Largest minimal inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where 1 \le i \textless{} j \le n, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Carath\'eodory numbers for certain abstract convexity structures on the $(n-1)$-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever $n \ge 4$

Largest minimally inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set Q of permutations of the integers 1 to n is inversion-complete (resp., pair-complete) if for every inversion (j; i), where 1 j), where i 6= j), there exists a permutation in Q where j is before i. It is minimally inversion-complete if in addition no proper subset of Q is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion- complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (n1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide when ever n>=4

"Combinatorial Bootstrap Inference IN in Prtially Identified Incomplete Structural Models"

We propose a computationally feasible inference method infinite games of complete information. Galichon and Henry (2011) and Beresteanu, Molchanov, and Molinari (2011) show that such models are equivalent to a collection of moment inequalities that increases exponentially with the number of discrete outcomes. We propose an equivalent characterization based on classical combinatorial optimization methods that alleviates this computational burden and allows the construction of confidence regions with an effcient combinatorial bootstrap procedure that runs in linear computing time. The method can also be applied to the empirical analysis of cooperative and noncooperative games, instrumental variable models of discrete choice and revealed preference analysis. We propose an application to the determinants of long term elderly care choices.

Theoretical efficiency of the algorithm "capacity" for the maximum flow problem

A bad network for capacity -- Convergence

Anneaux achevés d'ensembles et préordrés

Définitions -- Hérédité et anneaux achevés d'ensemble -- Hérédité et fermeture reflexive-transitive -- Anneaux achevés d'ensembles et préordres

An Asymptotically Optimal On-Line Algorithm for Parallel Machine Scheduling

Jobs arriving over time must be non-preemptively processed on one of m parallel machines, each of which running at its own speed, so as to minimize a weighted sum of the job completion times. In this on-line environment, the processing requirement and weight of a job are not known before the job arrives. The Weighted Shortest Processing Requirement (WSPR) on-line heuristic is a simple extension of the well known WSPT heuristic, which is optimal for the single machine problem without release dates. We prove that the WSPR heuristic is asymptotically optimal for all instances with bounded job processing requirements and weights. This implies that the WSPR algorithm generates a solution whose relative error approaches zero as the number of jobs increases. Our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem.Singapore-MIT Alliance (SMA