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A Logic for Reasoning about Probabilities
 In: Information and Computation 87
, 1990
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On approximation methods for the assignment problem
 Journal of the Association for Computing Machinery
, 1962
"... This paper is concerned with approximation methods for handling the classical assignment problem. These methods permit solution of large scale assignment ..."
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This paper is concerned with approximation methods for handling the classical assignment problem. These methods permit solution of large scale assignment
Scaling algorithms for approximate and exact maximum weight matching
, 2011
"... The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the ..."
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The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the following sense. For any ɛ> 0, we give an algorithm that computes a (1 − ɛ)approximate maximum weight matching in O(mɛ −1 log ɛ −1) time, that is, optimal linear time for any fixed ɛ. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error. Our second contribution is a new exact maximum weight matching algorithm for integerweighted bipartite graphs that runs in time O(m √ n log N). This improves on the O(Nm √ n)time and O(m √ n log(nN))time algorithms known since the mid 1980s, for 1 ≪ log N ≪ log n. Here N is the maximum integer edge weight. 1
LinearTime Approximation for Maximum Weight Matching
"... The maximum cardinality and maximum weight matching problems can be solved in Ã(mân) time, a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article, we demonstrate that this âm â n barrier â can be bypassed by ap ..."
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The maximum cardinality and maximum weight matching problems can be solved in Ã(mân) time, a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article, we demonstrate that this âm â n barrier â can be bypassed by approximation. For any É> 0, we give an algorithm that computes a (1 â É)approximate maximum weight matching in O(mÉâ1 log Éâ1) time, that is, optimal linear time for any fixed É. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error.
A NOTE ON THE DATADRIVEN CAPACITY OF P2P NETWORKS
, 2009
"... Abstract—We consider two capacity problems in P2P networks. In the first one, the nodes have an infinite amount of data to send and the goal is to optimally allocate their uplink bandwidths such that the demands of every peer in terms of receiving data rate are met. We solve this problem through a m ..."
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Abstract—We consider two capacity problems in P2P networks. In the first one, the nodes have an infinite amount of data to send and the goal is to optimally allocate their uplink bandwidths such that the demands of every peer in terms of receiving data rate are met. We solve this problem through a mapping from a nodeweighted graph featuring two labels per node to a max flow problem on an edgeweighted bipartite graph. In the second problem under consideration, the resource allocation is driven by the availability of the data resource that the peers are interested in sharing. That is a node cannot allocate its uplink resources unless it has data to transmit first. The problem of uplink bandwidth allocation is then equivalent to constructing a set of directed trees in the overlay such that the number of nodes receiving the data is maximized while the uplink capacities of the peers are not exceeded. We show that the problem is NPcomplete, and
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"... We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has ..."
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We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (ChainAntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey’s Theorem for pairs (RT2 2) splits into a stable version (SRT22) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases these versions are strictly weaker (which is not known to be the case for RT 2 2 and SRT2 2). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2, showing for instance that WKL0 is incomparable with all of the systems we study; and prove computabilitytheoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman,