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Randomized Simplex Algorithms on Klee-Minty Cubes

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Randomized Simplex Algorithms on Klee-Minty Cubes Randomized simplex algorithms on KleeMinty cub es Bernd G¨artner G¨unter M. Ziegler Institut f¨ur Informatik Konrad-Zuse-Zentrum f¨ur Freie Universit¨at Berlin Informationstechnik Berlin (ZIB) Abstract problem “Is there an algorithm which quickly finds a (short) path to the lowest vertex?”. We investigate the behavior of randomized simplex The diameter problem is closely related to the algorithms on special linear programs. “Hirsch conjecture” (from 1957) and its variants [5, For this, we develop combinatorial models for the 14, 26]. Currently there is no counterexample to the Klee-Minty cubes [16] and similar linear programs “Strong monotone Hirsch conjecture” [26] that there with exponential decreasing paths. The analysis of two always has to be a decreasing path, from the vertex randomized pivot rules on the Klee-Minty cubes leads which maximizes xn to the lowest vertex, of length at to (nearly) quadratic lower bounds for the complexity most m − n. On the other hand, the best arguments of linear programming with random pivots. Thus we known for upper bounds establish paths whose length disprove two bounds conjectured in the literature. n is roughly bounded by mlog2 2 [11]. At the same time, we establish quadratic upper The algorithm problem includes the quest for bounds for random pivots on the linear programs un- a strongly polynomial algorithm for linear program- der investigation. This motivates the question whether ming. Klee & Minty [16] showed in 1972 that linear some randomized pivot rules possibly have quadratic programs with exponentially long decreasing paths ex- worst-case behavior on general linear programs. ist, and that the “steepest descent” pivot rule can be tricked into selecting such a path. Using variations of 1 Introduction the Klee-Minty constructions, it has been shown that the simplex algorithm may take an exponential num- ber of steps for virtually every deterministic pivot rule Linear programming is the problem of minimizing n [14]. (A notable exception is Zadeh’s rule [25, 14], lo- a linear objective function over a polyhedron P ⊆ IR cally minimizing revisits, for which Zadeh’s $1,000.– given by a system of m linear inequalities. prize [14, p. 730] has not been collected, yet.) Without loss of generality [22] we may assume that No such evidence exists for some extremely natu- the problem is primally and dually nondegenerate, ral randomized pivot rules, among them the following that the feasible region is full-dimensional and boun- three rules: ded, and that the objective function is given by the last coordinate. In other words, we consider the prob- random-edge: At any nonoptimal vertex x of P ,fol- x lem of finding the “lowest vertex” (minimizing n)ofa low one of the decreasing edges leaving x with n P ⊆ n simple -dimensional polytope IR with at most equal probability. m x facets, where the last coordinate n is not constant random-facet: x on any edge, and thus the lowest vertex is unique. If admits only one decreasing edge, then take it. Otherwise restrict the pro- In this setting, the (geometric interpretation of the) x simplex algorithm proceeds from some starting vertex gram to a randomly chosen facet containing . of P along edges in such a way that the objective func- This yields a linear program of smaller dimension P in which x is nonoptimal, and which can be solved tion decreases, until the unique lowest vertex of is random-facet found. The (theoretical and practical) efficiency of the by recursive call to . simplex algorithm [23] depends on a suitable choice of random-shadow: Start at the unique vertex y ∈ P decreasing edges that “quickly leads to the lowest ver- which maximizes xn. Choose a random unit vec- tex”. Connected to this are two major problems of tor c orthogonal to en. Now take the path from linear programming: the diameter problem “Is there y to the lowest vertex given by {x ∈ P : cx ≤ cz a short path to the lowest vertex?”, and the algorithm for all z ∈ P with zn = xn}. 1 random-facet is a randomized version, due to Specifically, our analysis of random pivots on the Kalai [11], of Bland’s procedure A [2], which as- Klee-Minty cubes yields the following two theorems. sumes that the facets are numbered, and always re- Theorem 1. The random-facet simplex algo- stricts to the facet with the smallest index. Inter- rithm on the n-dimensional Klee-Minty cube, started estingly enough, very elementary arguments imply a at the vertex v “opposite” (on the n cube) to the op- recursion n timal vertex, takes a quadratic expected number of F v f n, m ≤ f n − ,m− 1 f n, m − i steps n( ): ( ) ( 1 1) + n ( ) n i=1 − k+1 n − k π ( 1) 1 2 f n, m Fn(v)=n +2 ≈ − n . for the maximal expected number of steps ( ) k +2 2 4 2 on an n-dimensional linear program with m inequal- k=1 ities. From this one√ can get subexponential upper Moreover, for a random starting vertex the expected O n m bounds of roughly e ( log ) for the number of steps number of steps is of random-facet — see Kalai [11], and (in a very n2 +3n similar dual setting) Matouˇsek, Sharir & Welzl [19]. Fn = . The random-shadow rule is a randomized ver- 8 sion of Borgwardt’s shadow vertex algorithm [1] (a.k.a.theGass-Saaty rule [15]), for which the We note that one easily gets a linear lower bound F x ≤ n2 n / auxiliary function c is deterministically obtained in and a quadratic upper bound n( ) ( +3 ) 4for such a way that it is minimized on the starting vertex. the expected number of steps from an arbitrary start- x Borgwardt [1] has successfully analyzed this algorithm ing vertex . Furthermore, there are starting points facet random under the assumption that P is random in a suitable for which the rule will take only lin- model (where the secondary objective function c can early many steps. The fact that for some starting ver- be fixed arbitrarily), and obtained polynomial upper tices the expected number of steps is quadratic follows bounds for the expected number of simplex steps. from an explicit formula for the expectation value, None of the available evidence contradicts the pos- given in Section 2, or from the bound for a random sibility that the expected running time of all three starting vertex. randomized algorithms we consider is bounded from A result very similar to Theorem 1, in the setting above by a polynomial, even a quadratic function, in n of dual simplex algorithms, was earlier obtained by and m. In this connection, we report investigations of Matouˇsek [18, Sect. 4], who analyzed the behavior of the performance of such algorithms on infinite fami- the Matouˇsek-Sharir-Welzl dual simplex algorithm on lies of “test problems”: specific linear programs which a special class of linear programs. Similarly, for random-edge one gets an upper have decreasing paths of exponential length. n+1 En x ≤ It is not generally believed that polynomial upper bound ( ) 2 for the expected number of x n bounds can be achieved; it is equally conceivable that steps starting at any vertex of the -dimensional subexponential bounds such as those by Kalai [11] are Klee-Minty cube, see Section 2. This was first ob- essentially best possible. An interesting open problem served by Kelly [12], see also [24]. in this context is to find linear programs on which the Theorem 2. The expected number En of steps that algorithms in [11, 19] actually behave superpolynomi- the random-edge rule will take, starting at a ran- ally; Matouˇsek [18] has constructed abstract optimiza- dom vertex on the n-dimensional Klee-Minty cube, is tion problems — more general than linear programs bounded by — for which the subexponential analysis is tight. 2 In this extended abstract we concentrate on the n n +1 Θ( ) ≤ En ≤ . analysis of the “Klee-Minty cubes”, see Section 2. log n 2 These are very interesting linear programs whose poly- tope is a deformed n-cube, but for which some pivot This superlinear lower bound requires substantially rules follow a path through all the vertices and thus harder work, see Section 3. It implies that there is a need an exponential number of steps. vertex x with En(x)=Ω(n2/ log n), but compared to Our main results are quadratic, respectively nearly the case of random-facet we are not able to show quadratic, lower bounds for the expected number of this bound for a specific starting vertex, e.g. the top steps taken by the random-facet and the random- vertex. edge simplex algorithms. For the random-edge rule Our proof is based on a combinatorial model for the this seems to be the first superlinear bound. Klee-Minty cubes, which describes the random-edge 2 algorithm as a random walk on an acyclic directed graph (see Section 2). x 3 The combinatorial model also makes it possible to (0, 0, 1) do simulation experiments. Our tests in the range n ≤ , 1 000 suggest that the quadratic upper bound is close (1, 0, 1) to the truth. Also, it seems that a (nearly) quadratic lower bound is valid also if the starting vertex is chosen (0, 1, 1) to be the top vertex of the program, but as mentioned (1, 1, 1) above, our method does not prove this. Still, our result contradicts Exercise 8.10* in [21, (0, 1, 0) p. 188], where it is claimed that En(x)=O(n). It also disproves a conjecture of Kelly [12] that En(x)= O(n log2 n) for all starting vertices x.
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