The Minimum Cost Query Problem on Matroids with Uncertainty Areas

We study the minimum weight basis problem on matroid when elements\u27 weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base

Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty

We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral ? ? 2, we present algorithms that are ?-robust and (1+1/?)-consistent, meaning that they use at most ?OPT queries if the predictions are arbitrarily wrong and at most (1+1/?)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those

Sorting and Hypergraph Orientation under Uncertainty with Predictions

Learning-augmented algorithms have been attracting increasing interest, but have only recently been considered in the setting of explorable uncertainty where precise values of uncertain input elements can be obtained by a query and the goal is to minimize the number of queries needed to solve a problem. We study learning-augmented algorithms for sorting and hypergraph orientation under uncertainty, assuming access to untrusted predictions for the uncertain values. Our algorithms provide improved performance guarantees for accurate predictions while maintaining worst-case guarantees that are best possible without predictions. For sorting, our algorithm uses the optimal number of queries for accurate predictions and at most twice the optimal number for arbitrarily wrong predictions. For hypergraph orientation, for any γ ≥ 2, we give an algorithm that uses at most 1 + 1/γ times the optimal number of queries for accurate predictions and at most γ times the optimal number for arbitrarily wrong predictions. These tradeoffs are the best possible. We also consider different error metrics and show that the performance of our algorithms degrades smoothly with the prediction error in all the cases where this is possible

Discrete Optimization Methods for Segmentation and Matching

This dissertation studies discrete optimization methods for several computer vision problems. In the first part, a new objective function for superpixel segmentation is proposed. This objective function consists of two components: entropy rate of a random walk on a graph and a balancing term. The entropy rate favors formation of compact and homogeneous clusters, while the balancing function encourages clusters with similar sizes. I present a new graph construction for images and show that this construction induces a matroid. The segmentation is then given by the graph topology which maximizes the objective function under the matroid constraint. By exploiting submodular and monotonic properties of the objective function, I develop an efficient algorithm with a worst-case performance bound of $\frac{1}{2}$ for the superpixel segmentation problem. Extensive experiments on the Berkeley segmentation benchmark show the proposed algorithm outperforms the state of the art in all the standard evaluation metrics. Next, I propose a video segmentation algorithm by maximizing a submodular objective function subject to a matroid constraint. This function is similar to the standard energy function in computer vision with unary terms, pairwise terms from the Potts model, and a novel higher-order term based on appearance histograms. I show that the standard Potts model prior, which becomes non-submodular for multi-label problems, still induces a submodular function in a maximization framework. A new higher-order prior further enforces consistency in the appearance histograms both spatially and temporally across the video. The matroid constraint leads to a simple algorithm with a performance bound of $\frac{1}{2}$. A branch and bound procedure is also presented to improve the solution computed by the algorithm. The last part of the dissertation studies the object localization problem in images given a single hand-drawn example or a gallery of shapes as the object model. Although many shape matching algorithms have been proposed for the problem, chamfer matching remains to be the preferred method when speed and robustness are considered. In this dissertation, I significantly improve the accuracy of chamfer matching while reducing the computational time from linear to sublinear (shown empirically). It is achieved by incorporating edge orientation information in the matching algorithm so the resulting cost function is piecewise smooth and the cost variation is tightly bounded. Moreover, I present a sublinear time algorithm for exact computation of the directional chamfer matching score using techniques from 3D distance transforms and directional integral images. In addition, the smooth cost function allows one to bound the cost distribution of large neighborhoods and skip the bad hypotheses. Experiments show that the proposed approach improves the speed of the original chamfer matching up to an order of 45 times, and it is much faster than many state of art techniques while the accuracy is comparable. I further demonstrate the application of the proposed algorithm in providing seamless operation for a robotic bin picking system

Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a difference between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the mul- tiplicative inapproximability of minimizing the difference between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the difference between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.Comment: 17 pages, 8 figures. A shorter version of this appeared in Proc. Uncertainty in Artificial Intelligence (UAI), Catalina Islands, 201

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations