27,931 research outputs found
A unified way of analyzing some greedy algorithms
A unified way of analyzing different greedy-type algorithms in Banach spaces
is presented. We define a class of Weak Biorthogonal Greedy Algorithms and
prove convergence and rate of convergence results for algorithms from this
class. In particular, the following well known algorithms --- Weak Chebyshev
Greedy Algorithm and Weak Greedy Algorithm with Free Relaxation --- belong to
this class. We consider here one more algorithm --- Rescaled Weak Relaxed
Greedy Algorithm --- from the above class. We also discuss modifications of
these algorithms, which are motivated by applications. We analyze convergence
and rate of convergence of the algorithms under assumption that we may perform
steps of these algorithms with some errors. We call such algorithms approximate
greedy algorithms. We prove convergence and rate of convergence results for the
Approximate Weak Biorthogonal Greedy Algorithms. These results guarantee
stability of Weak Biorthogonal Greedy Algorithms
The Power of Verification for Greedy Mechanism Design
Greedy algorithms are known to provide, in polynomial time, near optimal approximation guarantees for Combinatorial Auctions (CAs) with multidimensional bidders. It is known that truthful greedy-like mechanisms for CAs with multi-minded bidders do not achieve good approximation guarantees.
In this work, we seek a deeper understanding of greedy mechanism design and investigate under which general assumptions, we can have efficient and truthful greedy mechanisms for CAs. Towards this goal, we use the framework of priority algorithms and weak and strong verification, where the bidders are not allowed to overbid on their winning set or on any subset of this set, respectively. We provide a complete characterization of the power of weak verification showing that it is sufficient and necessary for any greedy fixed priority algorithm to become truthful with the use of money or not, depending on the ordering of the bids. Moreover, we show that strong verification is sufficient and necessary to obtain a 2-approximate truthful mechanism with money, based on a known greedy algorithm, for the problem of submodular CAs in finite bidding domains. Our proof is based on an interesting structural analysis of the strongly connected components of the declaration graph
The Power of Verification for Greedy Mechanism Design
Greedy algorithms are known to provide near optimal approximation guarantees for Combinatorial Auctions (CAs) with multidimensional bidders, ignoring incentive compatibility. Borodin and Lucier [5] however proved that truthful greedy-like mechanisms for CAs with multi-minded bidders do not achieve good approximation guarantees. In this work, we seek a deeper understanding of greedy mechanism design and investigate under which general assumptions, we can have efficient and truthful greedy mechanisms for CAs. Towards this goal, we use the framework of priority algorithms and weak and strong verification, where the bidders are not allowed to overbid on their winning set or on any subsets of this set, respectively. We provide a complete characterization of the power of weak verification showing that it is sufficient and necessary for any greedy fixed priority algorithm to become truthful with the use of money or not, depending on the ordering of the bids. Moreover, we show that strong verification is sufficient and necessary for the greedy algorithm of [20], which is 2-approximate for submodular CAs, to become truthful with money in finite bidding domains. Our proof is based on an interesting structural analysis of the strongly connected components of the declaration graph
Independent sets in bounded-degree hypergraphs
AbstractIn this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter Δ. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a maximum weak (strong) independent set in hypergraphs, respectively. We propose a general technique that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs. This technique allows us to show that the greedy algorithm for MIS that corresponds to the classical greedy set cover algorithm has a performance ratio of (Δ+1)/2. It also allows us to apply results on local search algorithms on graphs to obtain a (Δ+1)/2 approximation for the weighted MIS and (Δ+3)/5−ϵ approximation for the unweighted case. We improve the bound in the weighted case to ⌈(Δ+1)/3⌉ using a simple partitioning algorithm. We also consider another natural greedy algorithm for MIS that adds vertices of minimum degree and achieves only a ratio of Δ−1, significantly worse than on ordinary graphs. For MSIS, we give two variations of the basic greedy algorithm and describe a family of hypergraphs where both algorithms approach the bound of Δ
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