Abstract not found

The Vickrey sealed bid auction occupies a central place in auction theory because of its efficiency and incentive properties. Implementing the auction requires the auctioneer to solve n + 1 optimization problems,where n is the number of bidders. In this paper we survey various environments (some old and some new) where the payments bidders make under the Vickrey auction correspond to dual variables in certain linear programs. Thus,in these environments,at most two optimization problems must be solved to determine the Vickrey outcome. Furthermore,primal-dual algorithms for some of these linear programs suggest ascending auctions that implement the Vickrey outcome.

As a common generalization of matchings and matroid intersections, W. H. Cunningham and J. F. Geelen introduced the notion of path-matchings. They proved a minmax formula for the maximum value of a path-matching, with the help of a linear algebraic method of Tutte and Lovász. Here we exibit a simplified version of their minmax theorem and provide a purely combinatorial proof.

We prove that every 6k-connected graph contains k edge-disjoint 2-connected spanning subgraphs. By using this result we can settle special cases of two conjectures, due to Kriesell and Thomassen, respectively: we show that every 12-connected graph G has a spanning tree T for which G-E(T) is 2-connected, and that every 18-connected graph has a 2-connected orientation.

We present new algebraic approaches for two well-known combinatorial problems: non-bipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions.

Let M = (Mr)r∈R be a system of matroids on a set S. Following the ideas of Nash-Williams [7], for every transfinite sequence f of distinct elements of S, we define a number η(f). We prove that the condition that η(f) ≥ 0 for every possible choice of f is necessary for M to have a system of mutually disjoint bases. Further, we show that this condition is sufficient if R is countable and Mr is a rank-finite transversal matroid for every r ∈ R. We also present conjectures about edge-disjoint spanning trees and detachments of countable graphs.

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In this note I show how very general and powerful results about the union and intersection of matroids due to J. Edmonds [19] may be deduced from a matroid generalisation of Hall's theorem by R. Rado [13]. Throughout, S, T, will denote finite sets, |X | will denote the cardinality of the

Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria – a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, westudy the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and singlecommodity games is strongly NP-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria.

Abstract. This is an introduction to the problems connecting frame theory