Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of @math independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact approximation algorithms to optimize our utility disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.
The Pandora's box solution can be written as a special case of the Gittins index theorem @cite_19 . @cite_22 consider a minimization variant of the Gittins index theorem when there is no discounting. Another very relevant paper is that of @cite_13 , while their results are to design auctions. Their proof of the Pandora's box problem inspired this work.
@cite_19 @cite_13 @cite_22
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General random coding theorems for lattices are derived from the Minkowski-Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski-Hlawka theorem itself is obtained as the limit, for p spl rarr spl infin , of a simple lemma for linear codes over GF(p) used with p-level amplitude modulation. The relation between the combinatorial packing of solid bodies and the information-theoretic "soft packing" with arbitrarily small, but positive, overlap is illuminated. The "soft-packing" results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda-Poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 (P N). We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields. The problem of computing category agnostic bounding box proposals is utilized as a core component in many computer vision tasks and thus has lately attracted a lot of attention. In this work we propose a new approach to tackle this problem that is based on an active strategy for generating box proposals that starts from a set of seed boxes, which are uniformly distributed on the image, and then progressively moves its attention on the promising image areas where it is more likely to discover well localized bounding box proposals. We call our approach AttractioNet and a core component of it is a CNN-based category agnostic object location refinement module that is capable of yielding accurate and robust bounding box predictions regardless of the object category. We extensively evaluate our AttractioNet approach on several image datasets (i.e. COCO, PASCAL, ImageNet detection and NYU-Depth V2 datasets) reporting on all of them state-of-the-art results that surpass the previous work in the field by a significant margin and also providing strong empirical evidence that our approach is capable to generalize to unseen categories. Furthermore, we evaluate our AttractioNet proposals in the context of the object detection task using a VGG16-Net based detector and the achieved detection performance on COCO manages to significantly surpass all other VGG16-Net based detectors while even being competitive with a heavily tuned ResNet-101 based detector. Code as well as box proposals computed for several datasets are available at:: this https URL We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger.