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A Logic for Reasoning about Probabilities
 Information and Computation
, 1990
"... We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable ( ..."
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Cited by 213 (20 self)
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We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by DempsterShafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satistiability is NPcomplete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.
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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Cited by 7 (0 self)
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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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Cited by 3 (0 self)
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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