No abstract

We show that every weighted connected graph G contains as a subgraph a spanning tree into which the edges of G can be embedded with average stretch O(log 2 n log log n). Moreover, we show that this tree can be constructed in time O(m log n + n log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique.Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from m2 (O( √ log n log log n)) to m log O(1) n, and to O(n log 2 n log log n) when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

We show that every weighted connected graph G contains as a subgraph a spanning tree into which the edges of G can be embedded with average stretch O(log 2 n log log n). Moreover, we show that this tree can be constructed in time O(m log n + n log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from m2 (O(√ log n log log n)) to m log O(1) n, and to O(n log 2 n log log n) when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

Signaling is an important topic in the study of asymmetric information in economic settings. In particular, the transparency of information available to a seller in an auction setting is a question of major interest. We introduce the study of signaling when conducting a second price auction of a probabilistic good whose actual instantiation is known to the auctioneer but not to the bidders. This framework can be used to model impressions selling in display advertising. We establish several results within this framework. First, we study the problem of computing a signaling scheme that maximizes the auctioneer's revenue in a Bayesian setting. We show that this problem is polynomially solvable for some interesting special cases, but computationally hard in general. Second, we establish a tight bound on the minimum number of signals required to implement an optimal signaling scheme. Finally, we show that at least half of the maximum social welfare can be preserved within such a scheme.