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COMPUTING AND PROVING WITH PIVOTS
, 2013
"... Abstract. 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 solvin ..."
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Abstract. 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. 1.
Course Match: A LargeScale Implementation of Approximate Competitive Equilibrium from Equal Incomes for Combinatorial Allocation.‖ Working Paper
, 2015
"... Combinatorial allocation involves assigning bundles of items to agents when the use of money is not allowed. Course allocation is one common application of combinatorial allocation, in which the bundles are schedules of courses and the assignees are students. Existing mechanisms used in practice hav ..."
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Combinatorial allocation involves assigning bundles of items to agents when the use of money is not allowed. Course allocation is one common application of combinatorial allocation, in which the bundles are schedules of courses and the assignees are students. Existing mechanisms used in practice have been shown to have serious flaws, which lead to allocations that are inecient, unfair, or both. A new mechanism proposed by Budish [2011] is attractive in theory, but has several features that limit its feasibility for practice: reporting complexity, computational complexity, and approximations that can lead to violations of capacity constraints. This paper reports on the design and implementation of a new course allocation mechanism, Course Match, that enhances the Budish [2011] mechanism in various ways to make it suitable for practice. To find allocations, Course Match performs a massive parallel heuristic search that solves billions of MixedInteger Programs to output an approximate competitive equilibrium in a fakemoney economy for courses. Quantitative summary statistics for two semesters of fullscale use at a large business school (Wharton, which has about 1,700 students and up to 350 courses in each semester) demonstrate that Course Match is both fair and ecient, a finding reinforced by student surveys showing large gains in satisfaction and perceived fairness.
in Border Gateway Protocol, the Internet’s interdomain
"... (ASes) on the Internet can result in persistent oscillations ..."
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Mathematical Social Sciences ( ) – Contents lists available at SciVerse ScienceDirect Mathematical Social Sciences
"... journal homepage: www.elsevier.com/locate/econbase ..."
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A LEMKELIKE ALGORITHM FOR THE MULTICLASS NETWORK EQUILIBRIUM PROBLEM
"... Abstract. We consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origi ..."
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Abstract. We consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origindestination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and propose an extension of Lemke’s algorithm able to solve this problem. At the same time, it provides a constructive proof of the existence of an equilibrium in this case. hal00857611, version 1 3 Sep 2013 1.