64,154 research outputs found

    Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard

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    State-of-the-art posted-price mechanisms for submodular bidders with mm items achieve approximation guarantees of O((loglogm)3)O((\log \log m)^3) [Assadi and Singla, 2019]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an m1/2εm^{1/2-\varepsilon}-approximation for any ε>0\varepsilon > 0 [Dobzinski and Vondr\'ak, 2016]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies, and that has been previously studied in relation to "Implementation in Undominated Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an α\alpha-approximation in implementation in advised strategies if there exists poly-time advice for each player such that an α\alpha-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [Assadi and Singla, 2019] mechanism achieves the same O((loglogm)3)O((\log \log m)^3)-approximation in implementation in advised strategies

    On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

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    We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is 1/21/2, which is achieved by any greedy algorithm. D\"urr et al. recently presented a 22-pass algorithm called Category-Advice that achieves approximation ratio 3/53/5. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the kk-pass Category-Advice algorithm for all k1k \ge 1, and show that the approximation ratio converges to the inverse of the golden ratio 2/(1+5)0.6182/(1+\sqrt{5}) \approx 0.618 as kk goes to infinity. The convergence is extremely fast --- the 55-pass Category-Advice algorithm is already within 0.01%0.01\% of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than 11/e1-1/e, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model

    Online Multi-Coloring with Advice

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    We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

    On the Closest Vector Problem with a Distance Guarantee

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    We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any α<1/2\alpha < 1/2, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most α\alpha times the length of the shortest nonzero lattice vector, requires as preprocessing advice only NO~(nexp(α2n/(12α)2))N \approx \widetilde{O}(n \exp(\alpha^2 n /(1-2\alpha)^2)) vectors, and runs in time O~(nN)\widetilde{O}(nN). As our second main contribution, we present reductions showing that it suffices to solve CVP, both in its plain and preprocessing versions, when the input target point is within some bounded distance of the lattice. The reductions are based on ideas due to Kannan and a recent sparsification technique due to Dadush and Kun. Combining our reductions with the LLM algorithm gives an approximation factor of O(n/logn)O(n/\sqrt{\log n}) for search CVPP, improving on the previous best of O(n1.5)O(n^{1.5}) due to Lagarias, Lenstra, and Schnorr. When combined with our improved algorithm we obtain, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance Decoding and the Closest Vector Problem with Preprocessing". Conference on Computational Complexity (2014

    Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

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    Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201
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