However, the simplex algorithm visits every vertex in the worst case of a problem whose feasible region is the Klee–Minty cube, so the number of steps rises exponentially with the dimension of the problem. In the case shown, the simplex algorithm takes five steps. The simplex algorithm traverses the edges between vertices until it reaches an optimal vertex. Because exponential functions eventually grow much faster than polynomial functions, an exponential complexity implies that an algorithm has slow performance on large problems.Īn illustration of a three-dimensional polytope which is the feasible region for a linear programming problem. For example, a generalization of Gaussian elimination called Buchberger's algorithm has for its complexity an exponential function of the problem data (the degree of the polynomials and the number of variables of the multivariate polynomials). There are examples of algorithms that do not have polynomial-time complexity. For example, Gaussian elimination requires on the order of D 3 operations, and so it is said to have polynomial time-complexity, because its complexity is bounded by a cubic polynomial. The time complexity of an algorithm counts the number of arithmetic operations sufficient for the algorithm to solve the problem. The Klee–Minty cube has been used to analyze the performance of many algorithms, both in the worst case and on average. The Klee–Minty polytope is given by x 1 ≤ 5 4 x 1 + x 2 ≤ 25 8 x 1 + 4 x 2 + x 3 ≤ 125 ⋮ 2 D x 1 + 2 D − 1 x 2 + ⋯ + 4 x D − 1 + x D ≤ 5 D x 1 ≥ 0, …, x D ≥ 0. Illustrations of the 'cube' have appeared besides algebraic descriptions. When the dimension is three, the 'cube' is a squashed cube. When the dimension is two, the 'cube' is a squashed square. The Klee–Minty cube was originally specified with a parameterized system of linear inequalities, with the dimension as the parameter. Later, modifications of the Klee–Minty cube have shown poor behavior both for other basis- exchange pivoting algorithms and also for interior-point algorithms. In 1973 Klee and Minty showed that Dantzig's simplex algorithm was not a polynomial-time algorithm when applied to their cube. In particular, many optimization algorithms for linear optimization exhibit poor performance when applied to the Klee–Minty cube. On the three-dimensional version, the simplex algorithm and the criss-cross algorithm visit all 8 corners in the worst case. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their 'squashed cube'. Minty) is a unit hypercube of variable dimension whose corners have been perturbed. The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Klee Minty cube for shadow vertex simplex method.

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