Loulou R.: Tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling,

*Math. Operations Res.***9**(1984), 142–150.May J. H. and Smith R. L.: Random polytopes: their definition, generation and aggregate properties,

*Mathematical Programming***24**(1982), 39–54.Meanti M., Rinnooy Kan A. H. G., Stougie L., and Vercellis C.: A probabilistic analysis of the multiknapsack value function, Report 8435, Erasmus University, Rotterdam, Econometric Institute, 1984.

Nicholson T. A. J.: Finding the shortest route between two points in a network,

*Computing J.***9**(1966), 275–280.Palmer E. M.:

*Graphical Evolution: An Introduction to the Theory of Random Graphs*, John Wiley New York, 1985.Posa L.: Hamiltonian circuits in random graphs,

*Discrete Math.***14**(1976), 359–364.Shamir R.: The efficiency of the simplex-method: a survey, Dept. of Industrial Engineering and Operations Research, University of California, Berkeley, 1984.

Smale, S.: The problem of the average speed of the simplex method,

*Proc. 11th Internat. Symp. Math. Programming*, Universität Bonn, August 1982, pp. 530–539.Smale S: On the average speed of the simplex method,

*Mathematical Programming***27**(1983), 241–262.Spira P. M.: A new algorithm for finding all shortest paths in a graph of positive arcs in average time

*O*(*n*^{2}log^{2}*n*),*SIAM J. Computation***2**(1973), 28–32.Stein D. M.: An asymptotic, probabilistic analysis of a routing problem,

*Math. Operations Res.***3**(1978), 89–101.Tinhofer, G.: Probabilistische Ansätze in der Optimierung, Berichte der Mathematisch-Statistischen Sektion in der Forschungsgesellschaft Joanneum, Bericht No. 230 (1984).

Todd M. J.: Polynomial expected behaviour of a pivoting algorithm for linear complimentarity and linear programming problems, Technical Report No. 595, School of Operations Research and Industrial Engineering. Cornell University, New York, 1983.

Walkup D. W.: On the expected value of a random assignment problem,

*SIAM J. Computation***8**(1979), 440–442.

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Borgwardt, K.H. Probabilistic analysis of optimization algorithms-some aspects from a practical point of view.
*Acta Appl Math* **10, **171–210 (1987). https://doi.org/10.1007/BF00046618

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DOI: https://doi.org/10.1007/BF00046618

### AMS subject classifications

- 68C25
- 90C05
- 90C10

### Key words

- Optimization
- probability theory
- complexity of algorithms
- probabilistic analysis of algorithms