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Minimum Cuts in NearLinear Time
, 1999
"... We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorit ..."
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Cited by 97 (12 self)
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We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an medge, nvertex graph with high probability in O(m log³ n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n² log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n² log³ n). Other applications of the treepacking approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a spaceefficient manner.
Matroids, Secretary Problems, and Online Mechanisms
"... We study a generalization of the classical secretary problem which we call the “matroid secretary problem”. In this problem, the elements of a matroid are presented to an online algorithm in random order. When an element arrives, the algorithm observes its value and must make an irrevocable decision ..."
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Cited by 36 (5 self)
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We study a generalization of the classical secretary problem which we call the “matroid secretary problem”. In this problem, the elements of a matroid are presented to an online algorithm in random order. When an element arrives, the algorithm observes its value and must make an irrevocable decision regarding whether or not to accept it. The accepted elements must form an independent set, and the objective is to maximize the combined value of these elements. This paper presents an O(log k)competitive algorithm for general matroids (where k is the rank of the matroid), and constantcompetitive algorithms for several special cases including graphic matroids, truncated partition matroids, and bounded degree transversal matroids. We leave as an open question the existence of constantcompetitive algorithms for general matroids. Our results have applications in welfaremaximizing online mechanism design for domains in which the sets of simultaneously satisfiable agents form a matroid.
Auctions for Structured Procurement
, 2007
"... This paper considers a general setting for structured procurement and the problem a buyer faces in designing a procurement mechanism to maximize profit. This brings together two agendas in algorithmic mechanism design, frugality in procurement mechanisms (e.g., for paths and spanning trees) and prof ..."
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Cited by 8 (0 self)
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This paper considers a general setting for structured procurement and the problem a buyer faces in designing a procurement mechanism to maximize profit. This brings together two agendas in algorithmic mechanism design, frugality in procurement mechanisms (e.g., for paths and spanning trees) and profit maximization in auctions (e.g., for digital goods). In the standard approach to frugality in procurement, a buyer attempts to purchase a set of elements which satisfy a feasibility requirement as cheaply as possible. For profit maximization in auctions, a seller wishes to sell some number of goods for as much as possible. We unify these objectives by endowing the buyer with a decreasing marginal benefit per feasible set purchased and then considering the problem of designing a mechanism to buy a number of sets which maximize the buyer’s profit, i.e., the difference between their benefit for the sets and the cost of procurement. For the case where the feasible sets are bases of a matroid, we follow the approach of reducing the mechanism design optimization problem to a mechanism design decision problem. We give a profit extraction mechanism that solves the decision problem for matroids and show that a reduction based on random sampling approximates the optimal profit. We also consider the problem of nonmatroid procurement and show that in this setting the approach does not succeed.
Backward analysis of the KargerKleinTarjan Algorithm for minimum spanning tree
 Information Processing Letters
, 1998
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Randomization in Graph Optimization Problems: A Survey
 OPTIMA
, 1998
"... Randomization has become a pervasive technique in combinatorial optimization. We survey our thesis and subsequent work, which uses four common randomization techniques to attack numerous optimization problems on undirected graphs. ..."
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Cited by 2 (0 self)
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Randomization has become a pervasive technique in combinatorial optimization. We survey our thesis and subsequent work, which uses four common randomization techniques to attack numerous optimization problems on undirected graphs.
Approximation Algorithms for Online Weighted Rank Function Maximization under Matroid Constraints
, 2012
"... Consider the following online version of the submodular maximization problem under a matroid constraint: We are given a set of elements over which a matroid is defined. The goal is to incrementally choose a subset that remains independent in the matroid over time. At each time, a new weighted ran ..."
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Cited by 1 (1 self)
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Consider the following online version of the submodular maximization problem under a matroid constraint: We are given a set of elements over which a matroid is defined. The goal is to incrementally choose a subset that remains independent in the matroid over time. At each time, a new weighted rank function of a different matroid (one per time) over the same elements is presented; the algorithm can add a few elements to the incrementally constructed set, and reaps a reward equal to the value of the new weighted rank function on the current set. The goal of the algorithm as it builds this independent set online is to maximize the sum of these (weighted rank) rewards. As in regular online analysis, we compare the rewards of our online algorithm to that of an offline optimum, namely a single independent set of the matroid that maximizes the sum of the weighted rank rewards that arrive over time. This problem is a natural extension of two wellstudied streams of earlier work: the first is on online set cover algorithms (in particular for the
Coping with Dynamic Membership, Selfishness, and Incomplete Information: Applications of Probabilistic Analysis and Game Theory
, 2008
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Competitive Weighted Matching
"... Consider a bipartite graph with a set of leftvertices and a set of rightvertices. All the edges adjacent to the same leftvertex have the same weight. We present an algorithm that, given the set of rightvertices and the number of leftvertices, processes a uniformly random permutation of the left ..."
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Consider a bipartite graph with a set of leftvertices and a set of rightvertices. All the edges adjacent to the same leftvertex have the same weight. We present an algorithm that, given the set of rightvertices and the number of leftvertices, processes a uniformly random permutation of the leftvertices, one leftvertex at a time. In processing a particular leftvertex, the algorithm either permanently matches the leftvertex to a thusfar unmatched rightvertex, or decides never to match the leftvertex. The weight of the matching returned by our algorithm is within a constant factor of that of a maximum weight matching, generalizing the recent results of Babaioff et al.