Results 1  10
of
21
An update on the Hirsch conjecture,
 Jahresber. Dtsch. Math.Ver.
, 2010
"... Abstract The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample t ..."
Abstract

Cited by 41 (3 self)
 Add to MetaCart
(Show Context)
Abstract The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5dimensional polytope with 48 facets that violates a certain generalization of the dstep conjecture of Klee and Walkup.
A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games
"... ..."
The simplex method is strongly polynomial for deterministic Markov Decision Processes
 In Proceedings of the 24th ACMSIAM Symposium on Discrete Algorithms, SODA
, 2013
"... We prove that the simplex method with the highest gain/mostnegativereduced cost pivoting rule converges in strongly polynomial time for deterministic Markov decision processes (MDPs) regardless of the discount factor. For a deterministic MDP with n states and m actions, we prove the simplex method ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
We prove that the simplex method with the highest gain/mostnegativereduced cost pivoting rule converges in strongly polynomial time for deterministic Markov decision processes (MDPs) regardless of the discount factor. For a deterministic MDP with n states and m actions, we prove the simplex method runs in O(n3m2 log2 n) iterations if the discount factor is uniform and O(n5m3 log2 n) iterations if each action has a distinct discount factor. Previously the simplex method was known to run in polynomial time only for discounted MDPs where the discount was bounded away from 1 [Ye11]. Unlike in the discounted case, the algorithm does not greedily converge to the optimum, and we require a more complex measure of progress. We identify a set of layers in which the values of primal variables must lie and show that the simplex method always makes progress optimizing one layer, and when the upper layer is updated the algorithm makes a substantial amount of progress. In the case of nonuniform discounts, we define a polynomial number of “milestone” policies and we prove that, while the objective function may not improve substantially overall, the value of at least one dual variable is always making progress towards some milestone, and the algorithm will reach the next milestone in a polynomial number of steps. 1
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
, 2013
"... ..."
A subexponential lower bound for the Least Recently Considered rule for solving linear programs and games
"... The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most pivoting rules are known, however, to need an exponential number of steps to solve some linear programs. No nonpolynomial lower bounds were known, prior to this work, for Cunningham’s Least ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most pivoting rules are known, however, to need an exponential number of steps to solve some linear programs. No nonpolynomial lower bounds were known, prior to this work, for Cunningham’s Least Recently Considered rule [5], which belongs to the family of historybased rules. Also known as the ROUNDROBIN rule, Cunningham’s pivoting method fixes an initial ordering on all variables first, and then selects the improving variables in a roundrobin fashion. We provide the first subexponential (i.e., of the form 2 Ω( √ n)) lower bound for this rule in a concrete setting. Our lower bound is obtained by utilizing connections between pivoting steps performed by simplexbased algorithms and improving switches performed by policy iteration algorithms for 1player and 2player games. We start by building 2player parity games (PGs) on which the policy iteration with the ROUNDROBIN rule performs a subexponential number of iterations. We then transform the parity games into 1player Markov Decision Processes (MDPs) which correspond almost immediately to concrete linear programs. 1
Errata for: A subexponential lower bound for the Random Facet algorithm for Parity Games
, 2014
"... In [Friedmann, Hansen, and Zwick (2011)] and we claimed that the expected number of pivoting steps performed by the RandomFacet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by RandomFacet∗, a variant of RandomFacet that bases i ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
In [Friedmann, Hansen, and Zwick (2011)] and we claimed that the expected number of pivoting steps performed by the RandomFacet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by RandomFacet∗, a variant of RandomFacet that bases its random decisions on one random permutation. We then obtained a lower bound on the expected number of pivoting steps performed by RandomFacet ∗ and claimed that the same lower bound holds also for RandomFacet. Unfortunately, the claim that the expected number of steps performed by RandomFacet and RandomFacet ∗ are the same is false. We provide here simple examples that show that the expected number of steps performed by the two algorithms is not the same. 1
N.K.: Towards polynomial simplexlike algorithms for market equilibria
, 2013
"... In this paper we consider the problem of computing market equilibria in the Fisher setting for utility models such as spending constraint and perfect, pricediscrimination. These models were inspired from modern ecommerce settings and attempt to bridge the gap between the computationally hard but ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In this paper we consider the problem of computing market equilibria in the Fisher setting for utility models such as spending constraint and perfect, pricediscrimination. These models were inspired from modern ecommerce settings and attempt to bridge the gap between the computationally hard but realistic separable, piecewiselinear and concave utility model and, the tractable but less relevant linear utility case. While there are polynomial time algorithms known for these problems, the question of whether there exist polynomial time Simplexlike algorithms has remained elusive, even for linear markets. Such algorithms are desirable due to their conceptual simplicity, ease of implementation and practicality. This paper takes a significant step towards this goal by presenting the first Simplexlike algorithms for these markets assuming a positive resolution of an algebraic problem of Cucker, Koiran and Smale. Unconditionally, our algorithms are FPTASs; they compute prices and allocations such that each buyer derives at least a 1 1+εfraction of the utility at a true market equilibrium, and their running times are polynomial in the input length and 1/ε. We start with convex programs which capture market equilibria in each setting and, in a systematic way, convert them into linear complementarity problem (LCP) formulations. Then, departing from previous approaches which try to pivot on a single polyhedron associated to the LCP obtained, we carefully construct a polynomiallength sequence of polyhedra, one containing the other, such that starting from an optimal solution to one allows us to obtain an optimal solution to the next in the sequence in a polynomial number of complementary pivot steps. Our framework to convert a convex program into an LCP and then come up with a Simplexlike algorithm that moves on a sequence of connected polyhedra may be of independent interest. 1