48,243 research outputs found
On the Tree Augmentation Problem
In the Tree Augmentation problem we are given a tree T=(V,F) and a set E of edges with positive integer costs {c_e:e in E}. The goal is to augment T by a minimum cost edge set J subseteq E such that T cup J is 2-edge-connected. We obtain the following results.
Recently, Adjiashvili [SODA 17] introduced a novel LP for the problem and used it to break the 2-approximation barrier for instances when the maximum cost M of an edge in E is bounded by a constant; his algorithm computes a 1.96418+epsilon approximate solution in time n^{{(M/epsilon^2)}^{O(1)}}. Using a simpler LP, we achieve ratio 12/7+epsilon in time ^{O(M/epsilon^2)}. This also gives ratio better than 2 for logarithmic costs, and not only for constant costs. In addition, we will show that (for arbitrary costs) the problem admits ratio 3/2 for trees of diameter <= 7.
One of the oldest open questions for the problem is whether for unit costs (when M=1) the standard LP-relaxation, so called Cut-LP, has integrality gap less than 2. We resolve this open question by proving that for unit costs the integrality gap of the Cut-LP is at most 28/15=2-2/15. In addition, we will suggest another natural LP-relaxation that is much simpler than the ones in previous work, and prove that it has integrality gap at most 7/4
Dynamic Data Augmentation via MCTS for Prostate MRI Segmentation
Medical image data are often limited due to the expensive acquisition and
annotation process. Hence, training a deep-learning model with only raw data
can easily lead to overfitting. One solution to this problem is to augment the
raw data with various transformations, improving the model's ability to
generalize to new data. However, manually configuring a generic augmentation
combination and parameters for different datasets is non-trivial due to
inconsistent acquisition approaches and data distributions. Therefore,
automatic data augmentation is proposed to learn favorable augmentation
strategies for different datasets while incurring large GPU overhead. To this
end, we present a novel method, called Dynamic Data Augmentation (DDAug), which
is efficient and has negligible computation cost. Our DDAug develops a
hierarchical tree structure to represent various augmentations and utilizes an
efficient Monte-Carlo tree searching algorithm to update, prune, and sample the
tree. As a result, the augmentation pipeline can be optimized for each dataset
automatically. Experiments on multiple Prostate MRI datasets show that our
method outperforms the current state-of-the-art data augmentation strategies
Tight Bounds for Online Weighted Tree Augmentation
The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex spanning tree T and an additional set L of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in T cup F. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic O(log^2 n)-competitive algorithm and showed an Omega(log n) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic O(log n)-competitive algorithm for online WTAP, which is tight up to constant factors
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