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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) algorithm that fi ..."
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Cited by 71 (10 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 21 (4 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.
Backwards Analysis of the KargerKleinTarjan Algorithm for Minimum Spanning Trees
 Inform. Proc. Letters
, 1998
"... This note gives a short proof of a sampling lemma used by Karger, Klein, and Tarjan in the analysis of their randomized lineartime algorithm for minimum spanning trees. Keywords: Minimum spanning trees; Randomized algorithms; Backwards analysis 1 Background The problem of computing the minimum sp ..."
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Cited by 5 (0 self)
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This note gives a short proof of a sampling lemma used by Karger, Klein, and Tarjan in the analysis of their randomized lineartime algorithm for minimum spanning trees. Keywords: Minimum spanning trees; Randomized algorithms; Backwards analysis 1 Background The problem of computing the minimum spanning tree in a weighted undirected graph has a long history, but a lineartime solution is realized only recently from the work of Karger, Klein, and Tarjan [2]. Their algorithm is randomized and works under a restricted RAM model of computation where the only allowable operations on the weights are comparisons. The basic idea is pruneandsearch: one recursively applies (i) Bor ffi uvka steps to reduce the number of vertices by a constant factor, and (ii) randomsampling steps to reduce the number of edges. The novelty of the algorithm lies in (ii), and to be more explicit, we need a definition. Let G = (V; E) be the given connected graph with n vertices and m edges. An edge (u; v) is s...
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 3 (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.
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. 1 Introduction Randomization has become a pervasive technique in com ..."
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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. 1 Introduction Randomization has become a pervasive technique in combinatorial optimization. Randomization has been used to develop algorithms that are faster, simpler, and/or betterperforming than previous deterministic algorithms. This article surveys our thesis [Kar94], which presents randomized algorithms for numerous problems on undirected graphs. Our work uses four important randomization techniques: Random Selection, which lets us easily choose a "typical" element of a set, avoiding rare "bad" elements; Random Sampling, which provides a quick way to build a small, representative subproblem of a larger problem for quick analysis; Randomized Rounding, which lets us transform fractional problem solutions into integral ones; and Monte Carlo Simulatio...