In defense of the Simplex Algorithm's worst-case behavior

In the early 1970s, by work of Klee and Minty (1972) and Zadeh (1973), the Simplex Method, the Network Simplex Method, and the Successive Shortest Path Algorithm have been proved guilty of exponential worst-case behavior (for certain pivot rules). Since then, the common perception is that these algorithms can be fooled into investing senseless effort by ‘bad instances’ such as, e. g., Klee-Minty cubes. This paper promotes a more favorable stance towards the algorithms’ worstcase behavior. We argue that the exponential worst-case performance is not necessarily a senseless waste of time, but may rather be due to the algorithms performing meaningful operations and solving difficult problems on their way. Given one of the above algorithms as a black box, we show that using this black box, with polynomial overhead and a limited interface, we can solve any problem in NP. This also allows us to derive NP-hardness results for some related problems.

[1]  Robert G. Busacker,et al.  A PROCEDURE FOR DETERMINING A FAMILY OF MINIMUM-COST NETWORK FLOW PATTERNS , 1960 .

[2]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[3]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[4]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .

[5]  M. Iri.  A NEW METHOD OF SOLVING TRANSPORTATION· NETWORK PROBLEMS , 1960 .

[6]  Bernd Gärtner,et al.  Understanding and using linear programming , 2007, Universitext.

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[8]  Kamiel Cornelissen,et al.  Smoothed Analysis of the Successive Shortest Path Algorithm , 2013, SIAM J. Comput..

[9]  Éva Tardos,et al.  A strongly polynomial minimum cost circulation algorithm , 1985, Comb..

[10]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[11]  Norman Zadeh,et al.  A bad network problem for the simplex method and other minimum cost flow algorithms , 1973, Math. Program..

[12]  Martin Skutella,et al.  An Introduction to Network Flows over Time , 2008, Bonn Workshop of Combinatorial Optimization.

[13]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[14]  N. Amenta,et al.  Deformed products and maximal shadows of polytopes , 1996 .

[15]  Oliver Friedmann,et al.  A Subexponential Lower Bound for Zadeh's Pivoting Rule for Solving Linear Programs and Games , 2011, IPCO.

[16]  Shang-Hua Teng,et al.  Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time , 2001, STOC '01.

[17]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[18]  H. D. Ratliff,et al.  Note—Some Equivalent Objectives for Dynamic Network Flow Problems , 1982 .

[19]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[20]  Norman Zadeh.  What is the Worst Case Behavior of the Simplex Algorithm , 1980 .

[21]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[22]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[23]  James B. Orlin,et al.  A polynomial time primal network simplex algorithm for minimum cost flows , 1996, SODA '96.

[24]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .