We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. We begin by reducing the input linear program to a special form in which we merely need to certify boundedness. As boundedness does not depend upon the right-hand-side vector, we run the shadow-vertex simplex method with a random right-hand-side vector. Thus, we do not need to bound the diameter of the original polytope. Our analysis rests on a geometric statement of independent interest: given a polytope Ax ≤ b in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial. 1.

Abstract. The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow-vertex simplex method had polynomial smoothed complexity. On a slight random perturbation of arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/σ. We show that the length of walk in the simplex method is actually polylogarithmic in the number of constraints n. Spielman-Teng’s bound on the walk was O(n 86 d 55 σ −30), up to logarithmic factors. We improve this to O(max(d 5 log 2 n, d 9 log 4 d, d 3 σ −4)). This shows that the tight Hirsch conjecture n − d on the the length of walk on polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. We propose a randomized phase-I for solving arbitrary linear programs. Instead of finding a vertex of a feasible set, we add a vertex at random to the feasible set. This does not affect the solution of the linear program with constant probability. So, in expectation it takes a constant number of independent trials until a correct solution is found. This overcomes one of the major difficulties of smoothed analysis of the simplex method – one can now statistically decouple the walk from the smoothed linear program. This yields a much better reduction of the smoothed complexity to a geometric quantity – the size of planar sections of random polytopes. We also improve upon the known estimates for that size. 1.

Many algorithms and heuristics work well on real data, despite having poor complexity under the standard worst-case measure. Smoothed analysis [36] is a step towards a theory that explains the behavior of algorithms in practice. It is based on the assumption that inputs to algorithms are subject to random perturbation and modification in their formation. A concrete example of such a smoothed analysis is a proof that the simplex algorithm for linear programming usually runs in polynomial time, when its input is subject to modeling or measurement noise. 1. MODELING REAL DATA “My experiences also strongly confirmed my previous opinion that the best theory is inspired by practice and the best practice is inspired by theory. ” [Donald E. Knuth: “Theory and Practice”, Theoretical Computer Science, 90 (1), 1–15, 1991.] Algorithms are high-level descriptions of how computational tasks are performed. Engineers and experimentalists design and implement algorithms, and generally consider them a success if they work in practice. However, an algorithm that works well in one practical domain might perform poorly in another. Theorists also design and analyze algorithms, with the goal of providing provable guarantees about their performance. The traditional goal of theoretical computer science is to prove that an algorithm performs well This material is based upon work supported by the National

The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...

Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [Multiple-source shortest paths in planar graphs. In Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest non-contractible or non-separating cycle in embedded, undirected graphs in O(g² n log n) time.

We devise a new simplex pivot rule which has interesting theoretical properties. Beginning with a basic feasible solution, and any nonbasic variable having a negative reduced cost, the pivot rule produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the basis, and all reduced costs which were originally nonnegative remain nonnegative. The pivot rule thus monotonically builds up to a dual feasible, and hence optimal, basis. A surprising property of the pivot rule is that the pivot sequence results in intermediate bases which are neither primal nor dual feasible. We prove correctness of the procedure, give a geometric interpretation, and relate it to other pivoting rules for linear programming.

Abstract. The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions – computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses. 1. Asyptotic convex geometry and Linear Programming Linear Programming studies the problem of maximizing a linear functional subject to linear constraints. Given an objective vector z ∈ R d and constraint vectors a1,...,an ∈ R d, we consider the linear program (LP) maximize 〈z, x〉 subject to 〈ai, x 〉 ≤ 1, i = 1,...,n. This linear program has d unknowns, represented by x, and n constraints. Every linear program can be reduced to this form by a simple interpolation argument [36]. The feasible set of the linear program is the polytope P: = {x ∈ R d: 〈ai, x 〉 ≤ 1, i = 1,..., n}. The solution of (LP) is then a vertex of P. We can thus look at (LP) from a geometric viewpoint: for a polytope P in R d given by n faces, and for a vector z, find the vertex that maximizes the linear functional 〈z, x〉. The oldest and still the most popular method to solve this problem is the simplex method. It starts at some vertex of P and generates a walk on the edges of P toward the solution vertex. At each step, a pivot rule determines a choice of the next vertex; so there are many variants of the simplex method with different pivot rules. (We are not concerned here with how to find the initial vertex, which is a nontrivial problem in itself).

Appl. Probab. 25A, 287-298] yields a practical scheduling rule for the versatile yet intractable multi-armed restless bandit problem, involving the optimal dynamic priority allocation to multiple stochastic projects, modeled as restless bandits, i.e., binary-action (active/passive) (semi-) Markov decision processes. A growing body of evidence shows that such a rule is nearly optimal in a wide variety of applications, which raises the need to efficiently compute the Whittle index and more general marginal productivity index (MPI) extensions in large-scale models. For such a purpose, this paper extends to restless bandits the parametric linear programming (LP) approach deployed 3 in [J. Niño-Mora. A ( 2 / 3) n fast-pivoting algorithm for the Gittins index and optimal stopping of a Markov chain, INFORMS J. Comp., in press], which yielded a fast Gittins-index algorithm. Yet the extension is not straightforward, as the MPI is only defined for the limited range of socalled indexable bandits, which motivates the quest for methods to establish indexability. This paper furnishes algorithmic and analytical tools to realize the potential of MPI policies in largescale applications, presenting the following contributions: (i) a complete algorithmic

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Abstract. This paper is devoted to a Pareto frontier generation technique, which is aimed at subsequent visualization of the Pareto frontier in an interaction with the user. This technique known as the Interactive Decision Maps technique was initiated about 30 years ago. Now it is applied for decision support in both convex and non-convex decision problems in various fields, from machinery design to environmental planning. The number of conflicting criteria explored with the help of the Interactive Decision Maps technique is usually between three and seven, but some users manage to apply the technique in the case of a larger number of criteria. Here we outline the main ideas of the technique, concentrating at nonlinear problems. Keywords. Multi-objective optimization, Pareto frontier, visualization 1