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15
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 ..."
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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.
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 ..."
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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
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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 ..."
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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
COMPUTING AND PROVING WITH PIVOTS
, 2013
"... A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear ..."
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A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going from one subset σ to a new one σ ′ by deleting an element inside σ and adding an element outside σ: σ ′ = σ \ {v} ∪ {u}, with v ∈ σ and u / ∈ σ. This simple principle combined with other ideas appears to be quite powerful for many problems. This present paper is a survey on algorithms in operations research and discrete mathematics using pivots. We give also examples where this principle allows not only to compute but also to prove some theorems in a constructive way. A formalisation is described, mainly based on ideas by Michael J. Todd.
Geometric random edge
, 2014
"... We show that a variant of the randomedge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{cTx: Ax 6 b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanne ..."
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We show that a variant of the randomedge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{cTx: Ax 6 b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanned by n − 1 other rows of A is at least δ. This property is a geometric generalization of A being integral and all subdeterminants of A being bounded by ∆ in absolute value (since δ> 1/(∆2n)). In particular, linear programs defined by totally unimodular matrices are captured in this famework (δ> 1/n) for which Dyer and Frieze previously described a strongly polynomialtime randomized algorithm. The number of pivots of the simplex algorithm is polynomial in the dimension and 1/δ and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al.
OPTIMA Mathematical Optimization Society Newsletter
, 2011
"... with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meet ..."
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with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meetings, we are now heading towards the high point of 2012: the ISMP in Berlin. I hear from good sources that preparations are progressing well, and that all augurs are favourable. As you all know, several prizes will be awarded at the ISMP opening ceremony, recognizing the contributions or both younger and more senior colleagues. You undoubtedly have seen the various calls for nominations for the Dantzig, Lagrange, Fulkerson, BealeOrchardHays and Tucker prizes as well as that for the Paul Tseng lectureship. I encourage you to seriously consider nominating one or more optimization researchers for these prizes. These awards and the high scientific standards of their recipients not only recognize
Symmetric Strategy Improvement
, 2015
"... Symmetry is inherent in the definition of most of the twoplayer zerosum games, including parity, meanpayoff, and discountedpayoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where ..."
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Symmetry is inherent in the definition of most of the twoplayer zerosum games, including parity, meanpayoff, and discountedpayoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann’s traps, which shook the belief in the potential of classic strategy improvement to be polynomial.